hi welcome to the cork Maps ultimate GCSE Foundation revision video in this video I'm going to go through every single topic on the GCSE Foundation Maps course and the aim is to spend two or three minutes in each one of those topics to make sure you're familiar with nfn that you can encounter now this video will be really useful for you if you've got upcoming exams so if you've got your GCSE Maps Foundation papers coming up this will be really useful video for you also be useful for anyone who just wants to brush up on their math skills I said I spent two or three minutes going through every single topic so it's quite a long video so you may want to watch it in chunks or you may want to make notes as you go through and watch the video as well as the video I've made a booklet and I'll talk more about that later but I've made a booklet and that booklet has a question every single topic that I cover and I think there's nearly 400 questions in that booklet so it's going to be really useful for you to try that booklet as you go through and watching the video or at the end so let's have a look at the topic so these are all the topics that I'm going to cover in this video and in red we've got the number or the ratio and proportion topics so these are the topics I'm going to go through first so topic such as words and figures your operations place values square numbers productive primes order and fractions fractions decimals percentages compound interest reverse percentages proportion Best Buys topics like that then in green we have got our geometry our shape space and measures topics so topics such as angles symmetry your volume area speed Transformations trigonometry Pythagoras topics such as those then we're going to go go through and below the algebra topics there's topics such as substitution equations factorizing expanding brackets drawing straight line graphs the equation of a line quadratics the nth term change in the subject simultaneous equations topics like those and then in Orange we're going to go through the topics that are the statistics or the probability topics so your bar charts pictograms frequency trees pie charts scatter graphs then we've got our averages our estimated mean Venn diagrams probabilities topics like this so we're going to go through all of these topics so all these topics are the topics you need to know for your GCSE Foundation Maps there are one or two subtle differences between the examples not much in terms of the topics they're pretty much identical in terms of the three examples and the only three topics that I can think are far stem Leaf diagrams and frequency polygons who are one of the exam board covers them and the other two don't but for the sake of two or three minutes I think it's worthwhile watching them and it's a good practice for the average in range anywhere so I'm going to go through all of these topics and as I said I'm going to spend about two or three minutes going through each of them so what I'd recommend you do is you watch this video and if you need any extra help on any one of these topics you've got the video tutorial on corporate Maps which can help you and this revision checklist will be really useful for you it's in the description below and I'd recommend that you print it you keep it in your notes or even stick it on your wallet as I've said this at a company in booklet so I've called it the ultimate GCSE Foundation revision question booklet very catchy title I know but in that booklet there's a question on every single topic as we go through this video and they're in the same order so that booklet would be really useful for you and there's a QR code which brings you to this video the revision video and there's a QR code that brings you to the answers so that booklet will be really useful for you to practice what you're watching in the video now in this video I take off for each one of those topics I do sometimes use the chord mods revision cards and I take some of the information or I just use the whole card these revision cards will be perfect for you whenever you're revising for your GCSE Maps so if you are interested in those there is a link to them in the description below but they're perfect to go alongside the video and that other booklet that will really help you with your GCC manager Vision as well as the revision cards of highly recommend our five a days rather than cramming your revision just before the exams a little and often approach is fantastic and if you're revising for your foundation exams I'd highly recommend the orange Foundation books and the yellow Foundation plus books and those two books will be perfect because they give you five questions for every single day of the year and that little unofficular approach to build your confidence let's have a look at our first topic so first topic is words and figures now every single topic we go through in this video in the top right hand corner we have got our video numbers so if after you watch the video you want to try some practice questions you can go to codemavs.com forward slash contents and you can scroll down to video 362 and 363 and there's some practice questions so you can see lots of questions on this topic and there's also the video tutorial if you need any extra help there's that video for you as well so as I said I'm going to spend two or three minutes going through each of the topics but if you need a longer tutorial the video tutorials are there for you as well so first topic is words and figures so our first question says write 6840 in words so whenever a writing figures in words we write down what we say so it's six thousand eight hundred and forty so we would write that down we would write down 6000 comma 840 and that's it and if you want to write words and figures you write down what you read so if we've seen 6840 we would write that down okay let's have a look at our next topic well our next topic is actually Four topics and they are addition subtraction application and Division and there are operations and it's very important you can carry out written methods for those so let's have a look at addition so we've got 381 plus 64. I'm just lining the numbers up in columns so we've got the ones or the units the tens and then the hundreds so let's line them up and then let's start with the right hand side the ones and let's add those 1 plus 4 is equal to five eight plus six is equal to 14. so let's put our four down and carry our one and three plus one is equal to four so 381 plus 64 would be 445. okay let's have a look at subtraction so we've got 514 take away 175 so again lining them up in hundreds tens and ones or units and we're going to subtract them so we're going to do four take away five well we can't do that so we're going to borrow so we'll borrow from the one and call that zero now we've got 14. so we've got 14 take away five that's equal to nine zero take away seven again we can't do that so we're going to borrow so we'll cross that out and call it four and then that's ten ten take away seven is equal to three and four take away ones equal to three so that's where be 339 so it's very important you can do these written methods for addition and subtraction now let's have a look at multiplication and division so multiplication so we've got e83 multiplied by 29 so we get 83 multiplied by 29 and we're just going to line them up like this and I'm going to use the column method for multiplication so I'm going to multiply 83 by 9 and then going to do 83 times 20 and then add them together so let's do 83 times 9 to begin with so 9 times 3 is equal to 27 so let's put our 7 down and carry our 2 and then 9 times 8 is equal to 72 plus 2 is equal to 74. so if we do 83 times 9 the answer would be 747. now we're going to do 83 multiply by 20. so let's put our zero down and we're going to do 2 times 3 is equal to 6. so put our 6 down and 2 times 8 is equal to 16 so 16. and now we just need to add these together 7 plus 0 is equal to seven four plus six is equal to 10 put our zero down carrier one seven plus six is equal to 13 plus 1 is equal to 14 put our 4 down carrier one and finally one plus one is equal to two so 80 3 multiplied by 29 will be equal to 2407 now let's have a look at divisions so we've got 1032 divided by six so we've got 1032 and then we've got our bus shelter divided by six so we say how many sixes go into one at zero remainder one how many sixes go into ten well there's one six in ten remainder of four how many sixes go into forty three well that's seven because seven times six is forty two so that's seven remainder one because 42 the remainder would be one and finally how many six is going to twelve the answer would be two so one thousand and thirty two divided by 6 is equal to 172. and that's it so we've gone through addition subtraction multiplication and division and if you need a video tutorial any of those topics that are found here and also remember in that bumper booklet you've got a question each of those or you've got a few questions in each of those so feel free to do those now if you want to as well okay let's have a look at our next topic okay let's have a look at our next topic so our next topic is order of operations now whenever you're given something to work out it's very important everyone works out in the same way and sometimes on social media LCDs questions and people are answering it in different ways and they're getting different answers and a lot of the time it just depends on this order of operations now the video is 211 on corporate Maps now some people call it bod Mass some people call it bid Mass um you know it's got different names I tend to call it order of operations um I just remember there's an order like a hierarchy it starts off with any brackets so you look for any brackets then you look for any orders or indices that's like a that's a sort of sounds complicated but it's just anything with a squared or a cubed or a square root then you look for any divisions and multiplications and they're of the same importance to if you've got a question which has divisions and multiplications only you work from left to right and finally you work out any additions or subtractions and there again of the same importance as each other so if you've got a question which has just additions and subtractions you work from left to right so let's have a look at our first question so our first question says work out 5 plus 10 multiplied by two now whenever I'm looking at this the first thing I'm looking for is any brackets no then I'm looking for any squares or cubes or square roots no then I'm looking for any divisions or multiplications yes we've got this 10 multiplied by two that means we need to do 10 multiplied by two first so 10 multiplied by 2 is 20. so this is 20. so I'm going to write 20 and I'm just going to write it directly beneath the 10 multiplied by 2. but in front of that we still had five plus so I'm going to write five plus and we've got five plus twenty and five plus twenty is equal to 25. okay let's have a look at our next example so our next question says work out two plus eight squared so again we're going to look for any brackets no then we're going to look for any orders that's your powers your squares your cubes your square roots as you can see there's a squared there so we're going to work out that first of all we're going to work out 8 squared now later on the video we're going to go through squares so if you haven't got that far you know you might want to come back to this and if whenever you've done that but squared means to multiply by itself so we're going to do 8 squared that's 8 times 8 which is equal to 64. we write it directly beneath it so 64. and we still have the 2 plus in front so 2 plus 64. and 2 plus 64 is equal to 66 and that's it so it's very important you follow the correct order of operation okay let's have a look at our next topic okay it's very important that you know how to run numbers so our first question says round 235 to the nearest hundred and this is the quarter mileage revision card so we're rounding 2 235 to the nearest hundred so because 235 is in between 200 and 300 so our answer will either be 200 or 300. so if we consider where it is on the number line we've got 250 in the middle and 235 would be below 250 so 235 on the number line will be somewhere like here so that means that it's closer to 200 than it is to 300. now some people will just like if you round into the nearest hundred they would look in the tens column and see it's a three and because that's below 250 below the five that means you round down so if the number in the tens column is a zero a one two three or four you would run down here if it was a five or six or seven an eight or nine you would round up but I tend to just think what's in the middle of 200 and 300 because we know it's in between 200 and 300 which is 250 that's below it so we're going to round done okay next question our next question says round 7680 to the nearest thousand so because it's to the nearest thousand it's either going to be 7 000 or 8 000 because that's the two thousands is in between so we've got a number line here between seven thousand and eight thousand in the middle would be seven thousand five hundred and that number is clearly above it so it would look something like this where the number is above seven thousand five hundred so that means that our number is closer to eight thousand that is to seven thousand so answer would be eight thousand again another way to look at it is because we're rounding to the nearest thousand we're looking at the hundreds column we've got a six there which means we round up so the answer would be eight thousand and finally it's important to be able to run to decimal places as well so we're gonna round five point one eight to one decimal place so we've got 5.1 and 5.2 because 5.18 would be in between 5.1 and 5.2 so we want to figure out if 5.18 is closer to 5.1 or 5.2 well in the middle would be 5.15 so 5.18 is much closer to 5.2 than it is to 5.1 so answer would be 5.2 and again because we're rounding to one decimal place we could have just looked at the second decimal place which is an eight which then would tell us that we're going to round up so it's going to be 5.2 to be able to round to one signal significant figure it's quite useful for estimation as well and we've got some numbers here this is again the core mouse revision card and we've got a number 394 1273 and so on and we're going to round each of these numbers to one significant figure surrounding to one significant figure well it makes the number much more easier to deal with because it's just going to be one number followed by zeros so if we had 394 that's closer to 400 that is the 300. so answer would just be 400. we've got 1273 where we're only largest one number followed by zero so we're either going to have one thousand or two thousand this is closer to one thousand so our answer would just be one thousand then we'll get 7961 so again we'll add just one number followed by zeros so we could have seven thousand or eight thousand this is closer to eight thousand so to one significant figure we would have eight thousand and again like with rounding some people if you're rounding to one significant figure they look at the second digit and they figure if it's five or above you round up if it's four below you run down so here we've got 394 the second significant figure is the second digit that's the nine so we round up to four hundred one thousand two hundred and seventy three well the second significant figure the second digit is the two so we round down to one thousand seven thousand nine hundred and sixty one the second significant figure is in nine so we're rounding up okay let's have a look at the next one so our next one's a decimal number and whenever we're dealing with significant figures and decimals you ignore the note points and so on all the zeros at the front and you're looking for the first digit that's not a zero whenever it's a decimal number so we've got 0.618 so first significant figure here would be the six you ignore the zero and our second one would be a one because it's a one it's below for it's four below so you round down so it's going to be 0.6 another way to look at it would be if you're 0.618 it's either going to be 0.6 or not 0.7 this is closer to 0.6 so answer would be 0.6 okay let's have a look at our last one so last one we've got 20.501 because we've got a two at the front that's significant and then that's our first significant figure and then it's followed by zero so that would be significant it's only not significant if it's at the front the zeros so like here is no point so we just ignored that and look at the six but here we had our two so that's that's significant that's the first one second one third one fourth one fifth one so we want to round it to one significant figure so we just want one number followed by zeros so this number is 20.501 so to one significant figure you could have 20 or 30 this is much closer to twenty so our answer would be 20 20. so we've now looked at how to round numbers to one significant figure but we may want to be a little bit more accurate sometimes so we may want to round our answers to two significant figures of three significant figures and so on so let's have a look and see how we run numbers the same numbers to two significant figures so let's have a look at our first one so we've got 394. so we want two significant figures so for this number that would mean we would want two digits and Then followed by zeros so options would either be 390 or 400 because they are two digits so three nine and Then followed by zeros or four hundred that's four zero followed by zeros and we were on around the two significant figures so let's choose the closest one to receive a 390 or 400 so it's going to be 390 because this is closer to 390 than it is to 400. so answer would be 390. or an alternative way to look at it is if we have 394 we wanted to run this to two significant figures where we want to have two numbers Then followed by zeros so we look at the third significant figure which is a four and that means that we round down so it'd be 3 190 no 400 okay next one is 1273 we want to run that to two significant figures so we either want to have one thousand two hundred or one thousand three hundred but we've got two digits Then followed by zeros as you can see 1 273 would be closer to one thousand three hundred than it is to 1 200. it's also be one thousand three hundred or again we could look at the very significant figure we've got the first two and then the third significant figure is a seven so we round up so answer would be one thousand three hundred okay our next one our next one is seven thousand nine hundred and sixty one and we wanna run that to two significant figures so that means we want to have two digits followed by zeros so we could either have seven thousand nine hundred or eight thousand because eight zero and Then followed by zeros so because this is seven thousand nine hundred and sixty one it's closer to eight thousand than it is to seven thousand nine hundred so that means the answer would be eight thousand or again another way to look at it is if we want to render this to two significant figures we look at the third significant figure which would be a six and that means we round up so B eight thousand okay let's have a look at our next one our next one is 0.618 so we want to round this to two significant figures now remember with decimal numbers we ignore the note point and the zeros in the front so we're just going to look at our six one eight and we want to run this to two significant figures that means we either want 0.61 or 0.62 and because it's 0.618 we're going to round it up so it's going to be 0.62 okay the last number is 20.501 we want to write this to two significant figures so we want two digits Then followed by zeros or if it's just the two digit number we just want a two digit your number so we've got 20.501 so our choice is either going to be 20 or 21 and because it's 20.501 we're going to round upside down so B21 or again another way to look at it is because we're 20.501 we look at our significant figures well our first significant figure is a two a second significant figure is a zero and then we look at our third significant figure which is a five that means we round up so answer B21 okay so we looked at rounding now we're going to look at finding what the highest number could have been or the lowest number could have been so here we've got the population of wells and that's 12 000 to the nearest thousand and we've been asked to find what was the lowest possible population of Wales and what's the highest possible population of Wales so we told the population of Wales is 12 000 to the nearest thousand and we've been asked to find the lowest possible population of Wales so let's consider the numbers below 12 000 that would Round Up to be 12 000 to the nearest thousand so the population of Wales could have been 1199 and that number would obviously Round Up to be 12 000 to the nearest thousand it could have been even lower it could mean eleven thousand nine hundred if we rounded that to the nearest thousand it'd be twelve thousand it could be even lower it could have been eleven thousand six hundred if we rounded that to the nearest thousand our answer would be twelve thousand it could even be as low as eleven thousand five hundred that number would round to twelve thousand to the nearest thousand but it couldn't be eleven thousand four hundred and ninety nine because that would run down to eleven thousand so the lowest possible population of Wales would be eleven thousand five hundred eleven thousand five hundred because that's the lowest possible number that would Round Up to be twelve thousand to the nearest thousand okay next question what's the highest possible population of Wales so let's consider the numbers above 12 000 that would round down to be twelve thousand at the nearest thousand so the population of Wales could have been twelve thousand and one obviously if we rounded that to the nearest thousand it would be twelve thousand it could be something even higher it could be twelve thousand two hundred if we round that to the nearest thousand it'd be twelve thousand not thirteen thousand it could be something even higher twelve thousand four hundred would run down to be twelve thousand but it wouldn't Round Up to be thirteen thousand so it could be twelve thousand four hundred and it could be anything as high as twelve thousand four hundred ninety nine because twelve thousand four hundred and ninety nine would run down to be twelve thousand and not up to thirteen thousand it couldn't be twelve thousand five hundred because twelve thousand five hundred if we rounded that to the nearest thousand because it's a five in the hundreds column we would round it up to be thirteen thousand so it couldn't be twelve thousand five so they also be 12 499 so that would be 12 499 so the lowest possible population of Wales would be eleven thousand five hundred and the highest possible population of Wells could have been 12 499. now in this question we've looked at what we call discrete data and that's data that can only take certain values whether being population we didn't need to consider decimal numbers later on this video we'll look at what we call continuous data and that's data that can take any value on a given scale and we'll look at that whenever we look at a topic called error intervals okay let's have a look at our next topic okay let's have a look at our next topic so next topic is estimation which is video 215 in corporate Maps so here we've got a typical question and we've got a shop sale 78 magazines that cost five pound fifteen each estimate the total cost of the magazine sold so whenever we're doing estimation questions we want to use nicer numbers to make it a bit easier for ourselves and this will typically be on the non-calculated paper and we've got 78 magazines now I know this is 78 magazines is quite close to 18 and that would be a nicer number to use than 78. and we've got a cost of 5.15 well it's quite close to Five Pound so what I'm going to do is I'm going to round these numbers to what we call one significant figure so rounding 78 to one second for configure is 80. it's a nice number it's quite close to 70 here and instead of using 5.15 I'm going to use five pound and what I'm going to do then is work out the total cost so if I sold 80 magazines at 5.8 I would do 80 multiply by five pound and that would tell me the total cost so 80 multiplied by 5 is 400 so that would be 400 pound now remember we have rounded these numbers of 80 and 5 pounds so this 400 pound wouldn't be the exact amount of money the magazines would sell for but this is our estimation so I like educate a guess by rounding the numbers to one significant figure or rather than to nice numbers so instead of 78 we chose ET and a set of 5.15 we chose five pound and we done 80 multiplied by five pound is 400 pounds that's our estimation and that's it and sometimes in estimation questions we're given questions like this we've been asked to work out an estimate for 40.18 multiplied by 6.87 all divided by 0.512 now whenever we're doing a question like this with estimation it can be useful to round our numbers to one significant figure so let's round our numbers to one significant figure so let's start off with our first number 40.18 so we're going to run that to one significant figure so but that means one digit followed by zeros so that means that we're even gonna have a choice of 40 or 50 because it's 40.18 I'm going to choose 40. it's much closer to 40 that is the 50. our next number we've got multiply by and then we've got 6.87 so our choices will be six or seven now 6.87 will be closer to seven than it is the six so we're going to choose seven and then finally we've got divided by and then we've got 0.512 we want to run this to one significant figure so our choices will be 0.5 or 0.6 this number is closer to 0.5 because it's 0.51 than it is the 0.6 or we're going to write then 0.5 now whenever I rounded my numbers to one significant figure obviously whenever I carry on the answer is not going to be the exact answer so instead of writing an equal sign here I do this curly equal sign it looks like this and that means it's approximately equal to so whenever you round your numbers whenever you do an estimation question you round your numbers rather than putting an equal sign after you've run them it can be useful to put this approximately equal to symbol it just shows uh your teacher your examiner or even just remind yourself that you've rounded the numbers so now what we're going to do is we're going to work this out so we're going to work out 40 multiplied by 7 well 4 times 7 is 28 so 40 times 7 would be equal to 280 and then we've still got divided by 0.5 so when I just need to work out 280 divided by 0.5 and that will give us our approximate answer to this question our estimate now our next topic is going to be ordering decimals and just after that will be arithmetic with decimals and I'm going to show you how to divide by decimals like this really nicely and easily and what you would do is you would just times both of these numbers by 10 and you would do 2 2800 divided by five and you could do your brush shelter method to get that answer or another way to do it is to just consider how many halves there would be in 200 nearly so if we had 280 pizzas how many halves would there be and then that would be 560. and you can check this using the bus shelter method we could do 2800 divided by five and hopefully we'll get 560. how many fives are there and two zero remainder two how many fives are there in 28 well that'd be five remainder three how many fives are there in 30 that's gonna be six and how many fives are there and zero zero so the answer is 560. so our estimate for the answer to this question would be 560. that's not going to be exactly 560 unless we're very very lucky um but the at our estimate that's our the proximate answer okay let's have a look at our next topic so our next topic is ordering numbers and ordering decimal numbers and there are videos 221 and 95 on corporate maps and this question says range in order starting with the smallest 7.81 7.49 7.3 7 point 0.007 7.102 and we've been asked to arrange them in order starting with the smallest so let's start off by finding the smallest number here now they all start with seven seven seven seven so that's not going to help us so let's look at the next one to the right of the decimal point which is the tenths column and we're looking for the number with the lowest number of tenths this one has eight temps this one has four attempts this one has three temps this one has zero temps so that's going to be her candidate for our smallest number and this one's got one tenth so this is going to be our smallest number so we'll find our smallest number now let's carry on let's keep looking at the temps column we have e attempts four attempts three temps and one temp so our next one would be 7.102 and then carrying on and again we're looking at the attempts column now if you did have some numbers which have the same digit in the tenths column you'd look at the hundredths so our next number well we've got 7.81 7.49 and 7.3 so it's going to three in the tenths column so that's our next smallest and next would be 7.49 because it's only got a four in the tenths column and finally our biggest number would be seven point it's one and that's it so we've arranged the numbers in order from smallest to largest so the next topic is arithmetic with decimals so we're going to look at add and subtracting decimals we're going to look at multiplying decimals and we're going to look at division involving decimals so let's start off with our first question so first question says work out 4.2 subtract 1.79 now whenever you add and subtracting decimals you're doing the same approach as addition and subtraction as you've seen earlier on in the video but it's just very important that you line up the numbers so that the decimal points are all in the line so we'd write 4.2 like subtraction you'd put the first number at the top and then you're going to put the next number beneath it now over 4.2 so we're then going to write 1.79 and we've lined them up in columns we've got our ones or our units the four and the one we've got our decimal points lined up we've got the tenths lined up and we've got the hundredths lined up as you can see there were no hundreds for the 4.2 so what I'm actually going to do here is just so there's a placeholder in there I'm going to put a zero there as well just so that whenever I'm subtracting these there's some something there okay so let's then put the line beneath so now what we're going to do is we're going to work out a subtraction now before I do that I tend to put the decimal point in the right place so I just line up and put a decimal point in my answer beneath the over decimal points and let's work out the subtraction so 0 take away 9 but we're going to borrow so that's going to be a one and a 10. 10 take away 9 is 1 1 take away seven again we can't do so we're going to borrow so it'll be a 3 and an eleven eleven take away 7 is 4 and 3 take away one is two so four point two subtract 1.79 is 2.41 and you use the same approach as addition where you again you line up the numbers with the decimal points and then you add them so let's have a look at our next question so our next question is multiplication involving decimals now there's two common approaches for this question one approach is to count the number of decimal places in the question and then that means that the answer will have the same number and to put the decimal point in like so or then this second approach which is my favorite I'll talk about that in a second so if I had 0.8 multiplied by 0.3 well ignoring the note point and the null point you be left with eight times three now eight times three is equal to 24. so the answer will have a two and a four in it and then if we look at the question you have one digit after decimal point here and then we have another digit after decimal point here so because two digits after the decimal point in the question there's going to be two digits after the decimal point in the answer so it's going to be 0.24 and as you can see here the two and the four are digits after the decimal point so that's one two so then that would be the right answer 0.8 multiplied by 0.3 is 0.24 another approach which is quite useful is instead of doing 0.8 multiplied by 0.3 is to change these into whole numbers so multiply 0.8 by 10 to get in and multiply 0.3 by 10 to get three and do 8 times 3 is equal to 24. now we made 0.8 10 times bigger and we made 0.3 10 times bigger so that means we've made our answer 100 times bigger we've multiplied by 10 and by 10 again so if we divide our answer by 100 or divided by 10 and by 10 again we'll get our answer so 24 divided by 10 will be 2.4 and divided by 10 again would be equal to 0.24 and that's it okay let's have a look at the next topics the next topic is division evolving decimals so we've got work at 11.4 divided by 3. now whenever I'm dividing a decimal number by a whole number well that's quite nice actually you can just use the brush shelter method as normal but just make sure that you put your decimal point in so 11.4 the first number goes under the bus shelter or a short division and we've got the decimal point here so we put it there in the answer and we're going to divide that by three so we'll put the three at the front three into one doesn't go so put the zero remainder one we've then got 3 into 11 well 11 divided by three well three times three is nine so that would be three remainder two and then we've got 24 divided by 3 well 24 divided by 3 is it so answer would be equal to 3.8 okay let's have a look at our next question so our next question says work out 15.7 divided by 0.2 now here we're dividing by a decimal number which is a wee bit trickier so what I I tend to want to change this number into a whole number now one thing to notice is if I had 6 divided by 3 well 6 divided by 3 is 2. If I multiply both these numbers by 10 and adding 60 divided by 30 well how many 30s go into 60 well that's also 2 or if I times both of these numbers by 100 and said about 600 divided by 300 that's also two so whenever you multiply both the number you're dividing and the number you're dividing by by 10 or 100 or a thousand after you do the division the answer will always be the same so what I'm going to do here is I'm going to multiply both of these numbers by 10 so that will give me 157 divided by 2. now what I'm going to do is I'm going to work out the answer to this question and whatever the answer will be will be the same as the answer to the question we were asked so we're going to do 157 divided by 2. so the answer would be 78.5 so 157 divided by 2 would be 78.5 so that means the answer Target question would also be 78.5 and that's it so our next topic is real life negatives or ordering negative numbers so that's video 208 and 209 in corporate maps and here's a typical question we've been asked to write the cities in order of temperature from coldest to warmest so here's the map of the UK and Ireland and we've got Belfast with negative 8 degrees Celsius Quark at negative seven degrees Celsius Cardiff at zero degrees Celsius London at 2 degrees Celsius Newcastle at negative 4 degrees Celsius and Aberdeen at negative 6 degrees Celsius so we've got those cities and we've been asked to order them in order of temperature from coldest to warmest so if we have a look Belfast is the coolest at negative it then we've got negative seven which is cork then we've got Aberdeen which is negative six so Aberdeen after that then we've got Newcastle Cardiff and London left well Newcastle is a temperature of negative four degrees Celsius so Newcastle and then we're left with Cardiff in London where Cardiff is zero degrees Celsius and finally the warmest city is London with a temperature of the two degrees Celsius to London so in order from coldest to warm as the cities would go Belfast cork Aberdeen Newcastle Cardiff and London okay let's have a look at our next topic so our next topic is arithmetic which involves negatives so our first question is six subtract 10 well six take away six is zero so if we do six take away 10 is going to be a negative number and if we take away 6 we get the zero we'd have another four to take away so our answer would be negative four so six take away ten is negative four our next question negative seven plus twelve well for a negative seven and we add seven we get to zero and then we'd have another five to add so our answer would be five so negative seven plus twelve is five our next question our next question is negative 13 take away four so that means we're going to go four more down from negative 13 so that'll be negative 14 negative 15 negative 16 negative 17. so answer would be negative 17. okay let's have a look at our next question so the next question says five plus negative three now when we add a positive number it goes up when we add a negative number it goes down so we're going to do five add negative three would be the same as five take away three and five take away three is equal to two our next question our next question is it subtract negative seven now when you subtract a positive number it goes down we subtract a negative number it goes up so here we're going to do 8 plus 7 and 8 plus 7 is equal to 15 because it minus minus seven is fifteen eight plus seven is fifteen and our last question here we've got negative 10 now we're adding negative 5. now when you add a positive number it goes up when you add a negative number it goes down so we're going to be adding a negative which is the same as taken away so we've got negative 10 take away five and negative 10 take away five would be negative 15. and that's it okay let's have a look at our next topic so our next topic is multiplication and division involving negatives so let's go through our rules well a positive times a positive is a positive a positive times a negative is a negative a negative times a positive is a negative and a negative times a negative is a positive and the way I remember this is if they're both positive when are you whenever you're multiplying you get a positive answer and if they're both negative you get a positive answer so if they're both the same you get a positive answer and whenever you're multiplying you've got one positive and one negative you get a negative answer so let's have a look at our first question so our first question is 8 multiplied by negative three well 8 times 3 is 24 but we've got a positive times a negative when we've got a positive times a negative we've got one of each is going to be a negative answer so instead of being 8 times 3 being 24 that would be negative 24. and let's have a look at our next multiplication we've got negative 5 multiplied by negative four well it's a negative times a negative so our answer is going to be positive and we'll go 5 times 4 which is 20. now the same rules apply for a division so let's have a look at our first question so our first question is negative 64 divided by 8. now it's a negative divided by a positive so we know it's one of H is going to be a negative answer and 64 divided by 8 is 8 because 8 times 8 is 64. so 64 divided by 8 is it so negative 64 divided by 8 would be negative 8. and our next question we've got negative 30 divided by negative 6. so it's a negative divided by a negative so we're going to get a positive answer and we'll go 30 divided by 6. that's five so answer would just be five and that's it so the next topic is place value so it's very important to know place value so you can do decimal point and then going to the left or the First Column I tend to call it units but it's also called ones so you've got your ones or units then you could do tens hundreds thousands tens of thousands hundred thousands and millions and going to the right of the decimal point you've got your tenths your hundredths your thousandths and so on and here's typical place value question so the question says write down the value of a in the answer to 183 multiplied by 100. so let's start off by working out the answer to 183 multiplied by 100. so we've got 183 so 183 and we're going to multiply it by 100 so that means we're going to move the digits two columns to the left each of the digits again 100 times larger so the one would move into the tens of thousands the eight will move into the thousands the three would move into the hundreds and then we've got zeros so the answer to this question would be would be eighteen thousand three hundred and the question says write down the value of the eight in the answer to that question to the eights in the thousands column so the answer would be eight thousand so the value of the eight and the answer to that question would be eight thousand so now next topic is to look at the inequality sign now there are four different inequality signs you're going to need to know they are the smaller than symbol the greater than symbol the smaller than or equal to symbol and the greater than or equal to symbol and these are our four inequality signs so we use these to show that either one number is perhaps bigger than that another number are smaller than another number and so on so let's have a look at this question the question says write the correct symbol in each box to make the statements correct so our first box we've got 58 and 55 now 58 is bigger than 55 so we're going to put in the greater amount symbol I'm going to avoid talking about crocodile so I'm having my daughter likes to say you know the crocodile eats the bigger number so this is the inequality sound is going to eat the 58 and so on um I remember as the grid of them you know the bigger side going towards the bigger number so here we've got 58 and then we've got the you know the the bigger part of the inequality sign here and the smaller PowerPoint towards the smaller number but however you want to remember it's important to know these signs so 58 is bigger than 55. next we've got 99 and 101 well a 101 is larger than 99 so we're going to put the larger side towards the 101 or the less than symbol so we've got 99 is less than 101 and finally we've got 151 and 149 with 151 is bigger than 149 so we're going to put the gorilla van symbol in which is this one with the bigger side towards the 151 so we've got 151 is greater than 100 149 and that's it Okay so we've looked at place value now we're going to look at the type of questions that I call place value using calculations so our first question so we've been given the 67 multiplied by 34 is equal to 2278. well here we've got 67 multiplied by 340. and as you can see in the question we're trying to work out the 340 is 10 times larger than 34. so that means that our answer will be 10 times larger than this so if we multiply 2278 by 10 that will give us the answer to this question so answer would be 22 700 nearly so that'll be answer okay let's have a look at our next one so again we know that 67 multiplied by 34 is equal to 2278 and we've been asked to work out 6.7 multiplied by 34. so if we have a look at our 67 if we divide that by 10 we are given 6.7 so that means that if we divide our answer by 10 we'll get the answer to this question so 2278 divided by 10 would be 227.8 so because one of the numbers is 10 times smaller the answer would be 10 times smaller so that means that answer would be 227.8 and finally we've been asked to work out the value of 670 multiplied by 340. well if we have a look at the calculation we've been given 670 is 10 times larger than 67 and 340 is 10 times larger than is 10 times larger than 34. so that means if we want to find the answer to this calculation we need to times this answer by 10 and by 10 again or we could just multiply by 100. so if we take our 2278 we multiply by 10 that'll be 22 700 and and if we multiply by 10 again that would be equal to 227 800 and that's it okay let's have a look at our next topic so our next topic is multiple so video 220 and here's part of the Court Master vision card on multiples so the multiples of 4 are 4 8 12 16 20 and so on so four times one is four four times two is eight four times three is twelve four times four is sixteen and so on or you could just start off with four and then add four add four add four and so on so it says work out the first five multiples of 12 well the first multiple of 12 is going to be 12 and then they could do well 12 times 2 is 24 12 times 3 is 36 12 times 4 is equal to 48 or I could just be adding 12s here and do 12 plus 12 is 24 plus 12 is 36 plus 12 is 48 and the add number 12 will be equal to 60. and that's it so the first five multiples of 12 will be 12 24 36 48 and 60. okay let's have a look at our next topics our next topic is common multiples which is video 218. so common multiple so if we wanted to find the common multiples of two numbers we were looking for numbers that are multiples of both of those numbers so we've been asked to find three common multiples of six in here so what I'm going to do is I'm going to list the multiples of six I'm going to list the multiples of here and then I'm going to look for some common multiples now I've been asked to find three common multiples of six and eight so we're looking for numbers that are on both of these lists in the multiples of six and in the multiples of it so first of all I notice that we've got 24 4 because 24 is a multiple of 6 but it's also multiple of 8. the next number that I notice is 48 that's in both of the lists and actually the next number I'd write down is a multiple of 6 would be we're adding 6 for 60 66 add number 6 is 72 and there we've got our three common multiples of the first three common multiples obviously there'd be loads more infinitely many more um but we were asked to list three of them and the first three that I can find are 24. 48 and 72. okay let's have a look at our next topic okay let's have a look at our next topic our next topic is called LCM or lowest common multiple and that's video 218 and 219 on corporate maps to find the lowest common multiple of two numbers you consider there are multiples so if we had six the multiples of six would be 6 12 18 24 30 36 and so on the multiples of 15 would be 15 30 45 and so on and the lowest common multiple or the LCM is the first number in both of those lists and as you can see 30 is the lowest common multiple it's the first number in both the multiples of 6 and the multiples of 15. so 30 is the lowest common multiple and that's it okay our next topic okay let's have a look at our next topic so our next topic is factors and that's video 216 on COBRA maps and this is part of the club mouse revision card on factors so factors of a number are whole numbers that divide into it without a remainder so it says find the factors of 20 well 1 times 20 is 20. 2 times 10 is 20 and 4 times 5 is 20. send me the factors of 20 are 1 and 20 2 and 10 and 4 and 5. and whenever we put them in order they would be one two four five ten and twenty and they're the whole numbers that divide into 20 without a remainder because you can divide 20 by 1 2 4 5 10 and 20 and you'll have no remainder okay so let's find the factors of 30. well 1 times 30 is equal to 30 then we could try 2 well 2 times 15 is equal to 30. let's try 3 well 3 times 10 is equal to 30. now 4 well 4 times 7 is 28 4 times 8 is 32 so 4 is not a factor of 30. let's try five well five times six is equal to 30. now we've done five times six that's actually all the numbers we can try because we've done one two three we've tried four we've tried five and then we've got to six so that's it and let's list them as Factor so let's list them in order one two three five six ten fifteen and thirty and remember if you do want to practice some questions on factors if you go into that booklet revision booklet practice question booklet and there will be questions there on factors and all the topics that we've done so far so it's very important after you've watched me go through a topic to pause the video maybe write some notes on it try some questions and then carry on okay our next topic is common factors so we've looked at common multiples now we're going to look for common factors so whenever you're finding common factors of two numbers you write the factors out for both of those numbers and you look for factors that are factors of both of those numbers so the factors of 20 of 20 well we had 1 times 20 we've got two times ten and we've got four times five they're all the numbers the whole the whole numbers are multiplied together to give you 20. so 1 2 4 5 10 and 20. and 30 we've just done 30 they were one two three five six 6 10 15 and 30. now we're looking for common factors of 20 and 30 well one's a common factor of both of them straight away two also now three is not on the list for 20. four is not on the list for 30. five is a common factor six isn't a common factor it's not the list for twenty ten is a common factor it's in both of the lists and then 20 15 and 30 none of those are common factors so the common factors of 20 and 30 would be one two five and ten our next topic is found in the highest common factor we're going to do this by considering the factors and finding the highest common factor and again video is 218 and 219 will go through the hcf highest common factor and the LCM so let's have a look at the highest common factor so let's consider 16. so 16 is equal to one times sixteen two times eight and four times four so the factors of 16 are 1 2 4 8 and 16. and if we consider 20 well 20 is equal to one times twenty two times ten and four times five so the factors of 20 are one two four five ten and twenty and let's look at our common factors our common factors are one two and four so there are common factors and the highest common factor but that's going to be four so the highest common factor of 16 and 20 is 4. and this topic is particularly useful if you want to cancel fractions and factorize and things like that okay our next topic okay so our next topic is prime numbers and this is the chord management vision card on prime numbers so we've got a prime number is a number of exactly two factors one on itself so 5 is a prime number as its factors are one and five so nine is not a prime number as is factors are one three and nine because obviously nine is one times nine but it's also three times three so it's actually got three factors whereas a prime number only has two factors one in itself and here's a list of our prime numbers two three five seven eleven thirteen Seventeen nineteen twenty three 29 31 and so on it's very important to know these prime numbers and if you want to recap in prime numbers watch video 225 and corporate Maps so the next topic are square numbers and again I'm using the code mods revision card and these revision cards are really useful particularly if you do have a set of them you can sort of set these you know the ones that I'm going through you can set these cards out and sort of pin them up or bring to take them onto the wall so they're there to revise so square numbers that's video 226 and corporate Maps but here's the revision card a square number is a number that you find by doing one times one two times two three times three and so on so one times one is one so one's a square number two times two or two squared is four so four is a square number three squared that's three times three which is nine so nine is a square number four times four is sixteen so that's the square number and so on and the square numbers are 1 4 9 16 25 36 49 64 81 100 121 144 and so on I would learn these first 12 square numbers off by heart it's just really useful to know them okay so these are your first 12 square numbers 1 4 9 16 25 36 49 64 81 100 121 and 144. okay let's have a look at our next topic okay let's have a look at our next topic so our next topic is squaring numbers which is video 227 seven so first question says work out 30 squared so that little 2 is squared we've looked at it already in order of operations and so this is the squared symbol it's a little two above the number and what that means is you have to multiply the number by itself so whenever it says work out 30 squared you do 30 multiply it by 30. so we do 3 times 3 is equal to 9 and it would have one two zeros so the answer would be 900. it's very important to know where the squared button is on your calculator so obviously there's lots of different types of calculator so the squared symbol on this type of calculator will be here we've got this x with the little two above it x squared so that's a squared button on this model of calculator again we've got the X with the little two symbol there the x squared symbol so that would be the squared button and on this type of calculator we've got the same instead of having an X with the turbo the x squared this time we've got a little white box a little white square with the two above it and that's the square button there so obviously with different calculators the position of the square but may be in different positions so my squared button looks something like that so I've got a little x with the squared symbol so if I wanted to work out some something such as 1.5 squared I would type in 1.5 and then so I type in 1.5 and then I would press the squared button like so and my calculator would look something like this and then I would press equals it would give me my answer of 2.25 so the square number on a calculator make sure you're familiar with this button here and you could try it out try it out on 7 squared equals and you should get 49 and so on so squaring a number you just multiply that number by itself okay our next topic okay so our next topic is to square root and square root is the inverse of square so it's the opposite operation so for instance if you know that 5 squared is 25 the square root of 25 is 5 it's going back to if it's finding the number that you would multiply by itself to give that answer and the square root symbol looks something like this it's this symbol here and our first question says work out the square root of 49 well 7 times 7 is equal to 49 so the square root of 49 is equal to 7. so make sure that you know where the square root button is on your calculator mines is this button here it's got the little square root and a little white square beneath it and if I wanted to work out the square root of a number I would press that button to get the square root symbol and then I would just type in so if my question said it'll work out the square root of 32.49 I'd press that button and then I would type in 32.49 and then it would press equals and then it would give me my answer of 5.7 that's it so the square root is the inverse of square and it's finding what number was squared to give you the number beneath the square root symbol so if you're the square root of 49 it would be 7 if you're the square root of 100 it would be 10 and there's a square root button on your calculator and if you want to watch more practice on this watch video 228 on corporate Maps right so our next topic is Cube numbers so the cube number is the result of multiplying a number by itself and by itself again so we've got one cubed that's the cube symbol is one times one times one which is one two cubed is 2 times 2 times 2 which is eight three cubed is 3 times 3 times 3 which is 27 4 cubed is 4 times 4 times 4 which is 64 and 5 cubed is 5 times 5 times 5 which is 125. I tend to learn these ones off by heart so the first five cube numbers are 1 8 27 64 and 125 also I tend to learn that cubed is a thousand just because 10 times 10 times 10 is a thousand so these are the cube numbers that I would learn off by heart okay next topic okay so next topic is cube root so the cube root is the inverse operation to cubing so in other words you're saying what number do you multiply by itself about itself again to give you the number under the cubed root so the first question says find the cube root of 8 well 2 times 2 is 4 multiplied by two is eight so the cube root of 8 is equal to 2. this is the cube root symbol so it's select the square root but it's got a little three above it there so it says work out the cube root of 125 well that's going to be 5 because 5 times 5 times 5 is 125 but it's also important to know how to use it on the calculator and just above the square root symbol you've got the cube root symbol here in yellow so to press this button on the calculator what you do first of all because it's in yellow you press shift so the button here shift and then you press the square root button and then on your calculator the cube root symbol will come up like so then you just type 125. so you've got the cube root of 125 and then press equals and you'll get the answer of 5 and that's it so that's it to find the cube root of a number you just figure out what number do you multiply by itself and by itself again to get that number and on the calculator it's usually above the square root button so you just press shift and then the square root button and then that would bring you up the cube root symbol topic is index notation and that's video 172 on corporate Maps so if I had 5 times 5 times 5 that is 5 cubed and if you remember your Cube numbers which is multiplying the number by itself and by itself again you may want to write an index notation which means write it as a number with a power so we could write that as 5 cubed because there's three fives multiplied together here we've got two times two times two times two times two times two so write to that in index form because there's one two three four five six twos we would write two to the power of six and if we had y times y times y times y because it's four y it's multiplied together we would say y to the power of four it's very important to be able to write these in index form particularly for a topical product of primes and I'll talk about that in a moment so here for instance you wanted to work out 2 to the power of 6 we would just do 2 times 2 which is 4 times 2 which is it times two which is 16 times 2 which is 32 times 2 which is 64. so 2 to the power of 6 would be 64. or you can use a calculator to help you so here's a calculator so what I would do is I'd press the two button so I'd press 2 and then this is the power button and on my calculator it's an x with little white square just above it like so so I'd press that button and on your calculator display above the two would be a little rectangular little box appears and then press six and then press equals and then you will get the answer so 2 to the power of 6 is 64. and that's just a quicker way of working out on your calculator rather than writing 2 times 2 times 2 times 2 times 2 times 2. so that's index notation okay let's have a look at our next topic so next topic is laws of indices and there's three very important laws of indices that I would recommend so you know and this is video 174 on corporate Maps so our first law well if you're multiplying things with the same base so for instance if you had M cubed multiply by m to the power of 4 well that would be M times M times m m cubed multiplied by m times M times M times M and all together the B7 of them so that would be m to the power of seven and a quick way to work that out is because we've got m to the power of 3 and we're multiplying by m to the power of four you can add these Powers you can do three plus four is equal to seven and this is the chord miles revision card on laws of indices so if you've got the revision card this revision card will be very useful for you okay next one if we're dividing if you had m to the power of 8 so if you had M times M times M times M times M times M times M times M and you divided that by m squared that's M times M two of the M's would cancel out with two of the m so you'd be left with M times M times M times M times M times M and that'd be m to the power of 6 which is m to the power of six and a quick way to work that out is if you're dividing and you've got the same bases you can take away the powers you can do eight take away two and that's m to the power of six and finally we've got a power over par so if you've got a power and then another Power you multiply the powers together and let's have a look and see why that works so if you had M cubed squared remember squared means multiplied by itself so we're doing M cubed multiplied by itself so that's M times M times M multiplied by itself so that's M times M times M and if you multiply all those together you get m to the power of 6 and a quick way of doing it is to multiply the power so if you go to power to a power you can multiply the two powers of three times two is equal to six so let's have a look at some examples so if I add 3 to the power of 4 multiply by 3 squared I'd add the powers together because multiplying that would be to the power of 6. if I had three to the power of 10 divided by 3 squared I would subtract the power so 10 take away 2 is equal to 8 and finally if I add 4 squared cubed I would multiply the powers because it's a power over power you've got a power and then an over power so you'd multiply the powers together 2 times 3 is equal to 6 so it'd be 4 to the power of 6. and that's it okay let's have a look at our next topic so next topic is negative indices and that's a number public because it's in red and this video 175 on corporate Maps now let's consider this pattern multiple 5 to the power of three five to the power of two five to the power of one five to the power of zero five to the power of negative one and five to the power of negative two so five cubed well 5 times 5 times 5 is 125. 5 squared plus 25 5 times 5 is 25 5 to the power of one that's just a five so our answer would be five okay next next we've got 5 to the power of zero well 5 to the power of zero is one okay and next one is five to the power of negative one well if we look at our numbers here we've got 125 25 5 and 1. now there's a pattern here to go down in this pattern we're dividing by five we're dividing by five we're dividing by five we're dividing by five and then if we want to find out our next one we would just divide by five again so if we do one divided by five well that'd be 0.2 or we could just write as a fraction which is one over five if we want to get our next answer we could divide by five again okay so we do one over five or fifth divided by five so that'd be one over twenty five now what's actually quite useful the spot here is because we've got five to the negative two that's the same as putting one over and five squared and five squared is 25. so that's our rule if we have x to the power of negative n that's the same as 1 over x to the N so you can just put one over and then just use the positive power so here if we were asked to work out 2 to the power of negative three we could just put one over and then just write 2 cubed change in our negative 3 to just three and then two cubed is eight so that would be just one over eight or one if so 2 to the power of negative three is one over eight next 10 to the power of negative 2 well we put 1 over and then just write 10 squared and 10 squared is 100 so answer would be 1 over 100. so if we had 10 to the power of negative 2 our answer would be 1 over 100. that's it so if we want to work out a negative power you just put 1 over and then just use the positive power on the denominator okay let's have a look at our next topic so our next topic is product of primes and that's video 223 on corporate Maps so every single whole number that is greater than one is either prime or can be written as a product of primes and the word product means to multiply so it e remains that every single number that's greater than one is either a prime number or is equal to prime numbers multiplied together so our first question our example says write 60 as a product of primes this is very important that you know your prime numbers and your prime numbers are 2 3 5 7 11 13 17 19 23 29 31 and so on and it's very important you know those prime numbers so we're going to write 60 as a product of primes and we're going to give our answer in index form so let's start off with 60 and I like to do this using a prime factor tree you can use another approach by using sort of an upside down by shelter whatever approach you prefer is totally fine but I like to use a prime factor tree like so so 60 I think of two numbers that multiply together to give me 60. so I'm going to go forward 2 times 30. so 2 times 30 is equal to 60. now we look and see if either of these numbers are prime and 2 is prime so I'm going to circle it but 30 is not prime so what I'm going to do is I'm going to think of two numbers at times together to give me 30. so 3 times 10. now 3 is prime so Circle it whereas ten is not so now you're going to think of two numbers that multiply together to give me 10 and that's going to be 2 times 5. I never choose one in the number itself because you don't get any further so 10 so 10 would be 2 times 5 so 2 times 5 and let's Circle them and that's it we're finished so 60 is equal to 2 times 3 times 2 times 5. so let's write that out 60 is equal to and let's just do it in order so instead of doing 2 times 3 times 2 times 5 I'm going to do 2 times 2 times 3 times 5. so 2 two times two times three times five and let's just check it works 2 times 2 is equal to 4 times 3 is equal to 12 times 5 is equal to sixty and as you can see we've written 60 as a product of primes but the question says to write an index form as you've just seen earlier on with index notation if you get something such as 2 times 2 that's 2 squared so we can write 60 is equal to 2 squared multiplied by 3 multiplied by 5. so we've written 60 as a product of primes in index form and that's it and it's very important to be able to do that and video 223 we'll go through that and also if you go to corporatemath.com and you go to videos and worksheets and you scroll down to 223 as well as having the video tutorial there will be some practice questions and textbook exercises on this topic so if you do want extra practice feel free to do those but also remember there's the revision booklet which you can print it's the links in the description below and if you have that there's some questions on this now okay let's have a look at our next topic so our next topic is apply and product of primes now this video 223a and Cobra Mars so let's start off by looking at this question so a number M has been written as a product of primes as 2 multiplied by 3 squared so this is m 2 multiplied by 3 squared and the first part A is to say what number is M so let's work this out now remember we've got our order of operations that is we do any brackets no orders that's another name for power so that's squared so we're going to work out the squared first of all so we're going to do 3 squared so 3 squared is 9 and then we're going to multiply by the 2 2 times 9 is equal to 18. so that means m is equal to 18. and Part B says write the number at 10 m as a product of primes so 10m well if we know that m is equal to 18 10 m would be equal to 10 times M so that would be 180 so you could take 180 and do that prime factor tree and find out what 180 is as a product of primes and what's really useful here is we know that 10 is equal to 2 times 5. so two prime numbers 2 and 5 multiplied together would give me ten two times five now we know that m is equal to 2 2 times 3 squared so m is equal to 2 times 3 squared so we can multiply this 10 by m which is 2 times 3 squared that would be our answer but we want to write this obviously in index form and as you can see we've got 2 times 5 times 2 times 3 squared I would look at the twos first of all you get 2 times 2 which is 2 squared we've then got our 3 squared so multiply by 3 squared and then multiply by 5. so 2 squared times 3 squared times 5 would be 10 m and we can check that because remember we knew that 10 times M was 180 let's check this is 180. 2 squared is 4 3 squared is 9 4 times 9 is 36. I multiply by 5 would be 180 so that means a 10 m as a product of primes would be 2 squared times 3 squared times 5. okay and next part now we can also use productive primes to find square numbers and Cube numbers so for instance if we knew the 280 was 2 cubed multiplied by 5 multiplied by seven and we were asked what the lowest whole number the 280 would need to be multiplied by to give a square number what we can do is we can consider what a square number is so a square number is a number multiplied by itself and that would give you a square number so if we take what we've been given here so 280 is 2 cubed that's 2 times 2 times 2 times 5 times 7. just write an iron fill if we share these prime numbers out as evenly as possible and then find out what extra ones we need then we can find out what number you can multiply 280 by to get a square number so let's share these numbers out as evenly as we can so we've got two two that's three twos so we're going to have to give one Circle two twos and one Circle one two we've got a five well unfortunately that's just going to have to go in one of the circles as well we can't you know we don't have two fives and we've got a seven so again that's going to have to go in one of the circles now we want the same thing in both of these circles so let's put some extra prime numbers in this circle so we had two twos so we're going to need number two we're going to need a five and we're going to need a seven if we multiply 280 by 2 5 and 7 that should give us a square number so let's see what 2 times 5 times 7 is so 2 times 5 times 7 is equal to well 2 times 5 is 10 times 7 is 70. so if we multiply 280 by 70 we should get a square number so 280 multiply by 70 is equal to 19600 and if we work out the square root of that we get that's equal to 140 so it is a square number so if you want to find a square number what you need to do is share out the parameters you've been given as evenly as possible and then add in the extra ones you need and there is a bit of a shortcut if we take all the numbers in the circuits here are two times two times five times seven times two times two times five times seven that would be 2 times 2 times 2 times 2 which is 2 to the power of 4 times pi and we've got 5 times 5 that's 5 squared and we've got 7 times 7 that's 7 squared if you notice all the powers are even so if you want a square number all you need to do it is to make all the powers even so if I was doing this question I would first of all just look and see well as 2 cubed I'm going to need an extra two I've got five what's five to the power of one so I need an extra five and I've got seven to the power of ones I need an extra sevens so if I multiply this by two and by five and by seven I will get a square number and likewise if I wanted to find a cube number all the powers would have to be multiples of three so for instance if I had 2 to the power of 9 multiplied by 5 to the power of 3 that would be a cube number because this Powers a multiple of three and this Powers multiple of three we use the product of primes to work out the LCM that's the lowest common multiple or the hcf the highest common factor so let's have a look at our first question so let's write some numbers as product or primes and then we'll use that information to find the lowest common multiple and the highest common factor so first part says right 92 is a product of primes so I'm going to do my prime factor tree so 92 that's 2 times 46 let's Circle the two and 46 that's 2 times 23 and they're both Prime so let's Circle them so 92 as a product of primes would be 2 times 2 times 23 and in index form that would be 2 squared times 23. now our next question is to write 48 as a product of primes so I'm actually going to use my calculator to write 48 as a product of primes so on my calculator I've got this yellow fact there so what I do is I press 48 then press equals and 48 comes up in my calculator display then I press shift and then where it says fact there and on the display it will show me 2 to the power 4 multiplied by 3. so 48 equals 2 to the power of 4 times 3 or 2 times 2 times 2 times 2 times 3. so we've got 92 as a product of primes and with 48 is a product of primes let's now use that information to work out the lowest common multiple and the highest common factor of those numbers so here we've got our question it says write down the highest common factor of 48.92 and write down the lowest common multiple of 48 92. so let's write down what they were as product or primes again so 48 was equal to 2 to the power of 4 multiplied by 3 or if we wrote it out in full it would be 2 times 2 times 2 times 2 times 3 and 92 92 is equal to 2 squared multiplied by 23 or full 2 times 2 times 23. now let's put those numbers in our Venn diagram so here we've got a Venn diagram and we've got one Circle for 48 and with one Circle for 92 and we're going to put their prime factors into this Venn diagram so first of all let's see if they share anything so as you can see 48 is 2 times 2 times 2 times 2 times 3 and 98 is 2 times 2 times 23. so they've both got two twos 92 has got two twos and 48 has got two twos so we're going to put two twos in the middle now as you can see here 48 had two more twos so let's put those two twos in the 48 side 48 also had a three but 92 doesn't so let's put the three there as well and finally 92 has a 23 so we'll put that into the 92 side so as you can see in both circles we've got 48 and we've got 2 times 2 times 2 times 2 times 3 and that's 48 and for the 92 Circle we've got 2 times 2 times 23. so you put what they share in the middle and then you put the extras onto each side so the first question says work at the highest common factor of 48.92 so to find the highest common factor we multiply the prime numbers in the middle so we're going to do two times two so the highest common factor will equal 2 times 2 and 2 times 2 is equal to four so the highest common factor is four okay our next question says to work at the lowest common multiple of 48.92 so to find the lowest common multiple we multiply all the numbers in the Venn diagram so we're going to do 2 times 2 times 3 times 2 times 2 times 23 or in order the lowest common multiple equals 2 times 2 times 2 times 2 because there's four twos times three times twenty three and when we do that we get that's equal to one thousand one hundred and four so we've worked out the highest common factor and the lowest common multiple of 48.92 really quickly and easily by using this Venn diagram so what you do is you write each number as a product of primes you put the numbers into the Venn diagram and make sure whichever one says share go in the middle and put the extras on each side and that's it and to find the highest common factor you multiply the prime numbers in the middle and to find the lowest common multiple you multiply all the numbers in the Venn diagram okay let's have a look at our next topic okay let's have a look at our next topic so our next topic is called standard form and this video is 300 to 303 in corporate Maps now standard form is a really useful way of writing very large or very small numbers very quickly and easily so rather than a large number around lots of times we can use standard form to help us and a numbers in standard form if it's in this format we've got a number between one and ten so it's a number bigger than or equal to one but less than 10 so for instance 1 or 4.22 or 9.9 multiplied by 10 to a certain power so maybe 10 cubed or 10 to the power of 8 or even 10 to the power of negative three and we're going to look at how to write some large numbers in standard form and some small numbers in standard form and I'm actually going to do this part of the video twice so I'm going to actually use two different techniques here because some teachers will use a technique where they add zeros or move decimal points and some teachers will move the digits and say the decimal points have to stay fixed my job is to help you and make sure that I cover the way that you are familiar with and I don't want to confuse You by using just one technique and then you think hold on my teacher talks about it in a slightly different way so to begin with I'm going to talk about adding zeros and moving decimal places and then I'm going to do the questions again by moving the digits rather than the decimal points so first of all we had seven thousand and we wanted to write it in standard form well a number between 1 and 10 to choose well I'm thinking seven would be a good choice and then we're going to multiply it by 10 to a certain power now whenever we have a whole number and we multiply by 10 we add a zero on so 7 times 10 is 7 take better than zero if here we want to add three zeros on so what we actually want to do is multiply seven by a thousand and that would be 10 cubed because 10 times 10 times 10 is a thousand now notice that if you have a whole number and you want to add one two three zeros on the power would just be three so that would be 7 multiplied by ten cubed if we had a number such as 80 000 which is 8 followed by four zeros we could have eight multiplied by it because this is a whole number we could say it's 10 to the power of 4 because it's one two three four zeros okay let's have a look at our next one so next one is to write two million five hundred thousand in standard form so we're going to choose a number between one and ten now we've got two five so we're going to choose 2.5 here because that's a sensible number to choose between 1 and 10. now what we're going to do is we're going to figure out if we have 2.5 how many times we need to move the decimal point together at the end of the number so we want to move the decimal point from here to here so we would move it one two three four five six times so that means we need to multiply two point five by ten to the power of six because that would move the decimal place one two three four five six places to the right okay next let's look at some small numbers so here we've got 0.4 so that would be four would be our number we would choose between one and ten we need to multiply by ten to the power off now we've got a small number here which is less than one so this is going to have a negative power and we need to figure out how many times we move the decimal point to get from four to be this number so if we had four the decimal point would be here and we would move it one two three four five six seven times so answer would be four times ten to the power of negative seven or shouldn't once actually told me a bit of a shortcut they said well so there's one two three four five six seven zeros in front so you can just write four times ten to the negative seven and I thought that was quite a neat way to do is just count the number of zeros okay and finally we've got 0.018 and if we want to write that in standard form well it's going to be 1.8 because that's the number we choose between one and ten multiplied by ten and now we've got one point here so the decimal points here and we want to move the decimal point one twice to the left so the power would be negative two alternative you might have spotted the pattern that my student mentioned to me and because it's a small number a note Point number and it's got two zeros it'll be to the power of negative two okay now let's have a look at these questions using the other approach so let's get rid of that okay let's have a look at our first question so first question is to write 7 000 in standard form so we want a number between one and ten and a sensible number to choose here would be seven and we're then going to have multiplied by 10 and to a certain power now if we have seven sevens in the units of the ones column and we want to move it one two three columns to the left so we want to move the seven three columns to the left so we would multiply by ten a hundred and a thousand so that's actually a thousand is ten cubed so if we want to move a three columns to the left we have a power of three so that would be seven multiplied by ten cubed next we've got two million five hundred thousand so we've got to choose a number between one and ten so a sensible Choice here would be 2.5 and then times ten and to a certain power so let's put 2.5 in so the two would be in the ones column of the units column there would have a decimal point and we'd have our five and we want to move the digits one two three four five six columns to the left so we would have to the power of six so two point five times ten to the power of six would be two million five hundred thousand okay let's have a look at some small numbers so we've got 0.4 so that would be well we need a number between one and ten so we're gonna have four times ten now because this is a small number we're going to have a negative power because 10 to the negative 1 is 0.1 so we'll be multiplied by 0.1 10 to the power of negative 2 would be 0.01 and so on so we want to we want a negative power a very small number and if we and if we put our 4 in our units or our ones column there we want to move the four one two three four five six seven columns to the right so because we want to move at seven columns to the right the power would be negative seven so answer would be four multiplied by 10 to the power of negative seven and finally 0.018 well a sensible number to choose between 1 and 10 would be 1.8 and we want to multiply it by 10 to the power of negative something because we want it to be smaller and we've put one R 1.8 in our columns our one is in our units column are one's column and we want to move it to our hundredths column so we want to move the one one two columns to the right so we want them times it by 1.8 times 10 to the power of negative two okay next now let's look at how we would write numbers that are almost in standard form in standard form so if we have a look at this number 562.8 times 10 to the power of 5. this number is almost in standard form and what I mean by that is we need a number between 1 and 10 and this number at the front isn't between 1 and 10. so we need to make it between 1 and 10. so if I had the number of 562.8 I would divide that by 100 because if I divide that by 100 I would get 5.628 and that's between 1 and 10. but if I divide one number by 100 so for instance if I had 700 multiplied by two the answer is equal to one thousand four hundred if I divide this number by 100 and get seven I would need to multiply this number by 100 to get 200 to still keep the same answer so if I've divided this by 100 I need to multiply this by 100 to make sure answer stays the same so it's going to be 5.628 multiplied by 10. so if I multiply this by 100 I'd multiply by 10 so that'll be 10 to the power of 6 and I'm multiplied by 10 again so that'll be 10 to the power of 7. so answer would be 5.628 times 10 to the power of 7. and that's it okay let's have a look at another one okay this time we've got 0.024 times 10 to the power of 9. now this isn't a standard form because this number needs to be between 1 and 10. so we're going to have to write 2.4 here because that would then make sure that we would have a number between 1 and 10 at the front and then we would do multiply by 10 to the power of something so we multiply this by we'll move the 2 1 2 3 places to the left so we've multiplied this by a thousand so if we multiply this by five and we need to divide this by a thousands so dividing it by 10 would be 10 to the power of 8 divided by 10 again would be 10 to the power of 7 and dividing it by 10 again would be 10 to the power of 6. whenever to look at it is if we dividing it by a thousand remember that's 10 cubed and then if we take away the powers we get 10 to the power of six and that's it so just make sure that if you have a number that's almost in standard form if you make the number at the front 10 times larger you need to make the Times by 10 bit 10 times smaller to make sure that the answer stays the same okay next okay let's look at how to do our arithmetic numbers in standard form so how to do adding numbers in standard form multiplying numbers and standard form dividing numbers and standard form and so on now with adding numbers in standard form there's no set rule so if I had 5.3 times 10 to the power of 7 plus 7.96 times 10 to the power of 8. the way I would do this question would be to ride the mountain full so I'm going to write R 5.3 times 10 to the power of 7 I'm full so that would be 5 3 and then one two three four five six zeros and then with 7.96 times 10 to the power of 8 will be plus seven nine six that'd be 796 and then we would then have another six zeros one two three four five six and then we need to add these numbers together and whenever it happens together I'm just going to use the column method so let's line them up so we've got 7 9 6 and then one two three four five six zeros five three and then six zero so one two three four five six and then three five okay and then we're gonna add them so let's just add them and see what we get so it would be zero plus zero is zero zero zero zero zero zero we've then got six plus three is nine nine plus five is equal to fourteen so put the four down either one and seven plus one's equal to eight so answer would be 849 million or we might wanna write it in standard form again so it'll be eight point four nine times ten to the power of one two three four five six seven eight so beats part of it and that's it okay let's have a look at our next question so next question is multiply numbers in standard form and I like this because we can now put together our laws of indices if we have nine times ten to the power of four multiply by two times ten to the power of five so what we can do is we can multiply the numbers at the front we can do 9 times 2 is equal to eighteen and then we can multiply the tens to the powers so we get the 10 to the power 4 multiply by ten to the power of five and whenever we multiplying things with the same base they're both 10 to the power of something we add the powers so we would do 10 to the power of four plus fives to the power of nine so that's not our answer because that's 18 times 10 to the power of 9. remember if for our answer to be in standard form we want it to be between 1 and 10. so we're going to do 1.8 times 10. now we've made 18 10 times smaller so we need to make this 10 times larger so 10 to the power of 9 times my time would be 10 to the power of 10. so that would be 1.8 times 10 to the power of 10. now as well as being able to do this in a non-calculator paper it's also important to be able to do in a calculator paper so let's have a look and see how we would type this in on our calculator so on our calculator we've got our standard form button here down at the center there at the bottom we've got our multiply by 10 to the power of X there now whenever I'm doing a question like this I tend to like to use brackets um so I would write it like this so I would open up my brackets to begin with I would press my 9 because that's the number between 1 and 10 we're going to use and then I would press the standard form button there and then you would get it would come up with 9 times 10 and then you press 4 and that would be how you type in 9 times 10 to the power 4 it looks something like this and then close brackets so you press so you open up your brackets you press nine the standard form button and then press four to show that the power is 4 and then close brackets then press times and then for this 2 times 10 to the power of 5 you're going to do two the standard form button and then five and then close brackets and press equals and you should get this as your answer it'd be a good idea to practice that now and get your calculator and try typing this in and the answer in your display should look something like this where we had a 1.8 times 10 to the power of 10 and that was our answer and that's it okay let's have a look at our next question so this time we've got a division question and we've got 4.5 multiplied by 10 to the power of 7 divided by or over 9 times 10 to the power of negative 5. so we've got a large number divided by a very small number and whenever we divide by a very small number so getting even bigger so let's have a look at this question see what we're doing so we've got 4.5 times 10 to the power of 7 and we're dividing that by 9 times 10 to the power of negative 5. so we're going to do 4.5 divided by 9 let's actually do that quickly now I know the answer is 0.5 but I'm just going to show you how 4 divided by 9 is 0 remainder 4 and 45 divided by 9 it would be equal to 5. so that'd be equal to 0.5 so 4.5 divided by 9 is 0.5 then we'll put our multiplied by 10 parts so we've got 10 to the power of 7 divided by 10 to the power of negative 5. remember whenever we're dividing we take away the power so we've got 7 minus minus five and seven minus minus 5 we would add on the 5 would be multiplied by 10 to the power of 12 because 7 minus minus 5 would be 12. so we would get 0.5 multiplied by 10 to the power of 12. now this isn't a standard form because this needs to be between 1 and 10. so we're going to multiply that by 10 to get 5 but we need to divide this by 10 so 10 to the power of 12 take away one of those tens would be 10 to the power of 11. so that would be 5 multiplied by 10 to the power of 11. now to work out a fraction of an amount you divide by the denominator and you multiply by the numerator so let's have a look at two questions here we've been has to work out one third of 24. so to work at a third of 24 where the denominator is three so I'm going to take my 24 and I'm going to divide it by 3. we're dividing by the denominator so 24 divided by 3 is 8. so we've got 8 and then we would times that by the numerator now the numerator is just one here and 8 times 1 is just eight so the answer would be eight so to find a fraction of an amount you just divide by the denominator and multiply by the numerator if the numerator is one you could just divide by the denominator okay let's have a look at our next question we've been asked to work out four fifths of 35. so again we're going to take our 35 and we're going to divide by the denominator so we're going to do 35 divided by 5 and 35 divided by 5 is 7 because 5 times 7 is 35. now we're going to take that 7 and we're going to multiply by the numerator 7 multiplied by 4 and 7 times 4 is equal to 28. so four-fifths of 35 is equal to 28th and that's it and this is a very important topic you quite often ask to find fractions of amounts whether it's in the exam or whether it's even in real life in a up and so on and so what kind of fractions of our minds is very important and that's video 137 on corporate Maps so our next topic is to express as a fraction so sometimes you have to express something as a fraction so here's a typical question that says write two days as a fraction of three weeks so here we've got three weeks and three weeks well the seven days in a week so three times seven is equal to 21 days so we've been asked to write two days as a fraction of 21 days and so we would just write down two days out of 21 or 2 over 21 and that's it so to express something as a fraction whatever the total is you put that on the denominator and whatever you've been asked to write as a fraction of that total you put on the numerator and that's it okay let's have a look at our next topic so the next topic is to find a fraction of a ship so here we've got a grid and we've been asked to shade in two thirds of the grid now whenever you find in two thirds of this grid you can do it in different ways you could say well if I'm shading in two thirds that's two out of every three squares so if I had a look at this top row you've got three squares and you could shade in two of them share in sheared in and leave blank then go to the next row and sheared in sheared in and leave blank and the last row again you could go sheared in sheared in and leave blank and I'm actually this is terrible there um so that's one way you could do that question so you could just consider it as in if you've been associated in two thirds you'd share it in two of them believe one blank or if you were to share it in three quarters you'd share them three of them and leave one blank and so on alternatively you could have looked at the columns here and you could have said well there's three columns one two three I need to see it in two thirds so I could see it in two columns and leave one blank and then that would have worked as well alternatively you could have said well there's one two three four five six seven eight nine squares you could work out two thirds of nine and that would be six as well and then you would just see it in the six squared that's it okay so fraction of a ship and that's video 143 on corporate Maps right so let's have a look at our next topic so our next topic is equivalent fractions and that's video 135 in corporate Maps so here we've got two fractions that are equivalent to each other one half and five tenths and as you can see one half is the same as five tenths now for two fractions to be equivalent to each other the numerator and denominator must be multiplied or divided by the same number to get an equivalent fraction so as you can see here if we multiply one by five we get five and if we multiply the denominator of two by five we get that's equal to ten so we've multiplied the numerator by five and we've multiplied the denominator by five and that's given us an equivalent fraction and also works for division if we started with the five tenths if we divided by five and divided by five then you get a half so for two fractions to be equivalent to each other the numerators and denominators must be multiplied or divided by the same number so here we've got two thirds and that's equal to 8 over blank and we're trying to find this missing number for these equivalent fractions so to get from two to eight we multiply by four so multiply by four so we've multiplied the numerator by four so let's multiply the denominator by four so let's do three times four and three times four is equal to 12. so two-thirds is equal to eight twelfths they are equivalent to each other okay so let's have a look at our next topic so our next topic is simplifying fractions or canceling down fractions so here we've got three fractions and we're going to cancel them down we're going to simplify them so let's have a look at our first one so first question says to simplify six eighths so to simplify six eighths we see what we can divide them both by or what are the common factors now with the two numbers they'll always have a common factor of one but dividing them by one won't help us simplify it or make the numbers any smaller so we're going to look for common factors apart from one so with six and eight I can divide both of these by two so let's divide six and eight both by two and see what we get well 6 divided by two is three and eight divided by 2 is 4 so our answer would be three quarters so whenever you simplify six eighths that's the same as three quarters okay our next one so our next one is fifteen twenty-fifths so we're looking for a common factor what can we divide both of these numbers by and well 15 and 25 they're both in the five times tables so let's divide both of them by five well fifteen divided by 5 is 3 and 25 divided by five is five so then we get three fifths and three and five well we can't divide these by anything well you could divide them by one but that's not going to change the elephant so if simplified it as far as possible okay let's have a look at our last question so last question is to simplify 12 via teams Now twelvet teams you can actually divide both the numerator and the denominator here by different common factors we could divide both of them by two and that would give us six over nine but as you'll notice here six and nine are both divisible by three so then you divide both of them by three and that would give you two over three we could have divided twelve eight teams above by three because both of these numbers on the three times tables and if we divide both of them by three we get well 12 divided by 3 is 4 and 18 divided by 3 is equal to 6 so you get 4 6. now they're both even so you can divide them both by two to get two thirds or alternative you might have noticed that 12 and 18 are both in the six times tables so you can if you can divide them by the highest common factor that would be fantastic if you can spot it so 12 and 18 you can divide both of them by 6 12 divided by 6 is 2 and 18 divided by six is three so you get to two thirds so whenever you simplify fractions you might be able to cancel it down to the final answer straight away or sometimes you might want to just half both of the numbers to give you sort of smaller numbers then you can divide again and so on and that's it the next topic is to order fractions and this is video 144 in corporate maps and we've been asked to arrange in order smallest first three quarters two-thirds five sixths and seven twelfths so what we're going to do is we're going to make all these fractions have the same denominator so if we've got four three six and twelve we want to make all these denominators the same number now what's great is if you notice here we've got 7 12s well we can times four by three to get 12. we can multiply three by four to get 12 and we can double six to get 12. so what I'm going to do is I'm going to find equivalent fractions but all of them are going to have 12 on the denominator and then we can look at them and see which one's the biggest and so on so we've got three quarters well to get from 4 to 12 we multiply by three so we're going to need to multiply the numerator here by three well three times three is nine so three quarters is nine twelfths to get from 3 to 12 you multiply by four so you need to multiply this numerator by four two times four is equal to eight so two thirds is the same as 8 12. we've got five six well to get from 6 to 12 you double it or you multiply by two so doubling 5 would give us ten and then finally well we had seven twelves that's seven twelfths now we need to arrange an order now as you can see here we've got 9 12 8 12 10 12 and 7 12. as you can see the smallest one would be our seven twelve so in order it would be seven twelfths then our next smallest so we've done that one our next smallest would be this one which was 8 12 but in the question it was two-thirds so let's write what they gave us in the question which is two-thirds next we've got our 9 12 and again in the question they give us three quarters so let's write that down three quarters and finally the largest fraction was our 10 12 which is five six so we'll write that down five six so our answer is seven twelfths two-thirds three quarters and five six and just remember if you want to practice any more questions like this that booklet is really useful so if you go to the description and click on that you'll find that there'll be questionnaire and ordering fractions our next topic is adding fractions and that's video 133 on corporate Maps so to add fractions it's very important that the fractions have the same denominator a common denominator so for instance if we were asked to work out two-fifths plus a quarter as you can see the denominators the numbers on the bottom of the fraction aren't the same we've got a five and a four so what we're going to do is we're going to find the lowest common multiple of five and four well that's going to be 20 because 5 10 15 20 4 8 12 16 20. the lowest common multiple of both of them is 20 and what we're going to do is we're going to find two equivalent fractions now remember to find an equivalent fraction we consider what you multiply 5 by to get the 20 well that's four so you multiply the numerator by 4 as well so 4 times 2 is equal to 8. so two-fifths is the same as eight twentieths next our quarter we want to have 20 on the denominator so to get from 4 to 20 multiply by five so you need to multiply the one by five as well so that's going to be five so one quarter is five twentieths and finally we've got eight twentieths plus five twentieths when we're adding fractions with the same denominator we just add the numerators so if we had eight twentieths plus five twentieths that's going to be thirteen twentieths and that's it so two-fifths plus one quarter is thirteen twentieths okay let's have a look at our next question so our next question is to work out two and seven ninths plus one and two thirds so here we've got some mixed numbers and the mixed numbers where you've got a whole number and a fraction so for instance four and a half and we're going to be dealing with these mixed numbers and whenever we're dealing with mixed numbers whenever ad nurse subtracting or multiplying dividing fractions it can be very useful to turn them into top heavy fractions and that's where we write them as a fraction with a larger number on the numerator than on the denominator and to do that what we just consider 2 and we're dealing in lengths here so if I had a cake or a pizza and I cut it into ninths it'd be nine ninths in a pizza in two pizzas there would be 18 slices you would have 18 names so 2 is equal to 18 notes and then we've got another seven ninths so if I had 18 ninths and another seven ninths that would be 18 plus 7 is 25 ninths now the shortcut I take with this is so agitate the whole number which is two and times it by the denominator so 2 times 9 is 18 and add on the 7 is 25 so that gives me the 25 ninths next I take the whole number which is one and I do one times three is three and add on the numerator which is two so three plus two is equal to five so that would be Five Thirds and again let's just check this if I had a pizza and I cut it into thirds that's three slices and then over two slices is five slices so here we've written our mixed numbers as top heavy fractions so we want to have the same denominators so we get ninths and we've got thirds so the lowest common multiple of nine and three would just be nine we could multiply both the numerator and the denominator of this fraction by three and that would give us 9 on the denominator so let's do that so we've got 25 ninths and then we're going to add if we multiply both of these by three we get well 5 times 3 is 15 and 3 times 3 is equal to nine so we've got 25 ninths plus 15 knives and that's fantastic because they've both got the same denominator so if we just add the numerators 25 plus 15 is 40 so answer would be 40 ninths now if the questions are mixed numbers such as like you know two and seven ninths I would change this top heavy fraction or answer into a mixed number as well remember the line in a fraction means divided by so how many nines go into forty well four nines is equal to 36 so it's going to be 4 and 36 where we have 40 so our remainder would be four so answer would be four and four ninths and that's it our next topic is now subtracting fractions and that's the same Technique we get a common denominator and then we just take away the top numbers we take away the numerators instead so if we had nine tenths and two thirds so if you look at the denominators we put 10 and 3. now the first number in the 10 times tables and three times tables would be 30. so I'm going to write a common denominator of 30. so if I multiply both with the numerator and denominator of this fraction by 3 well 10 times 3 is 30 and 9 times 3 is 27. and with this fraction we had two thirds if we multiply the three by ten we get 30 and if we multiply the 2 by 10 we get 20. so we've got 27 30 take away 20 30ths well if you had 27 30ths and you took away 20 30 if you're left with 7 30 just taken away the numerators that's it okay next topic our next topic is multiplying fractions so multiplying fractions is really easy you just need to multiply the numerators and multiply the denominators and that's it so if you had two fifths times a quarter if you multiply the numerators two times one that's two and if you multiply the denominators 5 times 4 that's 20. now the answer here is two twentieths I'm going to cancel this down I'm going to simplify it we can divide both of these by two so our final answer would be one tenth okay next we've got 1 and 1 7 multiplied by two-thirds again this is a mixed number so let's make this a top heavy fraction one times seven is seven plus the one is eight so one and one seventh would be eight sevenths or another way to think if it is if you had a pizza and you cut it into seven slices then you had an extra one that's going to be eight slices so it's eight sevenths and then multiply it by two thirds so let's multiply the numerators so 8 times 2 is 16 and 7 times 3 is 21. so our answer would be 16 over 21. 16 and 21 have no common factors apart from one and that's not going to help us so if you want more practice and multiplying fractions if you watch video 142 on corporate maps you can watch the video tutorial on it you can do some practice questions so the textbook exercises but also remember you've got that practice question booklet so you can't do these questions down that booklet and just make sure you're you're happy with this topic okay let's have a look at our next topic our next topic is dividing fractions which is video 134 in corporate Maps so to divide fractions what we do is we multiply by the reciprocal of the number we're dividing by and that's really complicated well what it means is if you've got 7 15 divided by three quarters instead of doing 7 15 divided by three quarters what we can do is we can do 7 15 multiply by and then we can find the reciprocal of three quarters and the reciprocal is a fancy word for flipping over so instead of writing three quarters we could write four thirds Okay so we've at 7 15 multiplied by four thirds now we've got multiply so we can just multiply the numerators and multiply the denominators so 7 times 4 is equal to 28 and 15 times 3 is equal to 45. so answer is 28 40 fifths and apart from one they've got no over common factors so that's it so if you watch video 134 that would be quite a useful video to see why this works but in this video I'm just trying to make sure that you're familiar with each of these topics so to divide fractions what we do is instead of dividing by three quarters we multiply by its reciprocal so we change the divider multiply and you find the reciprocal of the second number and then you just multiply the fractions and that will give you the answer okay next question our next question is to work out two and a half divided by one and three-fifths so we've got mixed numbers here so let's turn them into top heavy fractions so two well we've got two times two is four plus one is five so it's going to be five halves and another way to think about it remember is if we're dealing with halves if you had two pizzas and cut them into half so you could four of them and another one will be five of them so be five halves and we're dividing by well if we look we've got one and three fifths so one times five is five plus three is eight so it's going to be eight fifths now remember we want to multiply by the reciprocal of the number we're dividing by so we're going to write five over two multiplied by five over eight next we're going to multiply so we've got 5 times 5 is 25 and 2 times 8 is 16. so our answer would be 25 over 16. now because the question is given in mixed numbers I'm going to write this top every fraction as a mixed number so the line means divide by so we've got 25 divided by 16. so there's 116 and 25 and the remainder is 9 so we've got 1 and 9 16. and that's it okay let's have a look at our next topic so our next topic is equivalent fractions decimals and percentages it's very important to know that if you had a half that's the same as 0.5 which is the same as 50 or a quarter is the same as 0.25 which is the same as 25 and so on so it's very important to know these key fractions decimals and percentages so for instance knowing that seven tenths is the same as 0.7 or 7 temps which is 70 so it's important to know these key equivalences here and the this is the chord Mars revision card and fractions decimals and percentages but it's also important to know how to change a fraction to a decimal or a fraction to a percentage decimal to a fraction a decimal to percentage and a percentage to a fraction the decimal and it's important to know how to change between these and if you watch video 121 all the way up to video 129 that will show you in detail how to change between each of them one way which I tend to do this I just learned to change a fraction to a decimal I did the numerator divided by the denominator that gives me 0.5 1 divided by two is not 0.5 the change from a decimal to a percentage at times about 100 so 0.5 times 100 is 50 so then I write 50 and then if I want to go the other way if I wanted to write a percentage as a decimal I just divide by 100 and then to write a percentage as a fraction I just because it's a percentage I know it's out of 100 so I would write 50 over 100 and I would cancel it down to one half that's it okay so let's have a look at some questions so here's a table and we've been asked to fill in the missing numbers so one-half well a half is the same as 0.5 and if you do one divided by two you get 0.5 as a percentage well 0.5 Times by 100 is 50 so it's going to be 50 percent here we've got 0.25 and 25 well I know straight away off my heart that's equal to a quarter but if it didn't I would write 25 over 100 because that's a percentage and then I would cancel that and simplify it so I could divide both of these by 25 so 25 divided by 25 is 1 and 100 but divided by 25 is 4 so that would be one quarter so that's one quarter here we've got a fifth so we've got one fifth and that's twenty percent so let's divide our twenty percent by a hundred together as a decimal so twenty divided by 100 is 0.2 so that would be 0.2 so one-fifth is not 0.2 which is twenty percent and finally we've got 0.17 well let's write that as a percentage to begin with so let's multiply this by one hundred so no point one seven multiply by one hundred so that would move the one two columns to the left which would be in the tens the seven would move two columns to the left so it would move into the units at once it would be 17 so that would be 17 and as a fraction that's 17 over 100 and I don't think that can be canceled down no so that's it so 17 over 100 and that's it okay so our next topic is expressing as a percentage now we've looked at expressing as a fraction so let's look at expressing as a percentage and that's video 237 on corporate maths so whenever I'm expressing someone as a percentage I tend to express it as a fraction first of all and then I change it into a percentage so the question says in a box there's 20 counters nine of the countries are blue what percentage of the counters are blue so I'm going to express that as a fraction to begin with I know that nine of the countries are blue so I know there's 20 countries in the box and nine of them are blue so that means that 920 of Sir blue now if I'm doing this on a non-calculator paper I want to get this to a percentage which means that I want to write this as a fraction with 100 on the denominator to get from 20 to 100 we multiply by 5 so 20 times 5 is 100 and then look at my numerator which is 9 and I times that by 5 as well and 9 times 5 is 45 so that means that if I know that nine twentieths of the countries are blue that would be the same as 45 out of 100 being blue which is 45 percent so that's how we do a value calculator but if I was doing this on a calculator what I would tend to do is I would write it as a fraction to begin with which is 9 over 20 like so and then I want to change that into a percentage so I would just do 9 divided by 20 on my calculator and 9 divided by 20 on my calculator is equal to 0.45 so I would just do the Division 9 divided by 20 and that gives us a decimal 0.45 and then I times about 100 so I Times by 100 and that would give me 45 okay our next topic is percentage of amounts and this is a very important topic and quite often we'll be asked to do either with or without a calculator I'm going to do it without a calculator to begin with and then I'm going to show you how to do it on a calculator so this is video is 234 and 235 in corporate Maps now whenever I'm working out percentage of amounts without a calculator I tend to remember these four building blocks you've got fifty percent to find fifty percent of something you divide by two to find 25 of something you divide by 4 or you half it and half it again and that will give you 25 so that's divided by four half and half and again then we've got to find 10 you divide by 10 and the found one percent you divide by 100 and with these four building blocks we should be able to work out these questions quite nicely so first question says work out 25 of 60. so to find 25 of something you divide by four so we're going to take our 60 and we're going to divide it by four so 60 divided by 4 well divided by 4 you could divide before using our short division or you could divide it by 2 and divide it by two or half and half it again so 60 half is 30 and half again is 15. so answer would be 15 centimeters okay next question our next question says find sixty percent of 800. so what I'm going to do is I'm going to find fifty percent I'm going to find 10 and then I'm going to add them together so if we take our 800 and we take 800 and we divide it by 2 we will find fifty percent so 800 divided by 2 is equal to 400. so that's equal to 400 so that's fifty percent now we want to find our ten percent so to find 10 we divide by 10 so we take our 800 and we divided by 10 and 800 divided by 10 is 80 and that's ten percent now we want to find what 60 is so we would add these two together so if we add our 50 and 10 we will find 60 now 400 plus 80 is 480. so 60 of 800 is equal to 480. okay and our last question our last question says find five percent of 90. now to find five percent I tend to want to find ten percent and then half it so let's find ten percent of 90 to begin with so to find ten percent we divide by 10 so we're going to do 90 divided by 10. and 90 divided by 10 is 9 so that's ten percent so we find 10 is equal to 9 but we want five percent which is half of ten percent so if we do 9 divided by two that's equal to 4.5 so answer would be 4.5 so we never found a percentage of amounts with a calculator there's two common approaches one of them is to find one percent and then multiply by the percent you want so for instance if we were asked to find 19 of 240 what I would do is I would divide 240 by 100 defined one percent so I would do 240 divided by 100. to get one percent and because it's a calculator you can type in 240 divided by 100 and you get 2.4 so 2.4 and then if we want to Define 19 of 240 I would just take our one percent which is 2.4 and I would multiply it by the percent we want which is 19. so then you do 2.4 multiplied by 19 and you'll find an answer of 45.6 and that's it so 19 of 240 is 45.6 so to find the percentage of an amount on a calculator I tend to divide by 100 dividend one percent and then multiply by the percent you want and there is another approach and that's by changing the percentage that you want to find of the number into a decimal so changing 19 to a decimal that's 0.19 this is called a multiplier and you multiply the 0.19 by the 240 and that would give you 45.6 as well so if you want to find a percentage of an amount on a calculator you can divide by 100 to find one percent and times where the percent you want or you can change the percentage of the amount you want to find so 19 into decimal and then you can multiply that decimal that multiplier by the number you're finding the percentage off so you could do 0.19 times for 240. okay let's have a look now finding a percentage of an amount where the percentage is greater than 100 so we're going to find 180 of 300 pounds there's lots of different ways to do this but I'm going to show you one way to begin with and then I'll talk about a few other approaches you could use at the and so we're going to find 180 of 300 pound so what I'm going to do is I'm going to find 80 of 300 pound and then add it on to 100 of 300 pound so let's find 80 of 300 pound to begin with so let's find 80 so to find the 80 what I'm going to do is I'm going to find 10 and then times it by ear and that will give me 80 so let's find ten percent off 300 pound so 10 of 300 pounds if I divide 300 pounds by 10 I will get 10 which is equal to 30 pound so 30 pound is 10 or 300 pound now we want 80 so I'm going to times that by eight to get eighty percent so I'm going to find eighty percent off 300 pound so 80 of 300 pound or if I times that by eight well eight times three is twenty four so eight times thirty would be 240 pounds so we've now found eighty percent of three hundred pound which is 240 pound now divide 180 I just need to add that on to 100 so 100 of 300 pounds with obviously you'd just be 300 pound so if I take our 300 pound or 100 and I add the 80 which is 240 pound that will tell me 180 percent of 300 pound so 300 plus 240 would be equal to 540 pounds so 180 or 300 pounds would be 545 so that's one way to do it there's over approaches that we could use we could take our 300 pound and we could divide it by a hundred to get one percent which would be three pound and then we could just Times by 180 and that would tell us 180 which is equal to 540 pound okay and that's it okay let's have a look now increasing and decreasing by percentage and we're going to do these questions using a non-calculator approach if I was increasing or decreasing by a percentage using a calculator I would often use multipliers and that's mentioned later on in the video but in this question I'm going to show you how to increase amounts by a percentage and decrease amounts by a percentage without using a calculator and that's video 238 and corporate Maps so let's increase 75 centimeters by 20 so if I wanted to increase 75 centimeters by 20 first of all I want to find out what 20 of 75 centimeters is so let's find 10 so 10 percent off 75 centimeters so to find 10 of 75 centimeters we're going to divide by 10. so 75 divided by 10 would be 7.5 centimeters now 20 will be double that so to find 20 of 75 centimeters we'll just double that so double 7.5 would be equal to 15. so that would be 15 centimeters so that means that 20 of 75 centimeters would be 15 centimeters but we've been asked to increase 75 centimeters by 20 so what that means is we need to add that 20 on that 15 centimeters onto what we started with so we're going to take our 75 centimeters and we're going to add 15 centimeters and that's equal to 90 centimeters so if we were to increase 75 centimeters by 20 our answer would be 90 centimeters okay our next question says to decrease 9000 pounds by three percent so if I wanted to decrease 9 000 pound by three percent I'm gonna find one percent and then I'm going to times it by three to find three percent and then because we've been asked to decrease 9000 pounds by three percent whatever that three percent is I'll take it away from nine thousand to decrease nine thousand by three percent so let's find one percent to begin with so one percent off nine thousand pounds so to find one percent of an amount we would divide by a hundred so nine thousand divided by a hundred is equal to ninety so one percent is equal to 90 pound but we don't want one percent we want three percent so we could add three lots of one percent so we could add three lots of ninety and three lots of ninety would be 270 pound or we could just take up one percent and we could just times it by three and that would be equal to three percent so ninety times three would be equal to two hundred and seventy pound so three percent of nine thousand pound is two hundred and seventy pound but in this question we weren't asked what three percent of nine thousand pound is we were asked to decrease nine thousand pound by three percent so that means we need to take that 270 pound away from nine thousand pound and that will tell us our answer so if we take our nine thousand pound and we take away two hundred and seventy pound that will decrease nine thousand Prime by three percent so in nine thousand take away two hundred is eight thousand eight hundred and take away another seventy pound will be equal to eight thousand seven hundred and thirty pound and that's our answer eight thousand seven hundred and thirty pound and that's it okay so our next topic is percentage change and this is video 233 and corporate Maps so percentage changes calculated by change divided by original times a hundred sometimes in a question you might be asked to find the percentage increase percentage decrease percentage profit or percentage loss and they'll all be calculated the same way by using this formula so this is a typical question that says the height of a plant in increases from 30 centimeters to 46 centimeters work out the percentage increase so the formulas change so as you can see the change is 16 centimeters so it changes 16 divided by the original which would be 30 and then times 100. so if we do 16 divided by 30 times 100 we get we get that's equal to 53.333 and so on percent so the percentage increase would be 53.333 so one percent or we could round this sometimes in the question you ask to give your answer to one or two decimal places so let's round this to two decimal places so answer would be 53.33 and that's it so percentage change is changed divided by original times 100. okay our next topic okay let's have a look at our next topic so our next topic is simple interest so simple interest is where interest is added on to money in a bank account and simple interest is where the cmite is given every single year so the question says 600 Pounders invested for three years at five percent per year simple interest work out the total interest so in this question we're told that it's five percent so let's work out five percent of six hundred so let's take our 600 and divide it by 100 so I'll tell us one percent which is six and then take our six and multiply it by the percent we want which is five six times five is equal to 30. so every single year because it's simple interest every single year 30 pound of interest is earned so in the first year 30 pounds added on in the second year Freddie points out at all and so on and we've been asked to work at the tool to lamented interest so the money's been invested for three years so that means that three lots of 30 has been earned in interest so 30 multiplied by three is equal to 90. so the question asks us to work at the total interest earned that would be 90 pound and that's it if we were asked how much money would be in the bank account at the end of the three years that 90 would be added on to the 600 so that would be 690 pound but the question just asked us for the tool interest so that would be 90 pound right so our next topic is compound interest that's video 236 in corporate Maps so compound interest is where the interest is added at the end of every single year so our question says GM's invested eight thousand pound in the bank for two years and it earns compound interest at a rate of five percent per year calculate the total amount of money that James has in the bank at the end of two years there's two different ways to do this question and I'm going to show you both approaches now the first approach is to just treat it like a percentages question so we're going to increase it by five percent to see how much James has at the end of the first year and then to work out another five percent of that and add that on to see what he has at the end of two years so at the beginning he has eight thousand pound and let's work out five percent of that so let's divide that by a hundred and Times by five so when we divide by 100 that we get that's equal to 80 and if we do 80 multiplied by five because it's five percent that gives us 400 pound so five percent of eight thousand pound is 400 pound now if we add that on that's how much money James has at the end of the first year so the end of the first year James would have eight thousand plus four hundred that's eight thousand four hundred pounds now to see how much money James has at the end of the second year we now need to work out five percent of this and Aaron to find out what he has at the end of the second year so if we take our eight thousand four hundred pounds because that's how much he has divide that by 100 we get that's equal to 84 pound and then if we take our 84 pound and we times that by five we get five percent so 84 multiplied by 5 is equal to 420 pound so in a second year James earns more interest he earns 420 pound and if you add that onto the 8000 400 we'll see how much money GM says in the bank account at the end of two years so 8400 plus 420 is equal to eight thousand eight hundred and twenty pound so the end of two years GM size 820 pound in the bank so that's one approach that's quite useful approach for a small number of years maybe two years but as we go into three years four years five years it can get quite time consuming to do it that way so there is a quicker way to do that and to do that I'm going to show you another topic called multipliers and then come back to combat interest so we're going to look at multipliers which is video 239 in corporate Maps now increasing by a percentage and decreasing by percentage now we can use a thing called a multiplier to do that really quickly and easily so if we were asked to increase 500 by 8 we can use a multiplier now if we start off with 100 and we increase by eight percent we'll have 108 and 108 is the same as 1.08 as a decimal number so that would be our multiplier so if we increase by eight percent we'll have 108 and that's 1.08 so if we take our number 500 and we multiply by 1.08 that will tell us our answer straight away it will increase 500 by 8 so that would be if we do 500 multiplied by 1.08 that gives us 540. so if we increase 500 by 8 the answer would be 540 so we can use this multiplier really quickly and easily to do it so next one if we wanted to increase 3500 by 13 well then multiply well if we had 100 and we increased by 13 that's 113 so multiply is 1.13 that's an increase of 13 so if we take our 3500 and multiply it by 1.13 we get our answer so three thousand five hundred multiplied by 1.13 is equal to 3955. so these multipliers are really great for increasing by percentage really quickly and easily and that'll be useful for compound interest likewise we can decrease using a multiplier if we decrease something by two percent we're left with 98 because we would have started with a hundred percent and decreasing by two percent leaves us with 98 so it's a multiplier 98 percent 98 is the same as 0.98 converting the percentage to a decimal so we can just multiply three thousand by 0.98 and that will tell us our answer right away so three thousand multiply by 0.98 is equal to 2940. so you can use a multiplier to decrease by a percentage as well so if we wanted to decrease eight thousand by twenty four percent we would take our eight thousand and we would multiply by well we would have started off with a hundred percent we're decreasing by 24 so 100 take away 2476 so I'll multiply would be 0.76 and if we multiply those we'll get our answers straight away so eight files and multiply by 0.76 is equal to 6000 and 80 and that's it so this topic of multipliers will be really useful whenever we're looking at compound interest because if we go back to our question we had James invested eight thousand pound in the bank account for two years it earned compound interest of five percent per year so five percent increase would be 1.05 as a multiplier because it's gone from 100 to 105 so we can use this multiplier really quickly and easily to work out our answer so if we want to work out how much money GM's had in the bank account after one year we could take the eight thousand and multiply it by 1.05 if we wanted to work out how much money he had at the end of two years we could then times buy another 1.05 and that would tell us how much money has in the bank after two years or a really quick and easy way to do is to do eight thousand multiplied by because we're multiplying by 1.05 and multiplying by 1.05 we could write 1.05 squared using that index notation so we could do eight thousand multiply it by 1.05 squared and let's see what we get so eight thousand multiplied by 1.05 squared equals 8 800 120 and that's the same answer as before so this is really quick and easy way of doing compound and address questions and it's got a formula it's an initial so the amount of money or whatever started with multiplied by the multiplier to the power of time so this formula will help us work out compounded interest questions really easily so our initial is the amount of money that was invested to begin with the 8000 are multiplier because it's a five percent increase we used 1.05 so that's the multiplier and because it was two years so we use squared if it was three years we would use cubed if it was four years reduced to the power four and so on and that's it so our next topic is to reverse percentages and that's video 240 in corporate maps and remember you can watch that video tutorial video 240 Accord maps and I will go through reverse percentages in a lot of detail also beside that video 240 in court Maps you'll find the practice questions which are really great because they will have a lot of curveball questions in there which are really fantastic to do they'll have the textbook exercises and I'll have the answers as well so video 240 and also remember this reverse percentages questions in that bumper pack of question which is in the description below so here's a typical reverse percentages question and it says Rebecca is given a 35 pay rise she has now paid 13.77 per hour what was Rebecca's pay before the pay rise so to do this what we need to do is consider the fact that we have increased her Pair by 35 so that means she now has 135 because she had 100 before her eyes and then she's given a 35 increase so she's now got 135 percent and 135 is equal to 13 pounds 77. now to find what her pair was before which we want to find 100 so let's find one percent and then find 100 so if we know what 135 is we can divide by 135 we can divide by 135 and we can divide by 135. and when we do that we get one percent and one percent is equal to 0.102 pounds and then we want to find what 100 is so we'll multiply by 100 so let's multiply by 100 and let's multiply by 100 and when we do that we'll get well one percent times 100 is 100 her pay before the pay rise and whenever we multiply 0.102 pounds by 100 we get that's equal to 10.20 so Rebecca's pair before the pay raise was 10.20 now there is another way to do this I went for a topic called multipliers and to increase by 35 we use a multiplier of we multiply by 1.35 so if you want to go backwards and find what number was increased by 35 we could divide by the multiplier so if you do 1377 divided by 1.35 you'll also find 10.20 and that's it okay next topic okay let's have a look at our next topic so next topic is simplifying ratios that's video 269 in corporate maps and we're told the ratio of oranges to Apples in a box is four to six and whenever you get those two little dots you read it as the word two so four two six and what that means is for every four oranges the six apples so as you can see here for every four oranges the six apples and sometimes we can cancel our issues down because here we've got simplify four to six well both of those numbers are divisible by two so we divide both of them by two so four divided by two is two and six divided by two is three so if we had the ratio 4 four to six we could simplify it two two to three and that would be an equivalent ratio and that's it and we've simplified these as far as we can go okay let's have a look at some more ratios and simplify those and our next one is simplify 35 to 10. so both of these numbers are divisible by five so let's divide both of them by five so 35 divided by 5 that's 7 and 10 divided by 5 is 2. so answer would be if we have if we were asked to simplify 35 to 10 our answer would be 7 to 2. okay let's have a look at our next question so the next question says simplify 28 to 40. now both of these numbers are divisible by four so we can divide 28 by 4 and we can divide 40 by 4. so 28 divided by 4 is 7 and 40 divided by 4 is 10. so 28 to 40 would be the same as 7 to 10. so if you're asked to simplify 28 to 40 the answer would be 7 to 10 and we can't simplify this any further because the only thing you only whole number you can divide these by is one and that would actually help okay let's have a look at our next topic so next topic is ratio and writing ratios in the form of 1 n or n to one and that means one to a certain number or a certain number to one and in certain situations it can be very useful to do that so here we've got a question says the ratio of adults to children on a skill trip is 5 to 24. so that means that for every five adults there's 24 children so there could be five adults and 24 children or it could be 10 adults and 48 children it could be 15 adults and 72 children it could even be 50 adults and 240 children and so on and we've been asked to write this ratio in the form of one to n so instead of being 5 to 24 we want in the form 1 to a number so we want the left hand number and the ratio to be one so to go from five to one we're gonna have to divide by five so five divided by five is one so if we divide the left-hand side of the ratio by five we're gonna have to divide the right hand side of the ratio by five so we're gonna have to do 24 divided by 5 as well so divided by five and 24 divided by 5 is equal to 4.8 so if we were asked to write the ratio of adults to children in the form 1 to n the answer would be 1 to 4.8 and that means that for every one adult there's 4.8 children obviously there can't be E1 out of 4.8 children but what it does is it lets us know that for every one adult there's just under five children or perhaps there's even a rule it might be there for a school trip there needs to be a ratio of one to four point five so for every one teacher there's 4.5 children and with there being one to 4.8 in this trip that they're actually going past that rule so writing a ratio in the form of one to a certain number or a certain number to one can be very useful and if in this question instead I've been asked to write the ratio in the form of one to n we were asked to write it in the form n to one what we would want to do is make the number on the right hand side of the ratio one so we'd divide both of these numbers by 24 and that would give us 0. something to one okay let's have a look at our next topic okay let's have a look now at fourman ratios so we're told there's red and purple countries in the bag and there's five times as many purple counters as red counters in the back write down the ratio of red counters to purple counters in the bag so we don't know the exact number of countries in the bag but we do know this five times as many purple counters as red counters so for instance there could be five purple counters on one red counter there could be 10 purple counters and two red counters there could even be 50 purple counters and 10 red counters so we don't know the exact numbers but we do know this five times as many purple counters as red countries in the back and we've been asked to write down the ratio of red counters to purple counters in the back so for instance if there was one red counter there would be five purple counters so that means the ratio of right counters the purple countries in the bag would be one to five for every one reg Hunter there's five purple counters and that's it now we could have chosen other numbers here we could have said for instance there were two red and ten purple and then we could simplify the ratio and when we divide both of these numbers by two we would get one to five so the ratio of red counters to purple counters in the bag is one to five so our next topic is ratios and fractions and this video 269 and Cobra maths and our question says a box contains white beads and gray beads and the ratio of white beads the gray beads is two to three so we don't actually know how many white beads and gray beads there are but what we do know is that for every two white beads there will always be three gray beads like so and we're asked what fraction if the beads are great well if we have a look at this diagram this little sketches will help us so if we work out what fraction of this is great that would be the answer to how many fractions of the beads in the boxer grow so we've got all together five beads here one two three four five so we'll put five on the denominator and for the numerator what fraction are gray well there's three that are gray so that means that three-fifths of these five beads are gray so means if three-fifths of the beads will be gray and that's it so whenever you ask to write a ratio as a fraction a diagram a little sketch like this can be very useful and our answer here would be three-fifths and instead of being asked what fraction the beads are great we could have been asked what percentage of the beads would grow so we would then read as a fraction as we've just done three-fifths and then we'll just change the Three-Fifths to a percentage so then that would be sixty percent because three-fifths is sixty percent if we're asked the question what percentage of the beads were white well all together there was five beads and two of the more weights we'd write that as a fraction two out of five two-fifths of the beads are white and then let's change that to a percentage that would be well two fifths is forty percent because one fifth is 20 so two-fifths is forty percent so that's would be forty percent so if you ask to change the ratio to a percentage I would often change it to a fraction to begin with and then change that fraction into a percentage now whenever we're dealing with ratio questions we may encounter questions to look like this almost says there's 91 beads in the Box explain why he must be incorrect so it's very useful to consider what the ratio means so we've got the ratio of white beads to gray beads in the boxes two to three so that means in the Box it could be we could have white beads and gray beads white and gray there could be two because the ratio is two to three there could be two white and three gray and all together the total number of beads would be five there now there may not be five beads in the box or maybe more so there could be another two white and another three great so another two white would be four another three gray would be six so that would be ten beads all together now there may not be ten so there may be more beads so maybe if we add another two white and another three gray that would be six and nine and six plus nine is equal to Fifteen and as you notice the total number of beads is a multiple of five because if we add the two and three together every time we're adding another five beads so Omar says there's 91 beads in the Box well there can't be 91 beads in the Box because if then the ratio of two to three that means it must be a multiple of five there must be a multiple of five number of beads in the box and 91 isn't a multiple of five okay let's have a look at our next topic so our next topic is sharing in a ratio in this video 270 in corporate maths so here's part of the chord miles revision card I'm with Jack and Chloe share 75 pound in the ratio two to three how much money do they each receive it's very important to be able to share a number in a ratio and a share a number and a ratio the first step is to add together the parts in the ratio because jackets two parts and Chloe gets three parts so all together two plus three would be five so there's five parts in total now what we're going to do is we're going to take the grand total of what they're sharing so in this case it's 75 pound and we're going to do 75 divided by whatever you get whenever you add the ratio together so we're going to divide it by five because two plus three is five so we know there's five parts in total so if we divide 75 by 5 we'll see how much money is in one part so 75 divided by 5 is 15. so that means it's 15 pound in one part now we know the Jack gets two parts and Chloe gets three parts so that means that Jack has two lots of 15 and Chloe gets three lots of 15. so for Jack we're going to do 15 multiplied by his number in the ratio which is two so 15 times 2 is 30 pound and for Chloe she gets three parts so we'll do the 15 pound multiply by three her number in the ratio so for Chloe we'll do 15 multiply by three and that's equal to 45. and what's great is if we add 30 pounds and 45 pounds we get the 75 pound that we started with so if you're asked to share something on a ratio the steps would be first of all add the parts together in the ratio then divide the grand total by the total number of parts and then times each of the numbers by how much is in one part and that's it okay let's have a look at our next topic which is whenever we're dealing with ratio questions whenever we're given one quantity and that's video 271 on corporate Maps so here we've got the ratio of lemon sweets to Strawberry sweets in a tub is five to three so in this tub for every five lemon sweets there's three strawberry sweets and we're told there's 120 lemon sweets in the tub so rather than the last time when we're told how many sweets were all together what we're told is how many lemon sweets there are and we're asked how many strawberry sweets will there be in the tub so in a question like this whenever you give them one quantity we know there's 120 lemon sweets now we're told the ratio of lemon to Strawberry is five to three so the lemon number in the ratio is five so if we divide 120 by five we'll find out how many sweets there are in one part so 120 divided by 5 is equal to 24. Sammy says 24 in one part and we find that by dividing how many lemon sweets there were by the lemon number and the ratio of the five now we're asked how many strawberry sweets are in the tub now strawberry is three parts so what we'll do is we'll multiply 24 by 3. so we'll do 24 multiply by three and that's equal to 72. so whenever you give them one quantity so in this case we were told there was 120 lemon sweets what you do is you divide that by the lemon part in the ratio which was the five and whenever you do 120 divided by five that tells you how many sweets are in one part and then you can times divide the other number in the ratio to see how many strawberry sweets there were in this case that's it and if you were asked how many sweets to wear in total you could add together the 120 and the 72 and I'll tell you how many sweets were all together so that's it okay so we're now going to look at what happens whenever we're given two ratios and this is video 271 a and corporate Maps so here we're told that the farmer keeps sheep cows and chickens on a farm and the ratio of sheep to cows so sheep the cows on the farm is four to three so that means that for every four sheep there's three cows and the ratio of Coast the chickens is ten to seven so that means that for every 10 cars the seven chickens we've been asked to find the ratio of sheep to cows to chickens so whenever I'm given a question like this I like to do a little table so I like to write down what we've been given so we've got shape so we've got sheep cows and chickens now it could have been in the question it may not have told us what we're looking at could have just been the ratio of a to B and B to C and then the headings of my table would be a b and c but in this case we were told what we're looking at sheep cows and chickens so the ratio of sheep to cows is four to three so sheep to cause is four to three so I'm going to write that ratio down four to three and then we're told the ratio of cars to check-ins is ten to seven so close to chickens and I'm just going to write that beneath is ten to seven so if we have a look we've got the ratio of sheep to cows is four to three and the ratio of cows to chickens is ten to seven so we've been asked to write this as one ratio of sheep to Coast to chickens so the key thing is going to be the cows because if we have like the cows are on both ratios for every four sheep there's three cows and for every ten cars there's seven chickens so if we can find a common multiple of three and ten that will help us combine these as one single ratio I'm going to find the lowest common multiple of three and ten with the lowest common multiple of three and ten would be 30 because if you write down the multiples of three three six nine twelve and so on and if you write down the multiples of 10 10 20 30 30 would be the first number before those lists so I'm going to write 30 beneath the 3 and the 10 because that's the lowest common multiple of 3 and 10. now what I'm going to do is I'm going to have a look at this top ratio and I want to instead of being four to three I want it to be something to 30. so to get from 3 to 30 we multiply by 10. so if we multiply the 4 by 10 as well then we will know what number this is in the ratio so if we Times by 10 and Times by 10 that would give us 4 times 10 is equal to 40. so 4 to 3 is the same as the ratio of 40 to 30. now if we have a look at our next ratio the the car is the chickens ratio which was ten to seven well we instead of having 10 to 7 we want to write 30 to something so if we multiply 10 by 3 we get 30. so if we multiply 7 by 3 we'll find this number in the ratio so 7 times 3 is equal to 21. so we've got that the ratio of course the chickens would be 30 to 21 and that's an equivalent ratio as 10 to 7. so now we've got one ratio we've got the ratio of sheep to college to chickens is 40 to 30 to 21. okay now let's have a look around ratios as equations and equations as ratios and that's video 271d on corporate Maps so we've been given the ratio X to Y is equal to one to four and we've been asked to write this as an equation Link in X and Y so here we've got the ratio and we know that the ratio of x to Y is one to four and that means that Y is four times larger than x because for every one of X is four of Y so it could be that X is equal to 1 and Y is equal to four it could be that X is equal to 2 and Y is equal to eight it could be x equal to 10 and Y is equal to 40 and so on but what we know is the value for y is four times larger than the value for x so let's write that as an equation Y is equal to 4 x because 4 times the smaller number x 4 times x is equal to Y because Y is four times larger and that's it and we can show this just look into this diagram here we've got Y and we know the ratio of x to Y is one to four so in other words if we've got y y is four times larger than x if we multiply X by 4 we would get Y is equal to 4X and that's it okay let's have a look at our next question so next question says given the ratio of x to Y is equal to two to three write an equation Link in X and Y so here we've got X and Y and X and Y are two numbers and when we simplify the ratio we get two to three now it could be that X is equal to 2 and Y is equal to three it could be that X is equal to 4 and Y is equal to 6. it could be the x equal to 20 and Y is equal to 30. but what we know is that when we simplify that ratio we go two to three and that means for every two of X is three of Y so Y is going to be bigger than x and I've represented it here as a diagram where I've got that my X is equal to two centimeters or more Y is equal to three centimeters so that's a little diagram here where x equals two centimeters and Y is equal to three centimeters so it represents this ratio now if we were to Double Y and we were to treble X it would be the same as each other if we have a look here if we had two y's and we had three X's they would be the same length and that's it that means we've got an equation that 2y is equal to 3x so that's our equation Link in X and Y now that is an equation Link in X and Y but if I had something like this I would tend to make y the subject or X is subject but if I was making y the subject I would divide both sides by two so divide by 2 and divide by two and my left hand side of the equation would just become y because 2y divided by 2 is just Y and if we have 3x divided by 2 we could write 3x over 2 like so so y equals 3x over 2. I would tend to write it like this as 3 over 2 x so y equals 3 over 2 x okay let's have a look at our next question so we're going to write an equation as a ratio so if we've got the equation y equals 2x and we've been asked to write down the equation of x to Y let's have a look at the equation to begin with we know that Y is equal to 2 times x that means that Y is two times larger than x for instance if x is equal to 10 y would be equal to 20. so let's write that as a ratio so for instance if x was equal to 10 then y would be equal to 20 and then you can just divide those by 10 and you get one to two we could have just written one to two to begin with for instance we know that Y is twice as big as X so if we just say x equal to 1 Y is equal to 2 and that's it so our ratio of x to Y would be one to two and that's video 271 D and code Maps Okay let's have a look at our next topic so our next topic is called the unitary method which is video 255a and corporate maps and we're told the four candles cost 13 pounds find the cost of seven candles now this is called the unitary method because we want to find the value for one so in this case we want to find the cost of one candle and then when we know the cost of one candle we can multiply by seven to find the cost of seven candles so if we know that four candles cost 13 pounds if we just Divide 13 by four we can find the cost of one candle so 13 divided by four is equal to 3.25 so that means that one candle costs 3.25 so that's one candle now we know the cost of one candle we can find the cost of seven candles really easily by just multiplying the cost of one candle by seven so if we do three Pi 25 multiply by seven that gives us 22 pied seventy five p and that's it okay let's have a look at our next topic which is exchange rates or currency and this video 214 and corporate maps and we're told Nicola went to Italy and she changed 800 pound into euros and the exchange rate was one pound is one Euro 40. so in other words for every pound the bank give her one Euro 40. and the question says to change 800 pound into Euros so to change from pounds into Euros or Pines into whatever currency you're converting into if you've got what one pound is you just need to multiply how many pounds there are the 800 by the number in the exchange rate so here we've got one part is one Euro 40. so if we do 800 multiplied by 1.4 or 1.40 well we don't really need the zero there so if we do 800 multiplied by 1.4 we'll see how many years she gets because for every punch she gets one Euro 40. so 800 multiplied by 1.4 equals one thousand to 120 euros and that's it now else is there Nicholas Cena watches she likes and it costs 105 euros and she wants to know how much that is in pounds well if she wants to convert back from euros into pounds we need to divide the 105 by the number and the exchange rate the 1.4 so if we do 105 divided by 1.4 that would tell us how many pounds it would be so if we do 105 divided by 1.4 that is 75 so the watch costs 75 pounds and that's it okay so our next topic is recipes and that's video 256 in corporate Maps so here we've been given a recipe or a list of ingredients and we've been told it serves five people and we've got our 500 grams of cod 400 grams of haddock 600 milliliters of milk 120 grams of butter 40 grams of flour and one kilogram of potatoes and Ben would like to cook the meal for four people so he would like instead of cooking it for five people he would like to cook it for four people and we've been asked how much of each ingredient should he use or how much should he use how much does he need of each ingredient so obviously he's not syrup cooking for five people he's cooking for four people if it was me I would just cooking for five people and just eat the extra bit but he won Hamilton how much of each ingredient should he use if he's been cooking the meal for four people so here we've got the list of ingredients and we've got how much of each of them now if it was for 10 people he wanted to cook before it'd be really easy we could just double all these measurements if it was for 15 people we could just time some more by three now we want to find for four people what I'm actually gonna do is I'm gonna find how much he would need for one person by dividing all these by five and that would tell us how much of each ingredient he would need for one person and then I'm going to times it by four and that'll tell us how much of each ingredient he would need for four people so let's write for one person and to find how much he needs for one person we're going to need to divide all of these by five so 500 divided by 5 would be a hundred grams of cod 400 divided by five well that's going to be 80 grams of haddock 600 milliliters of milk but we want to divide that by five so that's going to be 120 milliliters of milk 120 grams of butter if we divide that by five that would be 24 grams of butter 40 grams of flour if we divide that by five that'll be eight grams of flour and finally one kilogram of potatoes which is 1 000 grams if we divide that by five that'll be 200 grams of potatoes now we could have written it as 0.2 kilograms I've just changed it into grams because I just think it's a bit easier to to consider Okay so we've now got how much he needs for one person if he was cooking this meal for one person how much you would need but we've been asked to cook the meal for four people so we're going to multiply all these measurements by four so four people so we've divided by five now we're going to Times by four so 100 grams Times by four would be 400 grams of cod 80 grams of Carrick whenever we times that by four that'll be 320 grams of haddock 120 milliliters of milk when we times that by four that'll be 480 milliliters of milk 24 grams of butter when we times that by four that'll be 96 grams of butter 8 grams of flour when we times that by four that'll be 32 grams of flour and finally 200 grams of potatoes when we times that by four that would be 800 grams of potatoes and that's it so it can be useful in recipe questions to be able to just double it or half it but sometimes what we need to do is we need to find how much of each ingredient we need for one person and then to times about how many people we want to serve the meal for okay let's have a look at our next topic okay so our next topic is proportion that's video is 254 and 255 in corporate Maps so sometimes in proportion questions we're given a context and we're given a formula and here's an example the number of months M to complete a piece of research is found by m the number of months is equal to 2400 divided by n where n is the number of scientists we've been asked to work out how long the research to take if 30 scientists are working on it so we've got this formula and what this is telling us the amount of time it takes to complete the research is found by doing 2400 divided by n where ends the number of scientists now in this case the more scientists that there are the less time I would take to complete the research so this is inverse proportion and whenever we're dealing with an inverse proportion what we know is that as one number gets bigger the other number gets smaller so the more scientists there are whenever we divide 2400 by a bigger number what we'll find is that the number of months taken to complete the research will get smaller now in this question we've been asked to work out how long the research should take if 30 scientists are working on it so we just need in this question substitute 30 into the formula to work out m equals 2400 divided by 30 and if we do 2400 divided by 30 that's equal to 80 and we're dealing with moments here so it's 80 months so if there was 30 scientists working on this piece of research it would take early months for them to complete this research so sometimes the proportion questions you're given a context and you're given a formula and you just need to maybe substituting values into it but sometimes with direct proportion and inverse proportion we need to be able to recognize how it's represented as an equation or as a graph so let's start off with direct proportion so here we've got wires directly proportional to X so what that means is as the value for X increases the value for y increases and we represent that using this equation Y is equal to KX and that means Y is equal to k a certain number multiplied by X so for instance we could have y is equal to 2X or Y is equal to 10x or even Y is equal to 0.4 X so these equations could represent relationships that are directly proportional because as the value for X increases the value for y increases for instance if we multiply 2 by a larger number we're going to get a larger answer so it's important to recognize that y equals KX represents Y is directly proportional to X and as a graph if we represent it as a graph we would look something like this we've got our x-axis and y axis and we've got a line that's going up passing through the origin so as the value for X increases the value for y increases and that's it so if we wanted to represent y it's directly proportional to X as an equation it'll be y equals KX and as a graph it would look something like this so now let's have a look at inverse proportion so if something's inversely proportional to something else that means as one gets bigger the other one gets smaller and we can write that as an equation as something like this we've got Y is equal to k a certain number divided by X so as we divide by bigger numbers we get a smaller answer so if we had something like this where Y is equal to 20 divided by X if we divide that by 1 we get 20 divided by 1 which is 20 but then if we divided by 2 if we do 20 divided by 2 we get 10. so as the value for X increases the value for y decreases so why is inversely proportional to X is represented as y equals K Over X and and there's a graph it would look something like this where we've got our x and y axis and we have a curve that comes down like so and as we divide by bigger numbers we get a smaller and smaller answer and that's it okay now let's have a look at this question so we're told the Y is directly proportional to X so Y is equal to K times x or y equals KX and we're told that whenever Y is equal to 30 x is equal to 6 find y whenever X is equal to 11. so because we know Y is directly proportional to X we can write down that equation y equals K times x remember K is just a number and we're told two bits of information we're told that whenever Y is equal to 30 x equal to 6 so we've got Y is equal to K times x that'll be 30 is equal to K times 6. so if we have a look at this we've got 30 is equal to K times 6. so that means the K would have to be equal to 5 K is equal to five dividing both sides this equation by 6. so that means that our equation is y equals 5 times x or y equals 5X and then we've been asked to find what Y is whenever X is equal to 11 so we can then just say Y is equal to 5 times 11 and 5 times 11 is equal to 55. so y equals 55. and that's it okay let's have a look at another question which involves inverse proportion so we've got W is inversely proportional to X let's write that down W equals k a certain number divided by X and we're told that whenever W is equal to 10 so when w is equal to 10 x is equal to 4. so 10 equals K divided by 4. now let's find what K is by multiplying both sides of this equation by four so to multiply by 4 and multiply by 4 10 times 4 is equal to 40 on our right hand side K divided by 4 times by 4 would just be care so this number K that certain number is 40. so w is equal to 40 over X now we want to find W whenever X is equal to five so we just put X is equal to 5 in here so we get W equal to 40 divided by 5 and 40 divided by 5 is equal to 8 and that's it now sometimes whenever we're dealing with proportion we need to have questions that involve time and this is proportion dealing with time and this is video 256a and corporate Maps so here we've got our question it says it takes 20 hours for free pumps to fill a swimming pool how long would it take if four pumps were used so in a question like this we're told that it takes 20 hours for free pumps to fill a swimming pool so what I'm going to do is I'm going to consider how long it would take for one pump to fill a swimming pool well if there's three pumps one pump would take three times longer so if there's 20 hours if we multiply that by three that would tell us how long it would take one pump to fill the swimming pool so 20 multiplied by three would be equal to 60. so if there was one pump it would take 60 hours if there was two pumps it would take half of that time it would take 30 hours if there's three pumps you would divide it by three to get 20 hours and so on and the question says how long would it take of those four pumps used so if there's four pumps being used well if it takes one pump 60 hours four pumps would take four times less time we've divided by four so we're going to take 60 and divided by 4 and 60 divided by 4 is equal to 15 hours so if there was four pumps used instead of taking 20 hours for free pumps it would take for four pumps 15 hours that's it okay and that's video 256 a and corporate Maps Okay next topic is money now in with money there's lots of different possible questions you could be asked on money and if you're good to video 400 in corporate Maps you you find this video 400a 400b 400 C and so on and they've got lots of different types of money questions I'm going to show you one particular type of question now so it says a ruler cost 70p Adrian has 10 pound Adrian buys as many rulers as he can how much change does he receive now first of all we're told that each ruler costs 70p and he's got 10 pound now I'm going to change the 10 pound into Pence so each pound is 100 Pence so 10 pound would be one files and Pence so he's got 1 000 Pence and he's going to buy as many rulers as he can so let's get our calculator and we're going to divide the amount of money he has the 1000 by 70. and that'll tell us how many times 70 goes into 1000 and when I do that I get 14.285714 and so on so that means it's 70 goes into 1014 and a bit times now if he goes into the shop Adrian can't buy 14 in a bit rulers he can buy 14 orders he doesn't have enough money for his 15th thriller so we know that he buys 14 rulers now we know each ruler costs 70 Pence so if we do 14 times 70 we will find the total amount of money it costs for the 14 rulers so 70p multiplied by 14 rulers and when we do that we get that's equal to 980 Pence so nine pound deity but Adrian had 10 pounder at 1 000 Pence so if we take our 1000 Pence or our 10 pi and we take away the 980 pen so we see that this 20p left over that's how much change you're doing would receive so Adrian receives 20p change so here we've got a finance question so here's a typical question it says the value of a car decreases by five percent each year and so if we bought a car two years ago for ten thousand pounds work out the value now so we can use this formula initial multiplied by the multiplier to the power of time so initially the car cost ten thousand pound and then the multiplier well it's decreasing by five percent each year so if we have a hundred percent and we took away five percent we'll be left with ninety five percent so that's 0.95 and we're wanting to find the value of the car after two years so that's going to be to the power of two so we do ten thousand multiplied by 0.95 squared and that gives us nine thousand and twenty five pounds so the value of the car after two years would be nine thousand and twenty five pound and that's it okay so let's have a look at our next topic so our next topic is Best Buys which is video 210 and corporate Maps so here we've got a question a Best Buy's question that says packets of biscuits are sold in two sizes you've got a regular box of biscuits which has 10 biscuits for 95p or you've got a large box of biscuits and we've got a large box of biscuits which is 15 biscuits and it costs 1.50 and the question asks which packet of biscuits is better value for money now there's two ways which I typically answer questions like this so using the first approach we could divide the total cost the 95p by the 10 biscuits to find the cost per biscuit so I could do 95p divided by 10 and that would tell me how much it cost per biscuit in the regular box so 95 divided by 10 is 9.5 P so it's 9.5 P per biscuit here per biscuit and in the large box is 1.50 which is 150 Pence and that's for 15 biscuits so you can divide a 150 for 150 Pence by 15 biscuits and that would tell us that's equal to 10p per biscuit so 9.5 P perpet skin is cheaper than 10p per biscuit so that means that the regular box is better value for money so that's one approach to divide the total amount by how much you get another approach is to buy the same amount of Biscuits by either buying just the regular boxes or the large boxes so with the regular box you could buy 10 biscuits I could buy 20 biscuits that could buy 30 Biscuits by buying just regular packets or if the large box I could buy 15 biscuits or if I bought two packs that would be 15 plus 15 which is 30 biscuits ah so I could buy three regular boxes and that would be 30 biscuits so that would be three boxes at 95 pH so 3 times 95p which is equal to 285 P or 2.85 or alternatively if I was looking at just the large boxes to buy 30 biscuits well 30 biscuits would be two boxes of those so that would be two lots of 1.50 and two lots of 1.50 is three pound so if I wanted to buy 30 biscuits I could buy use the regular boxes which are 95 pH and that would cost me 2.85 alternatively I could use the large boxes to buy the very biscuits now cost me three pounds so as you can see it's better value for money to buy the regular boxes so the regular is better value so the next topic is use of a calculator which is very important first of all you've got your calculator that you're familiar with it you bring it to class every lesson and you're really confident with using that calculator and you could perhaps be asked to work something out on the calculator which will involve you knowing how to put in fractions or square roots and things like that so here we've got a question it says to work out 19.1 subtract 2.5 divided by the square root of 20. and we'd have to type this in our calculator so let's have a look at some of the calculator buttons which are quite useful now first of all if I wanted to type this in because you've got that line that fraction I would press the fraction button this button here whenever I press that button my display would look something like this next then I would type in the Top Line the numerator of the fraction the top line of the fraction that 19.1 subtract 2.5 so we'll just type in 19.1 subtract 2.5 then I would press the down arrow here to bring me down to the denominator and then I would want to type in this square root of 20 so I would press the square root button which is here which is next the fraction button this one so it Compares on the denominator here and then I would type in 20 and then whenever you type in 20 my calculator display should look something like this and then I would press equals now depending what mode your calculator is in you might get two different displays if you press equals you could get this display which says 83 times the square root of 5 divided by 50. if it comes up like that you want it as a decimal number you then press this button here SD this button here and whenever you press that it then would display as a decimal number for you and it would come up as 3.711872843 so you could be asked a question to work out something which involves you typing something into your calculator and if you want to recap that topic in corporate Maps it would be video 352. okay so let's have a look at our next topic so next topic is error intervals and that's video 377 on corporate Maps so we've looked at rounding and we've looked at finding the lowest number or the highest possible number whenever we've looked at data such as the population of Wells now let's have a look at what happens whenever we run numbers it can take any value at all and we often do that whenever we're looking at error intervals so here we've got the mass of a letter is 80 grams to the nearest 10 grams so because the mass of the letter it can take any value it can be whole numbers or it could even be decimals and we've been told to write down an error interval for the mass of the letter Y so the mass of the letter has been rounded to the nearest 10 grams so that means that the master letter may not be 80 grams it could have been 80 grams it could have been 79 grams it could have been 76 grams it could be anything as low as 75 grams because that's the lowest possible value that it could be because 75 grams would Round Up to early but anything below 75 grams so would be 74 point something and that would run down to be 70 grams to the nearest 10 grams so 75 grams would be the lowest possible mass of the letter now let's consider the masses above 80 grams because as I said the letter could be 80 grams but it could be something that's even higher than 80 grams as long as it runs to 80 to the nearest 10. so it could have been 81 grams it could have been 82 grams it could have been 85 4 grams it couldn't be 85 grams because 85 grams would Round Up to be 90 grams but it could be 84.7 grams it could be 84.9 grams it could be 84.99 grams and so on it could be anything that is up to but not including 85 grams so let's write this as an error interval so let's start off with the mass of the letter which is y so why and we know that it's bigger than or equal to 75 grams so it could be 75 grams or a numbers bigger than that so we're going to write bigger than or equal to 75 grams but it can go up to but not include 85 grams so we then write this less than 85 grams so this reads it why the mass of the letter is bigger than or equal to 75 grams but less than 85 grams and that's our air interval it shows all the possible values for the mass of the letter why okay let's have a look at another question okay let's have a look at our next example so Nigel runs the number X to one decimal place so there's a certain number X and the surrounded 2 one decimal place and his answer is 7.3 write down an error interval for X so the number could have been less than 7.3 it could have been 7.29 that would round to 7.3 to one decimal place it could have been 7.26 it could be anything as low as 7.25 because 7.25 when we round that to one decimal place because it's a five and above in the second decimal place when we round that to one decimal place it'll be 7.3 but it couldn't be anything lower than that so Nigel's number could be as low as 7.25 but it could have been a number bigger than 7.3 it could have been 7.3 it could have been 7.31 it could be 7.34 but it couldn't be 7.35 because then that would Round Up to be 7.4 so the naturals number could be n of and up to but not including 7.35 because it could be 7.348 and so on right up to but not including 7.35 so let's write that down X his number could be bigger than or equal to 7 0.25 but it can be up to but not including 7.35 and that's it so that's the error interval for X okay so let's have a look at our next question so next question says Chloe trunciator number W to one decimal place so Chloe's got a number actually truncated so she's not rounding it she's trunciating it and what that means is if you get a decimal number and you just chop it off so for instance if we had Pi which is 3.14159 and so on and if I rounded this to three decimal places it'll be 3.14 and then because the fourth decimal place is a 500 round up so it'll be 3.142 so that's rounding if I was to truncate this to three decimal places what that would mean is I just chop off I just ignore anything after the third decimal place so if I truncated this to three decimal places it'll be 3.141 so in this case Chloe has truncated a number W to one decimal place and her answer is 1.4 so she's got 1.4 so Chloe's number it could have been 1.4 it could have been 1.41 if we truncated that to one decimal place that'd be 1.4 it could have been 1.47259 if we truncated that to one decimal place it'll be 1.4 Chloe's number could have been 1.498972 and if we truncated that to one decimal place it'd be 1.4 but it couldn't be 1.5 so it'd be anything up to but not including 1.5 so let's write down the error interval for Chloe's number W so w it could have been 1.4 so it could be bigger than or equal to 1.4 but then it can go up to 1.5 but not be 1.5 so less than 1.5 so that would be the area integral for w that's it so our next topic is types of angle and here's the corporate Mouse revision card so we've got an acute angle and they're small angles they're between 0 and 90 degrees so this is an acute angle the next type of angle here we've got a right angle so it's a right angle if it's equal to 90 degrees so if it's 90 degree angle that's a right angle next we've got an obtuse angle obtuse angles are bigger than right angles they're bigger than 90 degrees but less than 180 degrees then you've got a straight line which is 180 degrees and angle is bigger than that so bigger than 180 and smaller than 360 Degrees would be a reflex angle so we've got an acute angle a right angle one that's equal to 90 an obtuse angle and then a reflex angle if it's bigger than a straight line and that's it okay let's have a look at our next topic so it's important that you're able to draw angles and measure angles so here we've got an angle and let's measure it so let's take our protractor and let's measure the cycle so if we put the protractor on the angle so like so we've got the cross off the protractor on the center of the angle so here with two lines of meat and then one of the lines is on zero and we're going to count around to whenever we get to the other line so as you can see we've got zero degrees 10 20 30 40 50 60 70 80 90 100 110 120 130 140 and we've got to 145 so this is a 145 degree angle 140 five degrees it's important whenever you're measuring angles that you start at zero and you come round to the other line if it was the way around you might need to look at the angles on the inside so start at zero and come around the other way okay let's do our next question our next question says to draw a 60 degree angle so we're going to draw 60 degree angle so let's put up a tractor so that it is the cross of the protractor is at the end of the line and then we've got zero on the line and we're going to count around to whenever we get to 60 degrees so as you can see the zeros on the inside this time so we're going to go around to 10 20 30 40 50 60. so 60 degrees would be here so there and then we're going to get a ruler and a pencil and you're going to draw a nice straight line starting at the point at the center here through that point there and that means that that would be a 60 degree angle so let's label it so let's put an arc in there and then write in 60 degrees and that's it we've drawn a 60 degree angle so it's very important to be able to measure and draw angles and therefore it's really important to have your protractor in every single one of your mouse lessons ready to use it okay our next topic our next topic is measuring lines sometimes we're asked to measure line segments such as this line a b and as you can see here we've got our line a b a is the beginning of the line so here and B is the end of the line here we've been asked to measure the length for the line so it's important whenever you're measuring lines you use the centimeters it's very rare the map so you'd use the inches side so we're going to get our centimeters so we're going to put our zero at the beginning of the line and we're going to see how long the line is so it's one two three four five six centimeters so this line is exactly six centimeters long so it'd be six centimeters so if I went a little bit forever it could be 6.1 or 6.2 centimeters or so on but this is exactly six centimeters so this line a b is six centimeters okay our next topic our next topic is angle facts and it's very important to know these angle facts so for instance that a right angle is 90 degrees a straight line would be 180 degrees the angle is going to triangle add up to 180 degrees the angle is on a quadrilateral out of the 360 and so on so it's a very important angle facts so here we've got a right angle and that means that these two angles will add up to 90 degrees so if we know that one of them is 70 degrees we can take that away from 90 to see what's left of the arrival so if we do 90 subtract 70 that's equal to well 90 subtract 70 is equal to 20. so that means that this angle X would be 20 degrees here we've got a straight line and this time we've got a little right angle marked in here so that means that this angle is 90 degrees so let's put in 90 there 90 degrees and we're trying to find out this angle here this x so we've got two angles and we know what's left would be this angle X so all three angles will have to add up together to give you 180 degrees so if you have the two that were given the 90 and the 55 we can see what's left so let's do 90 add 55 and that's equal to 145 degrees and if we take that away from 180 we'll see what's left for X so 180 take away 145 is equal to 35 degrees so that means that X is equal to 35 degrees so the angles on the right angle will add up to 90 degrees and the angles in a straight line would add up to 180 degrees and if you ever see a little right angle symbol right 90 beside it okay the next ones so here we've got a full circle the angles on a full circle or full term will add up to 360 degrees so here we've got one Angle an obtuse angle which is 140 degrees and we want to see what's left for this reflex angle so if we take the 140 degrees away from 360 we'll see what's left for X so do 360 take away 140 that's equal to 220 degrees so that means that this angle X is 220 degrees okay next we've got two lines across each other now when you've got two lines two straight lines across each other the opposite angles are equal to each other so that means that X here would be equal to 156 degrees and the Y would be equal to the angle opposite it these are called vertically opposite when you've got two straight lines across each other the opposite angles are always equal to each other and they're called vertically opposite so here x will equal 156 degrees because it's vertically opposite to the one we're given and then to find the Y well there's two ways we can find this one way is to just look at the straight line and say well Y and 156 is in a straight line so if we take 156 away from 180 we can find y or another way to do is to look at the full circle and say well if you've got 156 here and this is 156 you can add those two together to be 312 and you could take 312 away from the full circle which is 360 to see what's left for y and the one opposite it and when you know that amount divided by two half of it to see what Y is so I'm going to do it using just a straight line approach here that I know that Y and 150 sixes in a straight line so I'm going to do 180 minus 156 and that's equal to 24 degrees okay and if you do want extra practice in these angle facts the useful videos for you in corporate maps are 35 30 34 and 39. okay let's have a look at our next questions this time we're going to look at angles on a triangle which is video number 37 and the angles on the triangle add up to 180 degrees so here we've got a triangle and we've got two angles given to us at 75 and 80 and we've got our missing angle X that we need to find so if we add the two angles we're given the 80 and the 75 we can then take that away from 180 and see what's left so 80 plus 75 is equal to 155 degrees and then if you do 180 take away 155 that leaves us with 25 so it means that X is equal to 25 degrees next we've got an isosceles triangle with an isosceles triangles two of the angles are the same so as you can see here we've got our two lines that are the same length and the two angles here and here this one and the 35 will be equal to each other so means that this angle is 35 degrees so that means to find the angle X we'll add our 35 and 35 and then take that away from 180 so 35 Plus 35 is equal to 70 degrees and then if we do 180 take away 70 that's equal to 110 degrees so the angle X is equal here to 110 degrees and if you do want extra practice in these remember there is that practice booklet and find it in the description below our next topic is angles in a quadrilateral so here we've got a four-sided shape every single time you add on an extra side of a shape you add on 180 degrees to its angles so you know that with a triangle the angle is added to 180 if we add another side in that means the neither the angles on a quadrilateral will add up to 360 degrees so that means that if we add up the angles that we're given here the 70 the 50 and the 90 here's a right angle you can then take that answer away from 360 and see what's left for X so 70 plus 50 is equal to 120 degrees Plus 90 is equal to 210 degrees now we're going to take that away from 300 six days of 360 take away 210 is equal to 150 so that means the x is equal to 150 degrees that's it so the Angles and a quadrilateral are up to 360 degrees and that was video 33 in corporate Maps Okay next topic okay so let's have a look at our next topic and our next topic is topic 34 in corporate Mars and that's angles and polygons and a polygon is just a straight sided shape so we've got our triangles our quadrilaterals pentagons hexagons heptagons octagons and so on and this is the Cup match revision card on angles and polygons so it tells you the angles in a triangle add up to 180 degrees the angles on a quadrilateral add up to 360 Degrees the angles in a pentagon add up to 540 degrees and it's very useful to know that or alternatively know that if a quadrilateral adds up to 360 you can just add on over 180. if we add on another 180 that's 720 so the angles on a hexagon add up to 720 degrees the angles in a heptagon add up to 900 Degrees the angle is an octagon add up to 1080 degrees and so on and this is very useful this is the cool master revision card and if you want to get a set of those they're in the description below and so here we've got a polygon and it's got one two three four five angles so and it's also got one two three four five sides that means it's a pentagon and the angles on a pentagon add up to 540 degrees so if we want to find the size of this missing angle X we can add together the angles we're given the 80 the 140 the 95 and the 130 and we can take that away from 540 to see what's left for X so let's do that so 140 plus 80 plus 95 plus 130 is equal to and I'll just do that on my calculator 445 degrees and if we take that away from 540 we can see what x is so 540 take away 445 equals 95 degrees so that means the x is 95 degrees and that's it if you were given a hexagon they would give you five angles and you would add those up and take it away from 720 to find the size of the sixth form and so on now there's a formula that's really really useful and that if you want to find out that some of the interior angles or what the angles in the ship add up to you could use this formula which is n take away 2 so the number of sides take away 2 multiplied by 180. so for instance if I wanted to know what the angles added up to an octagon I could take the number of sides which is 8 take away two that gives me six and then multiply by 180 and that gives me an answer of 1080. likewise if we knew what the angles in a shape added up to and we needed to find out how many sides there were we could work backwards so we could divide by 180 and then add two that's it so this formula is really useful okay let's have a look at a question based on that so here we've got a question and it says find the sum of the interior angles of a 12 sided polygon so a 12-sided polygon that's a dodecagon I believe and we need to find the sum of the interior angles we need to find out what the angles in that 12-sided shape will add up to and the formula is to do n take away 2 times 180. so if we take our 12 sides and do 12 take away 2 that's equal to 10 and then do 10 multiplied by 180 that's equal to one thousand eight hundred so in a 12-sided polygon the angle goes without up to 1800 degrees okay next example so our next question says the sum of the interior angles of a polygon is four thousand five hundred degrees so it's got a few more sides and a 12-sided one and the question says how many sides does it have so to find this answer this 4 500 Degrees we've taken the number of sides we've subtracted two and then we've multiplied by 180 degrees so if we want to work backwards we're going to take our 4500 divided by 180 and then that will give us 25 but remember we've taken away two so because we're going backwards we're going to add two so we're going to do 25 plus 2 is equal to 27. so the angles in the 27-sided polygon will add up to 4500 degrees and that's it okay and that's it now this formula is really useful it's really useful for both regular polygons so regular polygons are where all the sides are the same length and the angles are the same size and it's also useful for irregular polygons polygons where the angles aren't all the same size so for instance if we knew that it was irregular we could add up the angles that we're given and then take it away and then find out the size of the missing one but if it's a regular polygon what we can do is so for instance with this 27-sided polygon if we divided 4 500 by 27 and we could find the size of each one of those angles now the sum of the rules with polygons and they are the exterior angles so if you've got the interior angle and you carry on that straight line the angle between the polygon and that straight line is what we call the exterior angle and I've marked them all on this diagram in Red so all of these red angles are exterior angles so the sum of all those exterior angles all of these red angles will be 360 degrees so so for no matter what polygon you've got if you add up all the exterior angles you will always add up to 360 degrees and then also the interior angle and the exterior angle because they make a straight line they will add up to 180 degrees so there are two very useful rules as well the sum of the exterior angles all those outside angles there all those red angles will add up to 360 degrees and the interior angle and the exterior angle allow up to 180. so let's do some questions based on that now so here we've got a typical question it says work out the size of each exterior angle for a 20-sided regular polygon so this is a regular polygon so I mean that all the interior angles are the same size and also all the exterior angles are the same size and it's got 20 sides and we know that all the exterior angles will always add up together to give you 360 degrees so if we divide 360 by 20 we will find the size of each exterior angle so if we do 360 divided by 20 that's 18. so that means that each of the exterior angles will be 18 degrees so we'll have 20 exterior angles and they'll all be 18 degrees each this time we're asked to work out the size of each interior angle of a 20 set of regular polygon now we've got two different ways to this we could use our first Formula where we take away two so that's 18 times that by 180 and that will tell you what the angles add up to in a 20 set a polygon and then divide that answer by 20 to find the size of each one of those angles so that's one approach alternatively if we know that all the exterior angles are 18 degrees each if we take that at 18 degrees away from 180 we will find the size of the interior angle because remember the interior angle and the extra angle will always add up together to give us a straight line 180 degrees so if we do 180 take away 18 that's equal to 162 degrees that's it our next topic is angles and parallel lines now whenever you get two parallel lines and a line that crosses them are straight line that crosses them you'll find that many of the angles are the same because as you can see you've got lots of vertically opposite angles like for instance here this obtuse angle is vertically opposite to this one so they're the same this acute one is equal to that one now as is a straight line that crosses the two parallel lines then the angles are below will be exactly the same you'll also have a 75 75 and so on now we've got some special knee mcu's whenever we're talking about these angles being the same as each other so if you've got two parallel lines and a line that crosses it and you've got this Z shape these are called alternate angles because they're alternate to each other so the 75 is the same as a 75 because there are alternate angles and it's very important to know that that word alternate angles try not to use Z angles use the term alternate angles so this angle is the same as this angle because they're alternate to each other here these angles these green angles are 130 degrees these obtuse angles and they're the same as each other so they're called corresponding because you've got if you look at it you've got the angles at the top and the angles at the bottom they're both in the bottom right corner here and here so they're corresponding to each other at the bottom right angle and the bottom right angle are the same because they're corresponding some people call it an F angle because you've got this F shape and the angles at the bottom below the f are the same but try to learn the word corresponding so as well as our alternate and corresponding angles we've also got co-interior and vertically opposite angles so if we go to two parallel lines this angle and this angle will add to 180 degrees and that's really useful because if we know one of them we can then work out the other one by taking it away from 100 nearly so this angle and this angle will add up to be 180 degrees so they are co-interior angles and also this angle and this angle will add together to be 180 degrees so their core interior angles as well that's really useful because if you know one of them you can then find the other one also we then go vertically opposite angles so remember if you've got two straight lines across each other the opposite angles will be equal to each other so if this angle is 135 degrees the angle opposite it will be 135 degrees also if this angle is 45 degrees the angle opposite is equal to 45 degrees so opposite angles where you've got two lines across each other the opposite angles are equal to each other and again that's really useful for core interior angles so if we could look at this diagram here we know that if this angle is 60 degrees this angle be 60 degrees because they're opposite each other and if this angle is 120 degrees this angle is 120 degrees because they're vertically opposite each other and that's it so here we've got a question just to practice we remember which ones were which so we've got our two parallel lines RS and T U and you've got that straight line that crosses them and we've been asked to find which angle is corresponding to a so we've got a here and there's the top left angle in this little section here so if we look at the angles below at the other line the angle on the top left there is e so A and E are corresponding to each other a b would be corresponding to f d would be corresponding to H and C would be corresponding to C our next question says which angle is alternate to C now when we see alternate we're thinking of that Z angle and you can see here we've got that c here and if you look at my pen here you've got this sort of Z shape here where you've got the c in there and the F there they're alternate to each other so C is alternate to F so C is alternate to f d would be alternate to e and they would be the alternate angle C is I'll turn it to F and D is alternate T and that's it okay so let's have a look at our next topic and our next topic is gills which is video 283 and 280 foreign and we've got a scale down the bottom and it says one centimeter represents or equals 30 miles so one centimeter on this diagram would be 30 miles in real life and the question says what is the actual distance between leak and Milton so if I was doing a question like that the first thing I would do is I would get a ruler on a pencil and I would join up the two places so I would join them up like so then I would get a ruler and measure the length of that line and our line is eight centimeters so that's an eight centimeter line so on our map we've got the distance between leak and Milton is eight centimeters now and obviously in real life it's going to be much bigger than eight centimeters and we're told that every centimeter represents 30 miles so if we have eight lots of 30 miles that would be the real distance between league and Milton so we need to do eight lots of 30 or 8 times 30. so if we do 8 times 30 well 8 times 3 is 24 so 8 times 30 would be 240 and that's in Miles because the question says every one centimeter is 30 miles so it's eight lots of 30 miles so answer would be 240 miles and that's it if you were given a distance between them in real life and you're asked what the distance was on the map then you would divide by the scale so for instance if I knew the distance between two times was 150 miles and the question said how far apart would they be on the map I would divide 150 by 30 which is five and then it would be five centimeters okay let's have a look at our next topic and our next topic is compass directions which is video 27b and corporate Maps so here we've got a compass we've got North then we've got East and we've got South and we've got West and in between north and east we've got North East and between south and east we've got Southeast between South and West we've got Southwest and between north and west we have North West so these are Compass directions they're very important to know some people remember saying such as never eat Shredded Wheat I do like Shredder weight but you've got never eat Shredded Wheat they could be your little mnemonic to help you remember the north east south and west but you need to know what what directions they go in and here we've got a little diagram and we've got some made up times we've got Castleton Lake Don Hampton Milton and cliff and the question says what direction is Milton from Sand Cliff so we're at sign Cliff because it says from sign cliff and we're asked which direction is Milton so Milton is in this direction so it's going to the left here now we've got North is going upwards so going to the left would be West so that would be West because if you're at cycliff and if upwards is north then it would be north east south and west and West would be Milton okay next question our next question says what direction is sand Cliff from leak so it says from leak so we're going to go to Lake so this little Lexus leak so we're here and we want to know what direction soundcliffe is in so we've got North going upwards then we've got Ace then we've got South so as you can see it's not one of our North East Side for West so it's going to be something like North East or Southeast and so on it's Everett Lake we've got north east south and west so you can see the cycliffe is in the south east Direction it's in the middle of siphon East this answer would be so faced so East that's it okay let's have a look at our next topic so next topic is Maps so obviously we've got our scales and our Compass directions now we're going to put them together in maps and that's video 283 and corporate maps and here we've got a map and we've got the same scale one centimeter it represents 30 miles so as like two topics ago whenever we looked at scales and we've got these towns Lee and Milton and we're told that red tone is 90 miles south of Milton so red tine is 90 miles south of Milton so we've got Milton we've got 19 miles now every centimeter is 30 miles so 90 divided by 30 well 30 we're going to 93 times so it would be three centimeters and we're told to show red tone on the map so we've got this map and we need to show where red tone is so we've got Milton North has gone upwards Ace is going to the right South is going down so it's going to be safe and it's three centimeters so we will get a roller pencil and we'd measure three centimeters downwards we'd find where that is and we'll put a little cross and then we'd ride beside it red tone and that's it bearings and here's a typical bearings question and bearings this video 26 in corporate Maps so if you're doing that full video talking through bearings if you watch that video 26 and cook them up so it'll go through bearings in a bit more detail but as I said the end of this video it's been about three or four minutes in each topic to make sure you're familiar with the topics so here we've got two times we've got Antrim and we've got Belfast and the question says right there in the three figure pairing of Belfast from Antrim so bearings are a direction of travel and they measured clockwise from North so in this question we've been asked to write down the three figure bearing of Belfast from antrum so if I was asked it out the first thing I would do is get a ruler and pencil and join up the two times so I'll join up Antrim and Belfast like so then next we'll draw offline at Antrim so I just get a ruling pencil and I would draw nor flank on upwards from Antrim like so because that's where we're measuring the bearing from it's the bearing off Belfast from Antrim so there's our North flan and I'm just going to put a little n beside it just to show that it's north next we want to Mark in the angle that we want to measure now bearings are measured clockwise from North so the bearing will be this clockwise angle going around from the north line to the line joining Antrim and Belfast and we're going to get our protractor so we're going to get our protractor and we're just going to rotate it so we've got our protractor and we're just going to put it on top of antrum so we've got the cross of it on top of Antrim and we've got the zero going along the north flat and we've got zero on the top now the bearing is measured clockwise from North so we just need to measure this angle so we've got zero at the top so we're looking at these outside angles and we're going to start from 0 10 20 30 40 50 60 70 80 90 100 and then we've got 101 two three four five so that is 105 degrees so if I move my protractor a 105 degree angle and the question said write down the three figure bearing off Belfast from antrum so that would be 105 degrees okay let's have a look at number one so we've got two made up times here we've got Castle town and Milton and we've been asked to write down the three figure bearing of Castle town from Milton so again our first step is to join up the two tones so get your ruler and pencil and join up Milton and Castleton like so so we've joined up our two times next we want to just make sure we know where we're measuring the bearing from so we want to measure the bearing off Castle tone from Milton so we're in Milton so let's draw an offline at Milton and we'll just put a little n at the top to show that it's north next let's get our protractor and we want to rotate it so that we've got the zero at the top so we've rotated our protractor and now let's just double check with angle we want to measure so we want to measure the angle going clockwise from North so mark it in the angle it's clockwise from North so that's the angle we want to measure and we're going to get our protractor and we're going to put the zero on the North Line and the cross on Milton so let's now measure our angle so we start at zero so it's 10 20 30 40. and as you can see we haven't reached 45 so we've actually got 44 degrees so that angle is 44 degrees so let's just move our protractor and Mark it into this 44 degrees so 44 degrees and we've been asked to write down the three figure bearing off Castle town from Milton but remember the bearing has to have three figures so it's going to be zero four four degrees we're just going to put a zero in front of that so zero four four degrees and that's it so the bearing of Castleton from Milton is zero for four degrees that's it okay our next question okay this time we've been asked to write down the three figure bearing off Port Rush from larne so this time we're going to join up the times as before so we've joined them up now we're going to make sure we know which way North is so North is going upwards and we want to just check where we're starting from so it's writing down the three figure bearing off point Rush from Lawrence we're in learn and we're going to do our North Planet larne so there's our North Planet let's just mark it with a little n now we've got a mark on the angle we want to measure so remember bearings are measured clockwise from North so we want to go clockwise from North all the way around to the line join in Port rush and learn so we need to measure this angle here it's a reflex angle so it's going to be bigger than 180 degrees but less than 360. so we want to measure that Angle now there's three different ways we can measure this angle the first way is my favorite way it's actually getting a 360 degree protractor so this protractor only goes up to 180 degrees so if you've got a 360 degree protractor that actually really really fantastic because you can put the zero on the North Line and you can just go around and read off even reflex angles like this one and if you imagine this one what you could do is so one approach is you could actually turn it the other way so we can measure this small angle here so we could actually get our protractor and measure the angle anti-clockwise so it's starting to zero on the inside 0 10 20 30. and we're in the middle here between 30 and 40 so that's 35 degrees so one approach is actually to measure this angle which is 35 degrees and because we know the angles on a full turn add up to 360. we could take the 35 away from 360 and do 360 subtract 35 which is equal to 325 degrees so that means that this angle is 325 degrees and then that would be our bearing because as the angle measured clockwise from North all the way around to the line join in the tones so our answer would be 325 degrees so that's one approach another approach is actually is to measure that reflex angle and what we could do is so answer 325 we can just go back to the beginning and we want to measure this reflex angle so what we can actually do is draw a line straight down like so and we know that a straight line the angles add up to 180 degrees so we do know this is 1 180 degrees on the right hand side so we just drawn the line straight down our suppose our South line which would be 180 degrees there and then we could just get our protractor and we could turn it around so that we've got zero at the bottom this time and we want to go from zero around to where the line is and as you can see if we start at zero which is at the bottom here we're 0 10 20 30 all the way around to in between 140 and 150 and it's in exactly the middle so that means that's a 145 degrees so if we wanted to measure this clockwise angle this reflex angle we know that we've got 180 degrees and another 145 degrees and if we add those two together we also get 325 degrees so that's it so the bearing measured clockwise from North from Lauren to Port Rush would be 325 degrees okay okay let's have a look at our next topic and that's actually bearings as well it's working out back bearings so in other words here's an example if we know the bearing off Nottingham from Dublin as 0.98 degrees what's the bearing off Dublin from Nottingham to go in the other way around so there's two different ways you can actually approach this the first is actually just a little sketch so the bearing off Nottingham from Dublin so let's do a little X for Dublin so let's just label that Dublin and then we've got Nottingham and it's on a bearing of 098 degrees so let's put our Northline in and 098 degrees or 98 Degrees if we started at the top we went around 98 Degrees it would be go past East and just a little bit beyond it so it means that it wouldn't be here it would just be a little bit further so let's put a little cross there and say that's Nottingham so we've got Dublin and we've got Nottingham let's join them up with a straight line now let's mark in the angle so we know that angle is 98 degrees because the bearing is 0.98 so that's 98 degrees there now let's also draw another nor flying at Nottingham so we're drawing our North line with the little land at Nottingham and we want to measure the bearing off Dublin from Nottingham so we want to measure the angle going clockwise from the north line at Nottingham going around to the line joining Dublin and Nottingham like so so we want to find the size of that angle now here we've got two North lines and they are parallel lines I remember with parallel lines the two interior angles are called co-interior these two angles add up to 180 degrees and actually if we go back to our revision criteria we've got cointerior angles two angles inside our parallel lines add up to 180 degrees so if this angle is 60 that angle is 120. so that's quite important so if we go back to our bearings we've got this is 98 degrees so this angle and this angle here will add up together to give us 180 degrees so if we do 180 subtract 98 we get that's equal to 82 degrees that means this angle here is 82 degrees next we want to find the size of this reflex angle now we know the angles on a full tone add up to 360 degrees so if we take 82 away from 360 we can find the size of this angle so we can do 360 take away 82 and that's equal to 278 degrees and that's it so that means that this angle would be 278 degrees and that would be the bearing of Dublin from Nottingham and that's it or another way to do it is to actually seen notice a bit of a pattern so if you're doing back bearings question and the bearing that you're given in the question is less than 180 degrees to get the bearing going the other way you could just add a 180 degrees and that would give you your answer and if the bearing you're given in the question was more than 180 degrees if you take away 180 degrees you can get your answer and that's a bit of a shortcut now the way I do that is I visualize it if I was traveling from Dublin to Nottingham and then I wanted to go the other way I would route him through 180 degrees to go the other way if you want to watch that more detailed you can watch video 27A and chord Mavs and there's questions there on it as well our next topic is perimeter on the grid and here we've got a centimeter grid and we've been asked to find the perimeter of this ship so stronger than a centimeter grid that means each of the lines is one centimeter so we've got one two three centimeters at the top there three centimeters then another one two three centimeters going down and then another one two three centimeters going across and the number one going up one going across one going up one going across and one going up and we're back to the certain point so remember the perimeter of the shape is that distance around the outside of the ship so we just need to count those so we could just go one two three four five six and so on or you could add them up three plus three is six plus three is nine ten eleven twelve thirteen fourteen so the perimeter of the shape is 14 centimeters okay our next topic our next topic is perimeter so here we've got our rectangle and we've been asked to find the perimeter of this rectangle as we've seen earlier with a rectangle the opposite sides of the same length so if the tops equal to six centimeters the bottom would be equal to six centimeters and if the right hand side of the rectangle is 20 centimeters the left hand side would also be 20 centimeters and we've been asked to find the perimeter of the ship so we just need to add up those distances so if we do 20 plus 6 plus 20 plus 6. that would tell us the perimeter so 20 plus 6 is 26 plus 20 would be 46 plus 6 would be 52. so the perimeter of this ship is 52 and the units are centimeters so 52 centimeters our next question says find the perimeter of this isosceles triangle so we have an isosceles triangle two of the sides of the same length so as you can see we've got two sides with the dashes that means they're the same length so that means that the left hand side here is seven would be also the same as the right hand side here seven centimeters we've been asked to find the perimeter so we just need to add together seven ten and seven so ten plus seven is seventeen plus seven is equal to 24. so the perimeter of this triangle would be 24 centimeters okay so our next topic is area on a grid so we've looked at perimeter of a shape on a grid now we're going to look at the area of a shape on the grid so here we've got a shape drawn on a centimeter grid that means each of these squares has got an area of one centimeter squared so all we need to do is find the area of the shape is current the number of squares so one two three four five six seven eight and because it's eight squares and each one has an area of one centimeter squared the area would be eight centimeters squared and that's it okay our next topic so now we're going to find in the area of a rectangle so to find the area of a rectangle we multiply its length by its width so here we've got a rectangle its length is nine and it's width is five so we just do area is equal to length times width so it's going to be the length which is nine multiplied by the width which is five and nine times five is equal to forty five now make sure we put the right units on it's centimeters so this is going to be 45 centimeters squared and our next question is to find the area of a square so here we've got a square and it's got a side length of seven centimeters and because it's a square that means all sides are seven centimeters that means that if the length is seven the width is seven so to find the area we're just going to do seven multiply by seven and seven times seven is 49 centimeters squared and if you want to also called Mouse video on our rectangle watch video 45 okay our next topic so next topic is finding the area of a triangle and this is the part of the chord Max revision card that tells us there of a triangle so the area of a triangle is half the base times the height now we can do this in two different ways we can either half the base and then multiply by the height or we can times the base and the height first of all and then divide by two then half it so here we've got a triangle and it's got a base of seven centimeters and a height of four centimeters so if we want to find the area of this we can either half the seven which is 3.5 and then multiply that by four and then that's whatever 4 times 3.5 is or we could times the base and the height together first of all so seven times four so seven times four is equal to 28 and then do 28 divided by 2 and that's equal to 14. so the area for this triangle is 14 centimeters squared so to find the area of a triangle you can either do the base times the height and then half it or half the base times the height now here we've got another triangle and let's go to base of 14 centimeters and a height of 5. so I'm actually going to have the best to begin with so I'm going to find the area which is equal to half the base so half of 14 and then I'm going to multiply that by the height which is equal to five so half of 14 is 7 and then we're going to multiply that by 5. 7 times 5 is 35 so the area of this triangle would be 35 centimeters squared so to find the area of a triangle just do half the base times the height and then that's it okay let's have a look at our next topic so our next topic is video 44 which is the area of a parallelogram so here's a parallelogram and it's got a base of 11 centimeters and a height of 5 centimeters and the air for parallelogram is found by the base times the height because if you were to chop this part of the parallelogram off and move it across and put it here we would now have a rectangle and the base would still be 11 centimeters because this bit has moved to here so the base would still be 11 and the height is still 5. so to find the area hologram you just do the base times the height so for this parallelogram the base is 11 multiplied by the height and that is the distance between the two parallel lines which is 5 and 11 times 5 is 55 centimeters squared if you do have any diagonals labeled such as this one you just ignore this because you just need the height that's it okay let's have a look at our next topic so the next topic is area of a trapezium and this is part of the chord Mouse revision card I've actually got into little pieces so you've got the area for a trapezium it's given by the formula a half bracket a plus b so A and B are the two parallel sides of the trapezium so we're going to add those two together divide that answer by two and then times it by the height and that will give you the area of a trapezium so here's a trapezium so the area would be equal to a half a plus b they're the two parallel lines that's seven plus eleven and then we'll get that answer and we'll multiply it by the distance between the parallel lines which is five so when we do that we get a half and then 7 plus 11 is equal to 18. so half of 18 multiplied by five a half of 18 is nine so we've got nine times five that's equal to 45 centimeters squared that's it so Define the area for trapezium you add together the two parallel sides you divide by two and then you times but I have the height the distance between the two parallel lines and that's it and then if you want more practice on that that's video 48 in corporate Maps okay next topic is to find the area of compound ships so that's whenever ships have been put together now those shapes could be rectangles triangles semi-circles quarter circles sick of circles parallelograms um trapezia trapeziums um it could be it could be loads of different ships and they've just been put together and you need to find the area off them so if you have a look here we've got the ship it looks like an arrow but if we divide it into two we can see that there's two parallelograms so we've got two parallelograms and each of them has a length of 35 centimeters for the base and the height of them or the distance between the two parallel lines so as you can see the whole thing is 52. when you divide that by two we get 26 that means the higher this parallelogram is 26 centimeters and the height of this parallelogram is 26 centimeters so all we need to do is work out the area of this parallelogram and there of this parallelogram and then add them together therefore parallelogram is the base times the height so that's 35 multiplied by 26 and when we do that we get 910 centimeters squared now these two parallelograms are identical or congruent so that means that they both have the same area so it's going to be 910 plus 910 and that's equal to 1 820 centimeters squared and that's it our next topic is units and it's very important to be able to convert metric units so here we've got some metric units and I would write these down these are the conversion facts on the revision card again if you've got the revision card pin on the wall revise them bring it with you on the bus whatever make sure you learn these facts and their videos 347 or 349 on core maths also these are really important one kilometer is a thousand meters one meter is 100 centimeters and one centimeter is 10 millimeters it's important to learn those facts off by heart in terms of capacity one liter is a thousand milliliters and one liter is equal to one thousand centimeters cubed they're both quite important and in terms of mass one metric ton is a thousand kilograms and one kilogram is equal to one thousand grams and these are all important unit facts to learn off by heart okay in the next topic our next topic is sensible estimates and it's important to be able to make sensible estimates for maybe the height of a building or the height of a door or the length of a bus or the amount of liquid in a glass or so on and what I would do is I'd consider the atoms around you and make sure that you're aware of maybe some of the lengths of them or Heights of them or maybe how much liquid or something contains or the mass of certain objects so for instance the height of a person roughly 1.8 meters the length of a swimming pool my local swimmer pulls 25 meters long so I remember that um here we've got a large bottle of fizzy drink it would hold two liters you can get sort of smaller ones which might be maybe 1.5 liter shoes or you've got the little ones that you could be 500 milliliters or you know so on a kind of fizzy drink might be 330 milliliters things like that might help you estimate uh the capacities of different objects so in terms of the massive objects well a bag of sugar is one kilogram I've got some weights as well so I know in terms of you know like a 10 kilogram dumbbell a 20 kilogram dumbbell and so on 30 40 50 kilogram dumbbells and so on of course um so in terms of the sensible estimates they are some sort of everyday items which might be useful for you so we're now going to look at Imperial units so that's videos 347 and 348 in corporate Maps so we have our metric units which are centimeters meters grams kilograms milliliters liters and so on and the modern units we've then got our Imperial or older fashioned units um over not cause any offense anyone there but you've got an older fashion Dura imperial units such as Pines ounces inches feet miles and so on and we're going to need to be able to convert between them and here's a typical question where we're told the length of a table is 35 inches so we've been given it in inches which is going to be in Imperial and we're told that one inch is 2.54 centimeters so every inch is 2.54 centimeters and the question says how long is the table in centimeters so we want to convert this 35 inches into centimeters and we're told that every single inch is 2.54 centimeters so in other words we've got 35 lots of 2.54 centimeters so if we do 35 multiplied by 2.54 that will tell us then how many centimeters long the table is so we'll do 35 multiply by 2.54 and when we do that we get that's equal to 88.9 centimeters and that's it and this is a calculator question so you can just type in 35 multiply by 2.54 and then just make sure we're putting centimeters on the end okay let's look at our next question now before we do that I'm just going to go through some of the key facts so one kilogram is roughly 2.2 pounds okay number one another one will be 5 miles is roughly eight kilometers so if you've got five miles that's roughly eight kilometers and dividing both of these by five will give you one mileage roughly 1.6 kilometers okay let's have a look at our questions so a question says the distance between Newry and down Patrick is 30 miles right this distance in kilometers so here we've got the distance between Yuri and Dan Patrick is 30 miles five miles is roughly eight kilometers so what we're going to do is we're going to take our 30 and we're going to divide it by five because that will tell us how many lots of five miles there are in this distance so 30 divided by 5 is 6. that means there's six lots of five miles and every time we've got five miles that's roughly eight kilometers we know the six lots of eight kilometers so if we do six times eight that's equal to 48 kilometers and that's it so our next topic is line symmetry which is video 316 in corporatemals.com so an isosceles triangle has one line of symmetry then we've got a rectangle a rectangle has got two lines of symmetry a vertical one and a horizontal one it doesn't have a diagonal one because if you try to fold it over the corners wouldn't go to the right place so a rectangle's only got two lines of symmetry we're just Square it's got four lines of symmetry so vertical horizontal and the two diagonals so parallelogram it has no lines of symmetry an equilateral triangle has three lines of symmetry and a regular hexagon has six lines of symmetry so it's important to be able to tell if a ship has got lines of symmetry or not okay our next topic our next topic is rotational symmetry and the final order of rotational symmetry over shape whenever we spin it through 360 Degrees we count how many times it lands on itself so here we've got a rhombus and as we spin it from the 360 Degrees as you can see it lands on itself once when it's upside down and twice when it gets back to its original position so a rhombus has a order of rotational symmetry too now the question says Circle the ship which has order of rotational symmetry 4 or rotational symmetry order four so we've got a heart it would have order one it would only have its final position we've got a square as you turn that through a full circle and land on itself four times so that will be our shape that will be the shape which has order of rotational symmetry four this five-pointed star it would have order rotational symmetry five and this isosceles triangle would only have order rotational symmetry one as you turn it around for the full circle it would only land on itself once so a ship has got rotational symmetry whenever it lands on itself more than once whenever you turn it through a full circle and find the order of rotational symmetry you just kind of how many times it lands on itself whenever you spin it through a full circle and that's it okay let's have a look at our next topic okay let's have a look at our next topic and our next topic is constructions and they are video 78 72 and 79 on corporate maps and if you want to watch me do these constructions where I'm filming myself actually doing the constructions those videos will actually show that but here I'm going to show you step by step and how I would do the constructions so first of all our first question says construct the perpendicular bisector of a b so first of all water is a perpendicular bisector so perpendicular perpendicular means 90 degrees and bisector means the cut in half so if we were asked to construct the perpendicular bisector of a b it would be a straight line that cuts a b in half going through the middle at 90 degrees so we'll be cutting this line a b in half at 90 degrees and to do that by using a pair of compasses and a pencil what you do is first of all you would get your pair of composters and you would put the point of the compass on air and you put the pencil on the compass and you would set it over halfway of the line so I've just chosen this point it's definitely over halfway off the line and I've drawn this semi-circle starting at a and I've drawn the semicircle around so keeping the point here I'm making sure I draw a nice semicircle going around now lifting up your compass and your pencil and making sure it doesn't change size put the point of the compass on B and then what we're going to do is we're going to do the same thing so we're going to do another Arc we're going to put a point at the compass on B and we're going to do an overarch I'm making sure it's the same size as the arc we've done at a and then finally what we're going to do is get a ruling pencil draw a nice straight line through those and this line will be our perpendicular bisector so that line will cross our line a bit 90 degrees so it's perpendicular and it's a bisector because it cuts the line a be exactly in half so if that line a b was 10 centimeters you would have five centimeters to the left here and you'd have five centimeters to the right so that's it okay next we're going to construct the angle bisector so here's an angle a b c and we're going to cut this angle in half by using a compass a pencil and a ruler so the first thing we're going to do is we're going to put our point of our compass on B so we're going to put the point of our Compass here and we're going to do an arc on the line a b here and an arc on the line BC here and it's important that you keep the compass the same size for both of those arcs so you put the point here and you just move it so you've got an arc there on the line a b and Arc here on the line BC okay so next we're going to lift up our compass and we're going to put the point of the compass here so we're going to put the point of the compass where the arc and the line a b mean and we're now going to do an arc in this direction so we're going to get our compass and we're going to do an arc in this direction and it'll look something like this so that's where the point of the compass is and we do this Arc looking like that now we're going to lift it up and we're now going to put the point of the compass here and we're going to do another Arc in that direction and I would look something like this and then finally what we're going to do is get a ruler and a pencil and we're going to join up the center of the angle ABC to where those two arcs meet and that will be our angle by sector that line will cut this angle exactly in half so if this was a 60 degree angle you would now have 30 degrees above it and 30 degrees below so that line is called the angle bisector that's a very important construction so we've constructed the perpendicular bisector and we've now constructed the angle bisector okay our next construction is the Constructor line perpendicular to a b so it's going to be align this at a right angles to a b and it passes through the point C so it's going to be a line that passes through C and there's a right angles to a b and we're going to do that again with our Compass our pencil and our ruler so first of all what we're going to do is get our compass and we're going to put the point of the compass on C and we're going to set it so that it's longer than the distance between C and the line so it's a bit further than that so it's not just going to reach the line it's going to be a bit further but it also needs to make sure that it reaches the line on two different places so if I set it too large so if I set it over here it would cross somewhere over here but it wouldn't actually meet the line so I've set it about here so it's a bit longer than the distance to the line and we're going to do two arcs one here to the left and one Arc to the right and again it's very important that you keep your compost the same size the whole way through next what we're going to do is keeping the compass the same size we're going to now lift it up and we're going to put it on the point here okay so putting the compass here what we're doing is an arc below the line and it's going to be below C so it's going to be an arc in this direction and then we're going to lift it up and we're going to repeat it here so we're going to put our point of our Compass here and we're going to do another Arc below and if you get your ruling pencil and join up from C to where those talks meet at the bottom here that line will be perpendicular to the line a b and it will obviously pass through C and that's it so we've constructed the perpendicular to the line a b that passes through the point C we're now going to construct a perpendicular to the line a b that passes through the point C which is on the line so in other words we're going to make a perpendicular line using our ruler our compass and our pencil and it's going to be a perpendicular line to a b so it can be at right angles and it's going to pass through the point C so first step is to put the point of the compass on C and we're going to set the compost so that it's getting close to the distance between C and B so I'm going to put it about this distance here and we're going to do an arc on this side and we're going to keep it the same size and do an arc on the other side so it would look something like this so we've put the point of the compass there and we've done an arc on this side and on that side and it's the same distance okay so now what we're going to do is we're actually going to look at this line going from here to here and we're now going to construct the perpendicular bisector of that so using the same steps as before so first step is to put the point of the compass here and set the pencil his past C and then we're going to do an arc above C and Below C and keeping the composite same size we're then going to swap over and we're going to put the point of the compass here and again keep it the same size we're going to do an arc above and below and then finally if we join up these two points there and there that line will pass through C at 90 degrees so there's a line that's perpendicular to a b and passing through C and that's it our last construction that we're going to look at is constructing an equilateral triangle and remember an equilateral triangle all the sides of the same length so this is quite a nice one that if you put your point of your compass here and you put your pencil of your compass on the end of the line that'll be five centimeters and then if you do an arc above it would look something like that so all of these points are five centimeters away from here now keeping the composition size and point putting the point of the compass on this side and putting the pencil here now doing the same thing and then we'd end up with something that looks like this and this point here would be five centimeters away from this end of the line and would also be five centimeters this end of the line so if we get a return pencil and join them up like so so there and there that will be an equilateral triangle so it'll be a triangle where the sides are five centimeters so that's five centimeters that's five centimeters and that's five centimeters and also the angles will be 60 degrees so that'll be 60 degrees that'll be 60 degrees that'll be 60 degrees and that's it okay so there were our constructions if you do want to watch full videos on them and be showing you how to do each one of them really clearly using a composter ruler and a pencil watch the videos above there okay let's have a look at our next topic so our next topic is lokai and that's video 75 76 and 77 on corporate maps and my question says to draw the locus of all the possible points that are two centimeters away from the line below so in other words we've been given a rule and we've got to draw on all the possible positions the points can be that follow that rule so we want to draw all the points that are two centimeters away from this line so the points above the line move quite straightforward you could be two centimeters just above the end you could be two centimeters above there you could be two centimeters above here and you could be anywhere that's two centimeters above the line like so alternatively it could be two centimeters below the line so it could be two centimeters there two centimeters there and you get your ruler and pencil you make sure all the points are two centimeters below the line so it looks something like this where you've got all the possible points above the line and all the possible points below the line and then what you do is get your ruler and join and go through the points they're all two centimeters above and below now obviously if you're using a ruler and pencil they would all be in a nice straight line so this point here wouldn't be there it would be actually up there okay and so these points above here and below here are all two centimeters away from the line but what about the end of the line so if we're at the end of the line here we could have two centimeters to the right so that would be there and then we could be two centimeters here or two centimeters here or two centimeters here and as you can see we're forming this semi-circle where they're all two centimeters away from the end of the line so it looks something like this and with a compass and a pencil you would put the point of your compass here and your pencil here you draw a nice semi-circle going around like so and that would join up perfectly uh excuse my freehand one but that's just a little Sketcher and likewise if we're in the end of this line here we could have two centimeters here and if we measure the points that are all two centimeters away from the end of the line again it would be another semi-circle and all of these points would be two centimeters away from this end of the line so all we'll do is again get the point of our compass and put it there and then put the pencil two centimeters away and then draw a nice semi-circle going around through those points so that's it okay let's have a look at our next question so this time we've got a diagram and we've got the points A and B and we've got a scale the one centimeter is one mile and we've got land and we've got C and the question says a boat is within eight miles of a so it's within eight miles of air and it's within five miles of B shared the possible positions off the boat and one thing I should have said in this question is the boat's at Sea so it's not on land it's at sea so we know that it's within eight miles of air and we know that one centimeter is one mile so what I would do is I would measure eight centimeters and I would set my compass so that is eight centimeters so I put the point of the compass on air and I would put the point my pencil so it's eight centimeters away so I put my pencil up with a c in the land meat and I would draw a circle and this circle has got a radius of eight centimeters and it would look something like this it's a freehand sketch sorry just got a radius of eight centimeters and that's a circle that looks something like that okay next we're now told that the ball is within five miles of beep So within five miles of B and that's five centimeters we get our Compass we'd measure it so the distance between the point and the pencil is five centimeters and we put the point here and it would be over here somewhere so we'll put the pencil with us seeing the landmate and we draw another Circle and this circle's a bit smaller it's only got a radius of five centimeters this time and it would look something like that and again sorry excuse my diagram so this circle has got a radius of five centimeters and this circle has a radius of eight centimeters and we were that the boats within eight miles of sea so in other words it's in here somewhere but it's also within this circle as well so that means the possible positions of the boat would be in this region here and that's it if you want more practice on lokai video 75 76 and 77 we'll go through more examples also there's some fantastic practice questions that you can do on corporate map so if you go to corporate maps and go down to worksheets and click on it and go down to those video numbers beside that you'll see practice questions and they'll be ideal to practice as well so this is video number one which is names of 2D shape it's very important to know the names of the different 2D shapes so let's start off with our polygons we've got our triangle which is a three-sided ship quadrilateral four sides Pentagon that's five sides if it's got six sides it's a hexagon seven sides a heptagon eight sides octagon nine sides nonagon and ten sides decagon also make sure you know your circle semicircle and so on okay our next topic and also important with the difference between your irregular polygons and your irregular polygons so a polygon is a straight sided shape regular polygon so where all the sides are the same length so your Square equilateral triangle is a regular pentagon a regular hexagon and so on also as well as the sides all be in the same length the angles are all the same size so for instance in your Square all the angles are 90 degrees and an equal actual triangle all of the angles are 60 degrees each and so on irregular polygons are where the sides and angles aren't all the same so the sides could vary in length and the angles can vary so the next topic is going through types of triangles so someone wants to know that you've got your right angle triangle so it's a triangle with a right angle our next type of triangle is the isosceles triangle the isosceles triangle has two equal length sides so for instance this side on the left and the side on the right for this one so for instance if this was 10 centimeters on this side this side here would also be 10 centimeters also as well as having two sides the same length we've got two angles the same size this angle in this angle would be the same you've put your equal out to a triangle all the sides are the same length and all the angles are the same they're all 60 degrees each and finally we've got a scalene triangle and scalene triangle is a triangle where none of the sides are the same length and none of the angles are the same and if you want to go for that more detailed video 327 a corporate Maps goes through the types of triangle our next topic is quadrilaterals that's video two incorbent jobs here are some quadrilaterals we've got our Square rectangle rhombus trapezium parallelogram and kite now in terms of the properties to square all sides of the same length it's got four right angles it's got four lines of symmetry down through the middle across and diagonally as well it's called order of rotational symmetry four then we've got a rectangle for the rectangle the opposite sides are the same length so the top and the bottom will be the same length and the left and the right hand side of this rectangle would have the same length each of the angles are 90 degrees it would have two lines of symmetry an order of rotational symmetry two now we've got a rhombus with a rhombus all the sides are the same length it's got two lines of symmetry so down through the middle and across it's got order of rotational symmetry two and in terms of the angles the opposite angles are the same size our neck shape so we've got a trapezium and the trapezium has one pair of parallel lines so the top and the bottom of this trapezium are parallel to each other also sometimes it has a line asymmetry so this trapezium does have a lamine symmetry A lot of the time it doesn't it would have of order of rotation symmetry one and yeah okay and next ship our next ship is a parallelogram a parallelogram has two pairs of parallel lines so the top line is parallel to the bottom and also the right hand side is parallel to the left hand side the opposite angles are the same as each inverse to the top left angle here would be the same as the bottom right and the top right would be equal to the bottom left um it would have order of rotational symmetry two and this parallelogram would have no lines of symmetry and finally we've got a kite a kite would have one line asymmetry going down through the middle uh the angle on the left and the right of this coat would be equal to each other and yeah okay and if you want more details in terms of quadrilaterals watch video turn corporate Maps Okay our next topic our next topic is 3D shapes here are some common 3D shapes you need to know the name of so you've got your Cube cuboid sphere cone cylinder triangular prism square based pyramid and pentagonal prism that's just an example of a prism and so we've got these different 3D shapes you're gonna need to know the names of them also you're going to need to know what edges are vertices are in faces I let's have a look at those now so here we've got a Vertex so the vertex is a corner so with this Cube it would have eight vertices vertex means one of them when you've got more than one that's vertices so it would have eight vertices that's eight Corners four in the top one two three and four and four in the bottom as well so we'd have eight vertices the cube would have six faces so here's a face at the front a facing the top a facing the right the bare face on the left the back and the bottom as well and just think of a dice it's got six faces and in terms of edges edges join the vertices it would have 12 edges so it would have one it'll go along the top here of one two three four then I'd have four going downwards one two three and one at the back and then have four in the bottom so it'd have 12 edges okay it's also important to know what Nets are and this is the chord Mouse revision card on net so here are the Nets of six 3D shapes you've got your Cube and out of the cube so it would fold round and then the two sides would fold up so it would be then out of a cube so here's the net of a cuboid the net of a square based pyramid so this Square would be the base and then the four triangles would fold up to meet at a point then you've got a triangular prism so this would be the base and then the two sides would fold up and then the two triangles would fold up to filler and those spaces so within the net of triangular prism here we've got the net of a cone and then out of a cylinder and it's important to know what Nets are if you want more practice on Nets watch video form chord Maps Okay so our next topic is parallel and perpendicular lines so two lines that are always the same distance apart or parallel lines so the two lines will never meet so here we've got an example of parallel lines people often think of railway tracks whenever they think of parallel lines and so it's a good one to think of in terms of perpendicular lines perpendicular lines are lines across at 90 degrees so if you've got two lines that are perpendicular to each other the angle between them will be 90 degrees and that's it so parallel lines are lines that they're always the same distance apart they never meet and lies across at 90 degrees or perpendicular so our next topic is views and elevations so if using elevations are whenever you're looking at 3D shapes looking at from different perspective and considering how the shape will look if you look at it from those angles of those perspectives see here we've got a ship it's a load of multi-link Cubes stuck together and I'm going to look at it from different angles we've been asked to draw what the front elevation would look like so if I was standing here if I was small and I was looking at it from here and this was the front of the shape I would just see this these four blocks this one this one this one and this one so I would see this shape here and I could get my pencil and ruler and draw these really tightly excuse this and that's what I would say out here rectangle that was one two three four blocks across some people like to put in the lines as well in between because you might see those lines joining the blocks but that's what I would see I would see that rectangle which is four rectangles wide and if you do want more practice of using elevations look at 354 video 354 in corporate maps.com okay let's look it up from a different perspective so again that's the front again now I'm going to draw the side elevation now there's two different side elevations here I could draw up from the left hand side or I could draw it from the right hand side and I'm actually going to draw both so let's start off with the left hand side so if I was here at the side and I was looking at it that way I would again see a rectangle and it'll be one two three four blocks across so it would be four blocks across like so and I would draw like that but you could put the lines in if you wanted to going down you might see where the blocks joined so some people would draw the lines in so but I would just draw like that alternatively I could draw from the other side so I could be standing over here and I could be looking at it from this side here and again I would see one two three four blocks now they're not all level with each other but that's what I would say I would see the block on the left the Block in the middle the next one back and the back one there so I would say again four blocks now this time if I was drawn I would draw the lines in like so just to show that I would see the one on the left hand side that won't be a further forward and then I might see that one a bit further back and that one a bit further back and that one a bit further right but again it would just be a rectangle four blocks across okay and finally we've been asked to draw the plan view the plan view is from above it's the bird's eye view as such so we're going to pretend that we're above the ship and we're looking down on it so if we were looking down at it from above we would see our four blocks at the bottom so our four blocks were going along the bottom like so one two three four and I'm looking straight down so then on the left hand side it would go up so it would go like one two three four like so if I'm looking down from the top then it will go across and down across and down across and down and across and down so it would look something like this from above the ship and that's what I would draw and again some people put the joints in but because they're all sort of flat I would tend to draw like so and that would be the plan view the view from above so whenever you're drawing shapes from different perspectives you've got the front elevation which is the view from the front you've got your side elevations which is the views from the sides and you've got your plan view which is the view from above okay the next topic we're going to look at is time calculations and before we look at an example let's look at some of the key information we need to know whenever we're dealing with time so one minute is 60 seconds one hour is 60 Minutes one day is 24 hours one week is seven days then you've got your months of the year and in a year there's 12 months and one year consists of either 365 days or 366 days if it's a leap year it's important to be able to work out time calculations as well so knowing the difference between 12r clock Am Pm 24 hour clock a video 322 goes through that so it's a typical question and the question says Ella finishes school usually at 3 pm and the time on her watch is 13 14. so that's 24 hour clock and let's change it to PM so it's obviously in the afternoon because it's past 12 so it's 13 so it's p.m and to find the time here we've got 13 well if you take away 12 you're left with one so it'd be 1 14 p.m and Elsa kind of watching seeing how long it's left until she finishes school I'm not sure why she's looking forward to the end of school and the question says how many minutes is it until Ella finishes school so Ellen's got well it's 1 14 p.m we've got 1 14 P.M and we want to get to 3 P.M so first of all let's get to 2PM well this is 14 minutes and there's 60 minutes in an hour so if we add six minutes to begin with so add six minutes it brings us to 1 20 p.m and then if we add another 40 minutes that will bring us to 2PM and then we've got another hour we've been asked how many minutes it is until Ella finishes school so instead of writing one hour here I'm going to write Adam over 60 minutes so let's find out how many minutes it is until Ella finishes school so she's got six minutes and then 40 and then an over 60. well 60 plus 40 is 100 plus 6 is 106. so 106 minutes and that's it so it's very important to be able to calculate questions involved in time so remember in the 60 seconds in a minute 60 minutes in an hour 24 hours in a day been able to do it for 12 hour clock 24 hour clock and so on so our next topic is timetables so here we've got a bus timetable and we've got our stops so for leak Milton Newtown Red Island sandville and Bakerstown and here are three buses each column represents a bus so the first bus go starts at so I feel at 9 18 it arrives at Lake at 9 28 and leaves at 9 28 gets to Milton at 9 41 and leaves at 9 41 and so on so this is the timetable this bus stops at every single stop this bus the second bus stops at every single stop and the last bus you can see some of them have got dashes in that means it's maybe an Express bar so it skips out some stops so this bus goes from sofil to Red Island to Bakerstown so the question says the Irish traveling to sandville so we've got sandville here that's where Dara's going and he arrives at Milton bus station so Milton bus station at 10 45 and the question says at what time should he arrive at samville so he arrives at 10 45 now if he arrives at Milton bus station he's already missed that first bus he's because it left at 9 41 so he's not going to get this bus so that bus is out of the question and then we've got the second bus and it's going to leave Milton at 1101. now he's arrived at 10 45 so that's great so he is going to get this bus and the question says it says what time should he arrive at samville so he should arrive at samville at 11 33 depending obviously if the bus is on time so the quick answer would be 11 33. so we could have been asked different questions here we could have been asked how long is that Journey well if it leaves at 1101 and I like it arrives at 11 33 that would take 32 minutes it could be how long does he need to wait until he can get the bus and so on but that's it we've answered our question what time should you arrive at samville 11 33 okay our next topic is distance charge so here's the distance chart and you've got your towns built in Newtown portsville league and Castleton actually saw one of these the other day in an apple green service station we had all the different kinds of cities and I had the distance between them and here we've got some times that I've just made up and the distance between them in kilometers and we've got built-in and if you wanted to find for instance the distance between built-in and Lake you would just go to Bilton and you would go down until you got to the road the lake was in and you can see that's 95 kilometers you could have started at leak and went across until you got to Bilton which again would be 95. okay so the question says Jessica travels from Bilton so here to portsville which is here so if we look at it that's going to be 12 kilometers then she travels from portsville to Castleton so she's at portsville and she's going to travel to Castleton so that's 63 kilometers 60 three kilometers and the question says how far does Jessica travel so she's traveled 12 kilometers to get from Bilton to portsville and then from portsville to Castleton 63 so if we had them together 12 plus 63 is equal to 75 kilometers and that's it okay our next topic okay our next topic is speed distance and time so this is a very important topic it's video 299 and I'm going to look at this topic in two different ways first of all just by considering a speed so for instance 30 miles per hour what that means and then I'm going to consider the formula that we use for okay so if I had a speed such as 30 miles per hour that means 30 miles each hour so if I had one R and I was traveling at 30 miles per hour I would a good distance of 30 miles if I was traveling for two hours would that would be two it's 30 miles an hour so that's two lots of 30 so I would travel 60 miles if I travel for three hours about three hours at 30 miles each hour so that's 90 miles and so on so that means that if I know the speed of 30 miles per hour if I want to find out how far I've traveled I would just multiply the speed 30 miles per hour by the time that I was traveling so 30 times 1 is 30 miles 30 times 2 is 60 miles and so on likewise if I knew the distance I traveled so for instance 300 miles and I knew it took 10 hours if I divided the Distance by the time so 300 divided by 10 that gives me 30. 120 divided by 4 is 30. 30 divided by 1 is 30. so if you divide the Distance by the time you get the speed and finally if you know the distance you travel and the speed you're traveling well 300 divided by 30 is 10 120 divided by 30 is 4 90 divided by 30 is 3. so if you divide the Distance by the speed you get the time taken so as we've seen speed is equal to distance divided by time distance is equal to speed times time and time is equal to distance divided by speed so a question says a car drives 180 miles in 4 hours calculate the average feet in miles per hour of the car so we want to find the average speed of the car so speed is equal to distance divided by time so if we divide the Distance by the time we'll find the speed so we're going to do the distance which is equal to 180 miles and we're going to divide that by the time which is four hours and if we divide 180 by 4 we'll find the speed so 180 divided by 4 is equal to 45. so the speed of the car the average speed of the car would be 45 miles per hour and that's it so it's very important to know the speed is equal to distance divided by time distance is equal to speed times time and time is equal to distance divided by speed okay let's have a look at our next topic okay let's have a look at our next topic so our next topic is called distance time graphs last video 171 on COBRA Maps so here we've got a distance time graph and we've got the distance from home so it starts at zero so that means they're at home and it goes up in two so it goes two kilometers four six eight ten and so on and horizontally we've got the time so it starts at 8 AM and we've got 9 A.M 10 a.m and so on and we're given some information we're told Rosie jogged our friend's house or Rosie starts at home and she jogs to her friend's house and after having a short rest Rosie then jogged home arriving at half past 10 and we were told to complete the distance time graph so let's find out where half past 10 would be so if we look at 8 AM and 9am let's find out what half past would be so half past is one two three four blocks going from 8 AM to 9 A.M and after two of them would be half past here so if we look at 10 AM we've got one two blocks so that would be half past 10 there so that is 10 30. and we've been asked to complete the distance time graph while she jogs home so we now need to draw a line a straight line going from here down to half past 10 and it would look something like this okay Part B so Part B says how long did Rosie rest for so as you can see she rested for one block and the one when I say block that's a collection of five of the little small squares one two three four five it's where the line is slightly thicker so you should rest for one block so as we spotted earlier we knew the two of them was half past so this should be quarter past let's check so we've got eight o'clock eight a.m quarter past eight half last year quarter to nine nine o'clock so each one of the smaller blocks of those collection of five squares is 15 minutes so Rosie rested for 15 minutes okay and part C says when did Rosie jog at the fastest speed so in other words did she jog to her friend's house at a faster speed than whenever she jogged home if it's steeper on a distance time graph it means it's a faster speed and as you can see the line going to her friend's house is slightly steeper than the line coming back this is a steeper line than this one so the jog to her friend's house is faster than the jog home because the line is steeper and I've just written that down going to a friend's house because the line is steeper okay let's have a look at our next topic so our next topic our next compound Majors topic is density and that's video 384 in corporate Maps so here's part of the Court Mouse revision card and density is equal to mass divided by volume so if you want to find the density of something you do it's mass divided by its volume but we can rearrange this we can multiply both sides of this formula by volume so if you multiply both sides of this formula by volume you'll get density times volume is equal to the mass so if you want to find the mass of something you can multiply its density by the volume and finally if you divide both sides this formula by density you get mass divided by density is equal to volume so if you want to find the volume of a material you can divide its mass by its density and that'll give you its volume so let's have a look at some questions now so first of all we've been given a piece of metal has a volume of 50 centimeters cubed and a mass of 900 grams calculate the density of the metal so we want to find the density of the metal so the density is equal to mass divided by volume so if we want to find density we do mass divided by volume so the mass of the metal is 900 grams we've got a new 900 divided by the volume of the metal which is 50 and if we do 900 divided by 50 we'll find the density of the metal so 900 divided by 50 is equal to 18. so it's going to be 18. now we're dividing Mass which is grams by the volume which is centimeters cubed so the units for density will be 18 grams per centimeter cubed so that means the density of this metal is 18 grams per centimeter cubed okay let's have a look at our next example okay let's have a look at the next question so next question says a glass pip way has got a mass of 420 grams and the density of the glass used is 2.5 grams per centimeter cubed find the volume of the paper weight so the volume is equal to mass divided by density so what we want to do is we want to find the volume of the property so we want to do its mass divided by its density so the volume is equal to the mass which is equal to 420 divided by the density which is 2.5 and when we do 420 divided by 2.5 we will find the volume of this paper width so 420 divided by 2.5 is equal to 168 and our units will be centimeters cubed and that's it okay let's have a look at one more example so this time we've been told Kylie has a solid glass cube so she's got a solid glass cube the length for each side of the cube is three centimeters so the cubes get a sideline for three centimeters the density of the glass uses 2.5 grams per centimeter cubed and it says find the mass of the cube so mass is equal to density multiplied by volume so we want to find the mass of this Cube so we know it's density as density is equal to 2.5 and we want to multiply that by the volume so to find the mass we're going to do 2.5 the density multiplied by the volume but in this question it's not obvious what the volume of the cube is we're told the side length of the cube is three centimeters but we don't know its volume but we know enough information to work it out the volume of a cube is the length multiplied by the way multiplied by the height and because there's a cube all of the side lengths are the same so it's going to be three multiplied by three multiplied by three and three times three is nine times three is twenty-seven so the volume of the cube is 27 centimeters cubed so to find the mass of the cube we're going to do the mass is equal to the density multiplied by the volume so we're going to do the density 2.5 multiplied by 27. so 2.5 multiplied by 27 equals 67.5 grams and that's it so the mass of this Cube will be 67.5 grams okay let's have a look at our next topic so our next topic is pressure and that's video 385 in Code maps so pressure is equal to force divided by area so this is part of the quart Mass revision card and pressure is equal to force divided by area so if we divide the force by the area we'll find the pressure and if we rearrange this if we multiply both sides of this formula by area we would get the pressure times area equals force so that means that the force is equal to the pressure times the area and if we divided both sides of this equation by the pressure we'll find the force divided by pressure is equal to area so it's very important to remember the pressure is equal to force divided by area so here we've got our question and it says a box exerts a force of 4000 newtons on a table so box has been placed on a table and the exerts a force of 4000 newtons on that table and the area of the base of the box is 250 centimeters squared work out the pressure on the table in Newtons per centimeter squared so you want to find the pressure on the table so pressure is equal to force divided by area pressure is equal to force divided by area so if we divide the force which is four thousand Newtons by the area which is 250 centimeters squared that would tell us the pressure so 4000 divided by 250 is equal to 16. that's measured in Newtons because we had 4 000 Newtons and we're divided by 250 centimeters squared so it's going to be nutrients per centimeter squared and that's it so the pressure on the table is 16 Newtons per centimeter squared and that's it so if you want any extra practice on pressure if you go to that ultimate GCSE Foundation practice booklet there's a question out on pressure for you okay let's have a look at our next topic okay let's have a look at our next topic so our next topic is translations so we're given a triangle translate B the ship B five squares to the right and two squares down so in other words we just need to slide this shape five squares to the right and two squares down and whenever I'm doing a question like this I just do one corner at a time so looking at the corner of the top of this triangle here I'm going to go five squares to the right so one two three four five would be here and then two down would be one two so we'll be moved to here this corner at the bottom right corner we're going to move it five to the right one two three four five and two down one two so it's going to move to here and finally this corner of the triangle this corner of B we're going to go five to the right so one two three four five and two down one two we move it to here and then we just get a ruler and a pencil and we join those up and that's it so we've translated triangle B five squares to the right and two squares down okay so our next topic is translations last video 325 in corporate maps and a vector looks something like this we're going to have a number on the top and a number beneath and some brackets around it the number on the top will tell you how to move the shape horizontally and it'll tell you how many squares to the right to move it so for instance if it's an e it you'll move eight squares to the right if it's a five it'd be five squares to the right if it's a zero you won't move it any Square to the right it'll stay where it is horizontally if it's a negative so for instance if it was negative three you would move it three squares to the left if it was negative nine it'd be nine squares to the left so the top number tells you how to move the shape horizontally and if it's positive it'll be to the right if it's negative is to the left the number beneath that will tell you how to move achieve vertically so if it's positive here it's a one so that's going to mean move the shape one square upwards if it's a four you move it's four squares upwards and if it's negative you move it downward so if it's negative three you'd move it three squares downwards so here we've got a typical question and we've got a little shape it's a little t-shape and the ship's called C and we've been asked to translate shape c by and we've got minus one five that will mean one square to the left so one left because it's negative it's going to go to the left so one left and then we've got a five that's going to be five up so we're going to move the shape one square left and five squares up and that's it so next up is the rotations so here we've got rotate shape a 90 degrees anti-clockwise about the point two minus one so let's mark on the point two minus one so this is going to be our Center of rotation and we're going to rotate chip a 90 degrees anti-clockwise so going this way anti-clockwise about that point so get your tracing paper and put it on top of the center of rotation and the shape like so and make sure your tracing paper is straight so it's straight and it's not diagonal like something like that so make sure in this case I put it landscape so going that way and what you're going to do is we're going to draw over the center of rotation and we're going to draw over the shape a so we've drawn over the center of rotation and we've drawn over a shape a so now what we're going to do is we're going to put our pencil on the center of rotation and we're going to rotate the tracing paper 90 degrees anti-clockwise so when we do that we turn it and it's important to keep the center of rotation on that point returning it like so until we've rotated the tracing paper in 90 degrees anti-clockwise and we can now see the position of where a would go after we rotated at 90 degrees anti-clockwise about 2 negative one so now what I would do is I would just draw over this rectangle a couple of times just to make sure that whenever I move my tracing paper there's a sort of a light impression on the paper so it looks something like this and that's it so we've repeated a 90 degrees anti-clockwise about the 0.2 negative one let's have a look at our next question so this time we've been given a triangle and we've got a set of coordinate axes we've got our x-axis and we've got our y axis and the question says reflect triangle a so we've got this triangle a and we've been asked to reflect it in the y-axis in other words this is where the mirror is Okay so we've got this y axis and this is where the mirror is we want to reflect this triangle across to the other side so let's do each point of the times so this top point of the triangle it's one line away from the y axis so we need to go another one to the other side there the point down here it's one away from the y axis so we need to go one the other way and the bottom right hand point of the triangle it's one two three four away from the mirror line so we need to go another four one two three four so it'll be there and if we get a rule on a pencil and join those up we'll have reflected triangle a in the y axis and that's it okay let's have a look at our next question so we've got the same triangle and we've been taught this time to reflect triangle a and the x-axis so we've got this x-axis so this is where the mirror line is this time and we're going to reflect the triangle downwards so it's going to move down here somewhere so again let's look at each corner of the triangle so let's start off with this corner and it's one two away from the x-axis so we go down to number two one two to here this corner is one two away from the x-axis so we're going over to one two to there and this point up here it's one two three four away from the x-axis so we need to go down four one two three four down to there and if we get a pencil and join those up that'll be answer if we reflecting triangle a and the x-axis and that's it so we've been asked to reflect this ship the ship C in the line x equals negative one so this line x equals negative one is going to be a line going straight through negative one on the x-axis so if we go to the x axis and go to negative one here it's going to be a vertical line going straight through that negative one so that's where our mirror line is going to be so in x equals line will be a vertical line going through whatever number that is on the x-axis now we have to reflect the ship C in that mirror line so let's start off with this corner so this corner is one two to the mirror line so we need to go another two one two so it's going to move to here the bottom right hand corner of C well it's one two three four to the mirror line so we're going to go one two three four to here and the top of triangle sees here so it's one two three four to the mirror line so we need to go another four one two three four so it's going to be there and we get a ruler and pencil and we just join up that ship and we have a reflected C in the line x equals a negative one next we're going to reflects the ship's C and the line at y equals one so let's just draw about what we've done so we've got our ship C again and we're going to reflect it this time in the line Y equals one so what we're going to do is we're going to go to the y axis and the point one so the y axis and the point one and we're going to draw a horizontal line going through that point so going through one on the y axis so if you've got a y equals and a number it'll be a horizontal line passing through that number on the y axis and we have to reflect the shape this triangle in that mirror line so let's start with this bottom left hand corner of C it's one above the mirror line so let's reflect it to one below this point is one above the mirrorland so we're going to go another one downwards so there and finally at this point to get to the mirror line we go one two three down so let's go number three one two three to there and let's join up those points and that's it we need to be able to reflect shapes in the lines of Y equals X so the diagonal line going upwards this way and the line Y equals negative X the line going down this way here okay and it would look something like this so it's very important to know what the lines Y equals X and Y equals negative X look like so that whenever you're doing Reflections you can then draw them really quickly and easily now to reflect the ship in that line it's quite easy we just count the diagonals so let's look at each corner so let's start off by looking at this corner the bottom left hand corner of B and as you can see if we go diagonally to our mirror line it's one diagonal so we need to go another diagonal the other way so it'll move to here this point in the bottom right hand corner if we count the diagonals it's one diagonal and then we've got half a diagonal so if we go another one and a half so a half and then one would be here so we've reflected these two points now let's look at the points at the top of the rectangle so let's look at the top left hand corner that would be one diagonal two diagonals so we need to go in over two diagonals that'd be one diagonal two diagonals would be there and finally the top right hand corner of B would be up here so if we cut the diagonalize that with one diagonal two diagonals and a half so we go a half one and two and as you can see we've got a rectangle shift now we just need to get a ruler and pencil draw a nice rectangle and that's it's a reflected shape be a rectangle B in the line Y equals negative X okay so let's have a look at our next question so this time we've been asked in large B this rectangle by skill factor two using the origin as the center of enlargement so sometimes you could be given a grid like this set of axes and you could be given a center of enlargement as the origin so that's the point zero zero and our second largest ship so we've got the ship B and we've been asking larger by scale factor two that means it's going to become twice as big but also twice as far away from the center of enlargement this origin so let's start off with each corner so let's start off with the bottom left hand corner of the rectangle here and it was one square to the right and one square up but whenever we enlarge it by scale factor two it's instead of being one across and one up it's going to become two across and two up so two to the right and two up so it's going to move to here next let's look at the bottom right hand corner of the shape this corner and it was two squares to the right and one square up but whenever we enlarge bicycle Factor two it'll become four squares to the right and two squares up so instead of being two across and one up it'll be four across and two up so it's going to move to here next the top left hand corner of the rectangle here it was one square to the right and three squared up but whenever we in large breasts give Factor two it's going to move two squares to the right and six squares up doubling it so instead of being one to the right and three up it'll be two to the right and six up so it'll be one two one two three four five six it's gonna move to here and finally the top right hand corner of the rectangle it was two squares to the right and three squared up and whenever we enlarge by skill factor two it become twice as far away so it'll be four squares to the right and six squares up so that's one two three four and then six up one two three four five six so it's gonna move to here so let's get our return petal and join those points up and that's it so we've enlarged B by scale factor two using the origin as the center of enlargement so it's become twice as big but it's also twice as far away from this Center of enlargement so here's a typical question it says enlarge by scale factor a half using negative five negative five is the center of enlargement so we've got a fractional scale factor okay so let's have a good this question so this is a large breast scale factor half using negative five negative five is the center of a larger so let's find negative five negative five so that's our Center of enlargement there and we've got our original shape our object and we're going to enlarge a rescue a factor of half that means all the points in the image will be half the distance away from the center of enlargement so for instance we looked at this point at the top of the kite this point at the top of the kite is one two three four five six squares to the right and one two three four five six seven eight squares upwards so because we're using a scale factor over half we're going to half those distances so instead of going six to the right we're going to go three to the right one two three and instead of going eight up we're going to go four up so one two three four so the top of our count we'll move to here okay now let's look at the left hand side of the card here so this point is one two three four squares to the right and one two three four five six squares upwards so we're gonna half those distances so instead of going four squares to the right we're gonna do two squares to the right one two and instead of going six up we're going to go three up one two three so that means that the left hand side of the kite will move to here the right hand side of the kite so we can kind of guess it's going to move to this point here but let's check it so going back to our Center for enlargement it is one two three four five six seven eight squares to the right and one two three four five six squares upwards we're gonna half those so that'll be four to the right and three up so one two three four and one two three so that's the right hand side of the code and let's be careful with the bottom of the cape because we need to make sure we get the right height here so it is whenever we look at our bottom of our code go back to the center of enlargement it's one two three four five six squares to the right and one up so we're gonna have our distances so instead of going six to the right we're going to go three to the right one two three and instead I've got one square we're going to go half a square up so it's going to be there and then we're gonna learn our pencil and we'll join those up and that's it so we've enlarged this year by scale factor of a half so it's got smaller using the center for enlargement negative five negative five okay let's have a look at our next question so this time we've been given a smaller rectangle and it's been enlarged to get this larger rectangle and we've been asked to find the scale factor of enlargement in other words how many times larger have we made the ship so if we look at the dimensions of this smaller rectangle it's got a width of two and it's got a height of one two three whereas the larger one it's got a width of one two three four five six and it's got a height of one two three four five six seven eight nine so if we want to find the scale factor of a larger one we just need to see how many times bigger the sides have become so the width has gone from two to six that's three times bigger and also in terms of the heights the height has gone from three to nine that's also three times bigger so the scale factor of enlargement would be three okay let's have a look at our next topic so our next topic is parts of the circle so here we've got some parts of the circle that you need to know off by heart we've got the radius and the radius is the distance from the center to the edge of the circle we've got the diameter that's the distance across the middle of the circle so through the center you've got the circumference that's the distance around the outside of the circle and you've got the chord the chords align the joins one part of the circle to another part of the circle now let's look at some more parts of the circle so we've got the arc which is part of the circumference we've got a tangent which is a straight line that touches the circle once and Carries On we've got a sector which is part of the circle this green region here or I like to think about it in terms of a slice of pizza and then we've got a segment and so a segment is if you've got that chord that goes from one part of the circle to the other part of the circle and this segment is one of those regions such as this okay so it's very important to know these parts of the circles as well and if you've got the chord Master vision card on that that's very useful the next topic we're going to look at is finding the circumference of the circles that the circumference is a fancy name for the perimeter of the circle it's the distance from are on the outside of the circle and the circumference of a circle is found by working out pi times diameter so if you multiply Pi by the diameter the distance across the circle going through the center that will tell you the circumference of a circle so it's very important to know where Pi is on your calculator and that's if you look down here on my calculator I've got a yellow Pi there so because it's in yellow that means I need to press the shift button first of all and then press the button here just beneath the pi symbol and that then will bring up Pan the calculator so in this question we've been asked to find the circumference of a circle that has got a diameter the whole way across of 14 centimeters so the circumference of the circle is pi times diameter so it's going to be pi times the distance across the circle which is 14. so that's going to be pi times 14 so we're on our calculator we press shift and then the pi button this one here it'll bring up Pi then you'll do times 14 and then press equals and it will tell us the circumference of a circle now sometimes it will come up like this 14 Pi depending on the mode of your calculator so you'd press the SD button there and then it will tell you the circumference of a circle and here we've got the circumference of the circle would be equal to 43.98229 and so on centimeters and I'm going to round this to two decimal places so I'm going to write 43.98 centimeters so the circumference of the circle would be 43.98 centimeters and always remember your units the circumference is the perimeter is the distance around the outside of the circle so if the diameter is 14 centimeters the circumference will be measured in centimeters also okay and also we might be given a circle like this where we've got the radius is 5 Centimeters so remember the circumference is pi times diameter so the diameter is the whole way across Circle so if the radius is 5 that means the diameter of the circle would be 10 so we would do circumferences pi times diameter so we'd do pi times 10 and then you would just work out what that is okay our next topic is the perimeter of a semicircle so here we've got a semicircle you can see it's got a radius the distance from the center of the circle if it was a full circle to the edge is three centimeters so the whole way across would be six centimeters and we've got this Arc at the top and we want to find this distance from here around to here and that will be half of a full circle because it's a semicircle so if I treat this as a full circle to begin with and I do pi times down which I get the circumference of the whole circle and then if we have and you'll get just this distance here just the length of this Arc at the top so we can then do six plus whatever that is so we're going to need to use the circumference formula so the circumference is equal to Pi times diameter so we're going to do PI on our calculator Times by the diameter of the circle and the diameter is six here the whole way across so we're going to do pi times six that's equal to 18.849 nine five five five nine two and so on and it's important not to round this because we're going to be doing more calculations with this and so you may just want to keep it on your calculator display so then we can do the next stage so whenever you do PI time 6 and press equals keep that on your calculator display so that's the circumference of the whole circle now we're dealing with a semicircle which is half of a circle so if we take that 18.849 and so on I'm just going to do some dots so that I don't have to write the whole thing down and if I divide that by two so in my calculator I've already got that on my calculator if I press divided by 2 so it comes up with answer A and S divided by 2 on my display and press equals I get it says 3 pi to begin with and then I press the SD button and that gives me 9.42477961 and so on so that 9.424 so on centimeters would be the distance from here all the way around the semicircle to here so that distance from here around to here would be 9.425 four seven seven and so on centimeters now we want to find the perimeter of the semicircle so that's going to be the length of all the sides so we've got the base and this Arc so we're going to do 9.4247 and so on and just keep it on your calculator display plus and we've got the base now the base of the shape is six centimeters we've got our three and a number three so plus 6 gives us 15.42477 and so on now I'm going to round this to two decimal places so whenever I round this to two decimal places that would give me 15.42 centimeters as perimeter and we're dealing in centimeter so our units were centimeters and that's why I use centimeters and that's it so the next topic is arc length and the formula to find the Arc Length so the length from here round to here is given by Vita the angle over 360. multiplied by pi multiplied by the diameter and this is video 58 in corporate maps and this is part of the core modular vision card and our question asks is to find the perimeter of this sector so here we've got a sector it's got a radius of 8 so the distance from the center to the edge of the circle is eight centimeters and we want to find its perimeter so we're going to do 8 plus 8 plus whatever this arc length is so the arc length is Vita so the angle is 14 degrees divided by 360. multiplied by pi and now we're going to multiply by the diameter of the whole circle so if we did have a whole circle here the diameter would be 16 the whole way across so multiply by 16. and that gives us 1.9547 and so on centimeters so we've been asked to find the perimeter of the sector so we're going to do 8 plus 8 plus 1.9547 and so on and we're going to get our answer so I've already got this on my calculator display so I'm just going to press plus 8 plus 8 and that gives me 17.9547 and so on centimeters or if I round it to two decimal places the answer would be 17.95 centimeters to two decimal places and that's it our next topic is the area of a circle and if you want to revise this topic and called Mouse and watch the film videos video 59 so the area of a circle is given by the formula pi r squared I remember our order of operations use Square you do any orders before you do any multiplications so what it means is we're going to square the radius and multiply it by pi so here we've got a circle and we want to find the area of this circle it's got a radius of six centimeters the distance from the centers to the edge is six centimeters so the area is equal to pi r squared so it's going to be area is equal to Pi times the radius 6 squared so what we could do is on our calculator we're just going to type this in High Times 6 squared the great thing is your calculator will know the order of operations whenever you type in pi times 6 squared and I'll work it out for you um alternatively you could do 6 squared which is 36 and then do pi times 36 and you get the same answer but I'm just going to type this in I'm going to type in pi so shift and then the pi button here and then that'll bring up pi and then Times by so multiply it by six and then squared pressing the square button and then I'm going to press equals my answer would be 113.0973355 so the area for the circle would be 113.973355 I'm going to run this to one decimal place so 113.1 centimeters and it's squared because this area that's it now if we were given the circle like this where we knew the diameter being 20 centimeters well it was pi r squared so it's pi times the radius squared so if you know the diameter you might need to half it so in this case the radius would be ten so do pi times 10 squared okay so let's have a look at finding the area of a semicircle so that's video number 47 on corporate Maps so here we've got a semicircle and using a similar approach to finding the perimeter of a semicircle we're going to consider the whole circle so here's a whole circle um yep there we go there's a filled Circle and you can see the radius of the whole circle is three centimeters so what we're going to do is we're going to find the area of the whole circle and then divide it by two so we're going to do pi times 3 squared because that'll give us the area of the whole circle and when we do that we get 28.27433 so on and then what we're going to do is we're going to divide there the whole circle by two so we'll take our 28.27433 so on and we'll divide that by two and we find that that's equal to 14.1371661 or 14.1 to one decimal place okay let's have a look at our next topic so the next topic is the area of a sector which is video 48 in corporate maps and just remember you do have those practice questions in the link below so if you do want to practice any of these questions as we're going through there's questions there for you to try Okay so for the area of a sector here's part of the corporation card so look at the area of the sector is Vita over 360. so the fraction of the circle Times by High Times by the radius squared so here we've got a sector and again it's got an angle which isn't necessarily a nice angle uh we've got an angle of 109 degrees so we're going to have 109 over 360 times by pi multiplied by the radius of the circle so if we had a whole circle here the radius would be 0.7 centimeters so no 0.7 squared and when we do that we get that's equal to 0.46609 and so on centimeters squared I'm going to round this to three decimal places so I'm going to write 0.466 centimeters squared to three decimal places and that's it okay so our next topic is Pythagoras and here's part of the chord Mass revision card on Pythagoras is theorem which is a squared plus b squared equals c squared where n b are the two shorter sides and C is the line for the longer side of the hypotenuse of the right angle triangle so here's a right angle triangle and we've been asked to find X the length of the hypotenuse the longest side so I label both the sides first also I label the shorter side a the next shortest B and the longer said c now in terms of A and B it doesn't actually matter which way around your label those as long as a and b are the two shorter sides and we're going to substitute those values into pythagorasis here so Pythagoras Theorem is a squared plus b squared equals c squared so if we substitute these values in well instead of a squared we're going to write 5 squared instead of B squared we're going to write 12 squared and instead of c squared we're going to write x squared so we know that 5 squared plus 12 squared equals x squared so now we're going to work out 5 squared and 12 squared so 5 times 5 is 25 and remember this is a calculator paper so you can't just write 5 squared plus 12 squared and press equals on your calculator plus and 12 squared is 144 and that equals x squared now 25 plus 144 is 169 so that gives us 169 equals x squared so we've got 169 equals x squared now obviously X is the length of the side here so we can square root both sides so it gives us the square root of 169 equals x and the square root of 1069 is 13. so 13 equals x that means that x equals 13 centimeters and that's it so that means that x equals 13 centimeters okay let's have a look at a right angle triangle where we're trying to find the length of one of the shorter sides so here's a right angle triangle and again Pythagoras is theorem is a squared plus b squared equals c squared so let's label our sides now we don't actually know which ones the shortest here so I'm just going to label the side the X okay I'm going to label the five centimeters B and obviously the longer side will be the hypotenuse the side opposite the right angle so that will be C so we write down Pythagoras is theorem that's a squared plus b squared equals c squared and so a squared about is going to be x squared plus and instead of B squared that will be 5 squared and that equals c squared which is 8 squared so x squared that's x squared plus I'm working out 5 squared well 5 times 5 is 25 so that's 25 equals 8 squared which is 64. now here we've got x squared plus 25 equals 64. so we want to get the x squared on its own so let's take away 25 from both sides so it gives us x squared equals and 64 take away 25 equals 39. so we've got x squared and to find X we're going to square root 39 so we'll work out the square root of 39 and that equals 6.244997998 and so on centimeters so let's round this to two decimal places that means that x equals 6.24 centimeters and that's it so Pythagoras Theorem is really useful for right angle triangles to find the length of missing sides if you know two of them and perfectionism is also useful in situations where you've got shapes that are made up of right angle triangles so for instance if there's a rectangle and it's cut across diagonally you may need to use Pythagoras's Theorem to work out the lengths of the sets there and so on okay let's have a look at our next topic so with Pythagoras we're dealing with right angle triangles and we're using the length of two sides to find the length for the third side with trigonometry we're going to be dealing with right angle triangles again but this time we're going to be involving angles we're either going to be using two sides to work out the size of one of the angles alternatively we would be using an angle and one of the sides to work at the length of another side and it's very important whenever you're dealing with trigonometry to know the trigonometric ratios or the trig ratios and they are sine is equal to opposite divided by hypotenuse the COS is equal to the adjacent divided by hypotenuse and that the tan is equal to the opposite divided by adjacent and some students remember sand such as Soca tour or I like to remember two old Angels skipped over Heaven carrying a harp so it's very important you remember these trig ratios whenever you're answering a trigonometry question so let's have a look at one now so here we've been given a right angle triangle we've been asked to find the size of this angle X and it's a right angle triangle we know that because it's been marked on and we've been given the length of the hypotenuse is 10 centimeters and we've been given the length of this side as eight centimeters and we've been asked to find the size of this angle so the first thing I do on a trigonometry question is to write down the trig ratios and they are two old Angels skipped over Heaven carrying a harp so two old Angels skipped over Heaven carrying a harp or Soca tour and they are the tan is equal to the opposite divided by adjacent sine is equal to the opposite divided by hypotenuse and that the COS is equal to the Json divided by hypotenuse now let's have a look at our triangle and figure out which trig ratio we're going to use in this question and I do that by labeling the sides as the opposite hypotenuse and adjacent so let's start off with the opposite so the opposite is the side opposite the angle we're using or trying to find in the question so we're trying to find this angle so the side opposite that is this one so this is the opposite then let's label the hypotenuse that is the longer side of a right angle triangle and it's a set opposite the right angle so this is the hypotenuse and that means the side that we've got left is the adjacent and it's adjacent to the angle we're trying to find but it's not the hypotenuse of that stage isn't so that is our opposite hypotenuse and adjacent so we've labeled the three sides of the right angle triangle so now we've labeled the triangle as the opposite hypotenuse and adjacent now we need to figure out which trigger issue we're going to use in the question so here we've been given the hypotenuse is 10 centimeters we've been given the adjacent is 8 centimeters so we're going to use those two and the opposite will not be given and we're not trying to find out so we're just going to cross it off we're not going to be using any trigger issue this involving the opposite so we're not going to be using the tan and we're not going to be using the sign we're going to use in the cause and this question the cos x is equal to the adjacent divided by the hypotenuse that's the trig ratio we're going to use in this question so let's substitute in the values for the adjacent and the hypotenuse into our trig ratio so we've got the cos x is equal to the adjacent which is 8 so 8 divided by the hypotenuse which is 10. so we've got the COS of the angle X is equal to eight tenths so we know that the COS of the angle is equal to 8 over 10 or no point in but this angle obviously isn't 0.8 degrees this angle is much bigger so what we want to do is we want to find out what x is so we want to get rid of this cos so we're going to do the inverse cos so the opposite of cos to both sides of the equation so whenever we do the inverse cos to both sides on the left hand side we'll just be given X the angle we're trying to find and then we want to do the inverse COS of eight temps now on your calculator just above the cause in yellow you've got a COS with a little minus one that's the inverse cause so we would then press shift and then the cause button and it will come up as cos with the little minus one and then you just type in 8 over 10 or 0.8 and then close brackets and press equals and that will give you the size of the angle and we find that X is equal to 36.86 nine eight nine and so on degrees and let's just run that to two decimal places that'll be the x is equal to 36.87 degrees and that's it so in this question we're asked to find the size of this angle X so my steps were first of all I wrote down the trig ratios then what I've done was I looked at my right angle triangle and I labeled the side so I label the side opposite the angle as the opposite the hypotenuse as the hypotenuse H and the adjacent as a then we knew we weren't dealing with the opposite in this question so we crossed off any of the trig ratios involving the opposite so we know it was a cause question we know that the COS of X the COS of this angle is equal to the adjacent divided by the hypotenuse so we found that cos x was equal to eight temps this angle obviously isn't the attempt so we need to do the inverse because of that so we press shift cos and then type in attempts and press equals and we get the size of this angle which is 36.87 degrees okay let's have a look at our next question so this time we're going to find the lengths of one of the sides so we've got this right angle triangle and we know that the hypotenuse of this right angle triangle is four kilometers and the size of this angle is 60 degrees we've been asked to find the size of X so again let's write down our trig ratio so we've got two old Angels skipped over Heaven carrying a harp so here we've got a right angle triangle let's label the sides so we have got the opposite so in the question we've been given the 60 degrees so the side opposite it is X so that's our opposite then we've got our hypothesis which is the side opposite the right angle so our four kilometers is the hypotenuse and the third left is the adjacent so this side is our adjacent next we cross off any side that has not been given or looked for in the question so in this question we're looking to find the opposite we've been given the hypotenuse and we don't need the adjacent order given the adjacent so we can cross it off so we then look at our trig ratios and we cross off any trig ratio that involves digestion so we're not using tan and we're not using the cause so in this question it's a sign question that sine X is equal to the opposite over a hypotenuse so let's substitute in our values so we've got the sine of our angle so that's going to be sine 60. so sine 60 is equal to the opposite which is equal to X and this question divided by the hypotenuse which is 4. so we've got sine 60 is equal to x divided by 4. so we want to find out what x is so we don't want this divided by 4 and this right hand side of this equation so let's multiply both sides of the equation by four so we'll get sine 60 times 4 so that's going to be sine 60. times 4 and that's going to be equal to and on the right hand side of the equation we had x divided by 4 we Times by 4 to get rid of the divided by 4 so we're just going to be left with X so we've got the x is equal to sine 60 times 4. so let's work that out so on our calculator press sine 60. now make sure you close brackets so close brackets multiply by 4 is equal to 3.4641 and so on kilometers and that's equal to X so X is equal to 3.464 kilometers to three decimal places and that's it so X is equal to 3.464 kilometers to three decimal places okay let's have a look at our next topic so our next topic is exact trig values and it can be useful at GCSE Foundation to remember some of your exact trig values so in other words if you were to work out the sine of zero degrees that's equal to zero the sine of 30 degrees that's equal to a half that's one I remember off by heart the sine of 30 degrees is equal to half the sine of 45 degrees that's equal to root 2 the square root of 2 divided by two the sine of 60 degrees is equal to the square root of 3 divided by 2 and the sine of 90 degrees is equal to one so it can be useful to remember those the COS of zero degrees is equal to one the COS of 30 degrees is equal to the square root of three divided by two so the COS of 30 degrees is equal to the sine of 60 degrees they're the same as each other because of 45 degrees is equal to the square root of 2 divided by two the same is the sine of 45 degrees and the COS of 60 degrees is equal to a half and I always remember that one as well always remember the sine of 30 degrees is equal a half and the COS of 60 degrees is equal to half and the cause of 90 degrees is equal to zero and finally tan the tile of zero is equal to zero the tan of 30 degrees is equal to the square root of three divided by three the tan of 45 degrees is equal to one and the tan of 60 degrees is equal to the square root of three and the tan of 90 degrees is undefined there's no answer for that okay so we want to write down the values of these so the sine of 30 degrees would be equal to a half the ton of 45 degrees would be equal to one and the COS of 30 degrees where the COS of 30 degrees is equal to the square root of 3 divided by two and that's it okay our next topic is congruence and you need to know what the word congruent means congruent a congruent means exactly the same shape and size so on this grid we've got some shapes that are congruent to each other so shape a is congruent to I so a and I are congruent to each other we've got some more shapes that are congruent we've got B and D B and D are congruent so b and d f and H would be congruent because they're right angle triangles and they've both got a height of two and they've both got a base of two so hit so F and H congruent as well so congruent means exactly the same shape and size so it means all the sides are the same length but also all the angles are the same size as well so we've looked at congruent shapes now let's have a look at similar ships so similar ships is where one ship is an enlargement of another so if you've got one shape and another ship that's similar to it then it's an enlargement of it so whenever you've got shapes that are similar such as this rectangle air and this rectangle B we know that to find the lengths we multiply by a certain scale factor now notice that the sides will become so many times bigger or smaller but the angles will all stay the same because obviously this shape to stay in the same ship so here we've got the core Mountain revision card part of it and we've got here two similar rectangles so we've got this rectangle rectangle air which is six centimeters long and four centimeters wide and we've got this rectangle B which is similar to it so it's an enlargement and it's got a width of 12 centimeters and a length of well we need to find that we've been asked to find the length of rectangle B so let's look at the sides so let's look at the widths of these rectangles we've gone from four centimeters to 12 centimeters so to find the scale factor of enlargement if we divide 12 by 4 we will find the scale factor of enlargement so if we do 12 divided by 4 12 divided by 4 is equal to three so that means that this width is three times larger than this width so multiply by three so if we multiply the width of rectangle a by three that means we're going to multiply the length of rectangle a by three as well and that will give us the length of rectangle B so if we do six multiply by three that's going to be equal to 18 centimeters so the length of rectangle B would be 18 centimeters so similar ships are where one ship is an enlargement of another that's it okay let's have a look at our next topics our next topic is congruent triangles and that's video number 67 on corporate Maps so congruent amounts we've looked at that previously at precongruent ships so congruent means identical it means the same shape and the same size so it means that if you've got two ships that are congruent it'll mean that the angles are the same and the sides are the same length as well now whenever we're dealing with triangles in a triangle we've got three angles and there's three sides now if you've got two triangles if you want to see if they're congruent to each other you don't actually need to know all six angles in both triangles and the lengths of all six sides what you can actually do is know if they're congruent by just know some of that information and these are the conditions to know if triangles are congruent or not so the first condition is what we call side side or SSS and what that means is that the sides are the same size so here we've got five centimeters and five centimeters then we've got seven centimeters and seven centimeters and nine centimeters and nine centimeters so if you've got two triangles that both of the same lengths of the sides so side side and side those two triangles will have to be congruent to each other so they will be the same shape and size okay let's look at the next condition so the next condition is what we call angle side angle or ASA and what that means is if you've got two angles and you know the side in between them for two triangles those two triangles will be congruent to each other so for instance here we've got 70 degrees 30 degrees and in between those is four centimeters so if you draw that triangle and then if you go to another triangle where you've got 30 degrees and 70 degrees so the same two angles and four centimeters in between them those two triangles will have to be congruent to each other because there's only one possible triangle that will have 70 degrees 30 degrees and four centimeters in between them so it's called angle side angle or angle side angle okay let's look at the next condition the next condition is SAS or side angle side so that means if you've got two sides and you know the angle in between them and they're the same for two triangles those triangles will be congruent so as you can see here we've got 12 centimeters and 14 centimeters and 30 degrees in between them and then this triangle here we've got 12 centimeters and 14 centimeters and 30 degrees in between them so these two triangles will have to be congruent to each other because there's only one possible triangle you can draw which has 12 centimeters and 14 centimeters and 30 degrees in between them so those two triangles will be congruent okay in another condition is what we call rhs which stands for right angle hypotenuse side so if you've got two right angle triangles and you know the hypotenuse and one of the shorter sides those two triangles will have to be congruent to each other because if you use Pythagoras film you can find the length of the third side and then it would be side side so these two triangles would have to be congruent to each other so this is the chord Master revision card and it's very useful to remember the conditions for congruent triangles whether it's side side side angle side angle side angle side or rhs and that's it okay let's have a look at our next topic okay next topic is to find the volume of a cuboid so here's a cuboid and we may need to find the volume of it and the volume of a cuboid this is from the core mileage revision card the volume is equal to length times width times height so if we look at this cuboid the length and the width and the height are given to us which is seven two and three I'm going because we're just multiplying them together you don't need to be really strict at what this was definitely the length on this one so we're from this one's the height because if we're just multiplying the three numbers it will give us the volume anyway so the volume is equal to the length which I'm just going to call Seven multiply with the width I'm going to call 2 and multiply by the height that's three and seven times two times three would be equal to 42. so our volume is 42 centimeters now it's volume so we measure that in centimeters cubed so the volume of this cuboid would be 42 centimeters cubed okay so let's have a look at our next topic so our next topic is the volume of a prism so to find the volume of a prism we find the cross-sectional area so whenever you've got a prism it's got that constant cross section so it's got that shape that's the same the whole way through it so for instance this triangular prism has a triangle as the cross section and it just sits the same the whole way through it so you get the area of that shape and you multiply that by how long it is so to find the volume of this triangular prism we're going to find the area of the triangle at the front and then just multiply it by four so to do that let's work out the area for the triangle so therefore triangle is half the base times the height so there this triangle would be half the base so half times seven multiplied by the height five so whenever we do that we get 7 times 5 is 35 and multiply by half we're dividing it by 2 would give us 17.5 meters squared so the area of this triangle at the front is 17.5 meters squared now we just need to multiply by how long the shape is so and the shape is four meters long so if we do 17.5 multiplied by 4 that gives us the volume of this triangular prism so multiply it by 4 equals 70 meters and the units would be cubed because it's the volume so 70 meters cubed so Define the volume of a prism you get the area for the cross section and then multiply by how long it is or alternatively it's standing upright you multiply by how tall it is our next topic is the volume of a cylinder and as you can see this cylinder standing upright so we're going to find the area of the circle and then multiply by how tall it is so the area of a circle area equals pi r squared so that's equal to High Times the radius so this is the radius it so pi times eight squared and when we do pi times 8 squared we get that's equal to 201.0619298 and so on centimeters squared so that's the area for the top now we're going to multiply that by how tall it is don't round this answer because we're multiplying it by 30. so if you round it that multiply by 30 you can you know that you're somewhere out from the actual answer so we're going to take our 201.0619 and so on if you've got a new calculator display just leave it there and press multiply by 30 and whenever you do that you get times where 30. 6031.857 and so on and one decimal place that would be 6031.9 and our units would be centimeters cubed and that's it okay so the next topic is looking at given our answers in terms of Pi now I've previously looked at topics such as circumference and over Circle volume of a cylinder and so on and in all of those questions so far we've been given answers as decimals now I've actually hinted in previous videos that we could actually give our answer in terms of Pi and that's working exactly so rather than given a decimal we're given an exact answer in terms of Pi so let's have a look at a typical question so we've got a cylinder let's get a diameter of eight centimeters so we've got a cylinder so for magic we've got a cylinder and let's get a diameter of eight centimeters so that means that the width of it is eight centimeters so the whole way across the circle at the top or the bottom is eight centimeters and the height of the cylinder is 15 centimeters so the height is 15 centimeters so we've now got our sketch now what we need to do is work out the volume of the cylinder and terms of Pi and what that means is rather than given our answer is a decimal number we're going to give it as perhaps something like 100 Pi or 75 pi and that means we've got an exact answer so rather than actually having to work it out and then rounding it we can actually just give it in terms of pack so the volume of a cylinder so we will find the area of the cross section so in this case the circle and then multiply by how tall the shape is so let's get the area of the circle so the circle let's get a diameter of eight so the radius of the circle is four centimeters and the area of a circle is pi r squared so pi times 4 squared now 4 squared is 16 so that means it would have pi times 16 and then if we just put them together like an algebra where would we write 16y or 16x we're just going to write that SQL to 16 Pi we're just going to put the number in front of Pi so it means the area of the circle is 16 Pi centimeters squared so that's the area of the circle and rather than rounding that as a decimal number and then multiplying it by 15 and so on what we're just going to do is use that 16 pack so Define the volume of a cylinder we'll find the of the circle now we need to times about how tall or how long the shape is so the height of the cylinder is 15 centimeters so to get the volume we're going to do 16 pipe there if the circle times about how tall the shape is which is 15. so what we're going to do is we're going to do 16 times 15 and then get our answer and just put Pi after it so 16 times 15 is equal to 240 and then Pi so answer would be 240 Pi centimeters cubed and that's it so we've left our answer in terms of Pi and our answer would be 240 Pi centimeters cubed so we're not actually having to work out with 240 times pi is and then write that as a decimal number we can actually just give it in terms of power and this is particularly useful on non-calculated papers our next topic nice to find the volume of a cone the volume of a cone is one third pi r squared H that means we're going to find a third of Pi times the radius squared multiplied by the height of the cone so in this cone we've got a radius of 20 and we've got a height of 21. so we're going to do one third multiplied by pi multiplied by the radius which is 20 squared and then multiply by 21 the height of the cone and when we do that we get an answer of 2800 Pi which would be 8796.46 centimeters cubed to two decimal places and that's video 359 according math so the volume of a cone is one third pi r squared hitch here's a pyramid and if you've been given a pyramid and you've asked to find the volume of it again the volume is equal to third the area the base times the height so it's going to be the volume is equal to Third times the area of the base well it's going to be nine times six so nine times six is equal to 54. so if it times 54 times its perpendicular height so that's going to be from the base straight up to the top which is seven centimeters and whenever we work that out we get that's equal to 126 centimeters cubed our next topic is the volume of a sphere the volume of a sphere four thirds pi R cubed and this is part of the code modular vision card here and of course Mazda's video 361 and here we've got our sphere and it's got a radius of 14 centimeters so to find its volume we will do four thirds multiply by pi multiplied by the radius which is 14 cubed so I'm going to press the fraction button on my calculator type in four thirds I'm then going to press multiply by and then pi times by 14 and then the cubed button eleven thousand four hundred and Ninety Four 0.04 centimeters cubed to two decimal places and that's it this time we've been asked to find the surface area and here's a cuboid and we've been asked to find the surface area of a cuboid and the surface area is the area of all the faces of that particular ship so this is a cuboid so I'll have six faces you've got the top which is green you've then got the bottom which would be the same size it would have the same area you've then got this rectangle on the right hand side here in blue and that would be the same as the rectangle on the left hand side and you've got the rectangle at the front the fist the front this red one and that'll be the same as the back so what we've got is we've got six rectangles we need to find the area often we need to add them all up now the great thing is that you've got pairs of them that are the same the top and the bottom the right the left and from the back are the same so if we work out the area of each of these ones so what we can do is then we consider two of them and then add them all up let's call the front one the right hand side two and the top three let's consider the front to begin with so for first one we've got the width of the rectangle is it multiplied by the head of the rectangle which would be five because here we've got a height of five so that means that the height here of this rectangle is five so we've got times five and eight times five is equal to 40 centimeters squared so the area of the front is 40. that means the area of the back would also be 40 as well so I'm just going to write it there as well and then two we've got the right hand side here if we look at this rectangle it's got a length of seven and a height of five so we're going to do 7 multiply by 5 and 7 times 5 is 35 so that's 35 centimeters squared so that means the area of this rectangle on the right hand side is 35 and that would be the same as the rectangle on the left hand side so there's another 35 centimeters squared and then the top which is face three we've got three that has a width of it and a length of seven because you've got it here so that's going to be it there and the 7 here will be there so we've got eight times seven and here times seven is 56 centimeters squared and the top would be the same as the bottom so we've got another 56 centimeters squared so to find the total surface area of this cuboid we're going to do 40 plus 40 plus 35 Plus 35 plus 56 plus 56 and that would tell us the total surface area of this cuboid and when we do that we get a total surface area of 262 centimeters squared so the surface area is in the area of all the faces of that 3D ship okay let's have a look at our next topic so our next topic is the surface area of a cylinder and that's video 315 on corporate Maps now here we've got a cylinder and we've been asked to find the surface area of the cylinder now this cylinder has three faces it's got a circle on the top a circle on the bottom and it's got this curved face going around the outside now there if the circles will be straightforward we'll do pi r squared for the top and then we'll do power squared for the bottom and they'll also be the same area so you could just do power squared for the top and multiply it by two or add it to itself now for this code fee it's going to run the outside if we were to straighten that out that would be a rectangle and the height of the rectangle would be six meters and the length of the rectangle that would be the circumference of the circle so we're going to get the circumference of the circle and multiply it by 6 the head of the cylinder and that would give us the area of the curved face and then we'll add together the two circles and the curved face so let's do that now so let's start off by getting the area of the circle so that would be pi times 5 squared and that will give us the area of the circle which is 78.53981 and so on then we've got the circle at the bottom and that'll have the same area so we'll have a circle beneath and that'll have the same area and then finally we need to find the area of this curved face going around the outside of the cylinder and to find that we're going to find the circumference of the circle so we'll do pi times diameter so we'll do pi times 10 because the diameter of the circle will be 10. so pi times 10 will be 31.4159 and so on and then we'll take the circumference of the circle and multiply it by the height of the cylinder and that will tell us the area of the curved phase so we're going to do 31.4159 and so on and I've got that in my calculator display so I'll just press times six and that will give me 188.4955 and so on so I'm going to add together now the area of the circle on the top the 78.53981 and so on I'll add another Circle to 78.53981 and so on and I'll also add the area of this face the curve face which was 188.4955 and so on and when I do that I get that's equal to 345.575 to three decimal places and then the units because of surface area and its meters will be meters squared and that's it okay let's have a look next topic so the next topic is a surface area of a cone so here we've got a cone and to find the surface area of a cone we have to get the area of the two faces and that's the area of the circle at the bottom which will be pi r squared that's straightforward and then we need to find the curved surface area so the area for the top of the cone and that's given by the formula curved surface area is pi RL where Pi is obviously pi r is the radius of the base of the cone so that will be six and then L is what we call the slant height so this diagonal length here of 11 centimeters so let's find the surface area of this cone so for this cone we would do pi r squared so pi times 6 squared for the base and when we do pi times 6 squared we get that's equal to 113.0973355 and so on and then just area of the base now we need to find the curved surface area so it's going to be pi times the radius 6 times the slant height which is 11. and when we do that we get that's equal to 66 Pi or 207.34 five one one five one and so on and when we have those together we'll find the surface area of this current which would be 320.44 centimeters squared to two decimal places and that's it okay our next topic our next topic is the surface area of sphere and so the surface area is 4 pi r squared so here we've got a sphere and we're going to find a surface area so we're going to do 4 multiplied by pi multiply by the radius which is 5 squared so we're now just going to work this out on our calculator so we're going to do 4 multiplied by pi multiplied by 5 squared and that gives us 100 Pi which is equal to 314.16 centimeters squared to two decimal places and that's it so our next topic is convert metric units for area and volume so earlier on on the video with converted metric units for length now let's look at what happens whenever we convert the metric units for area and volume so let's start off with converting metric units for area so here we've got a rectangle and it measures three meters by two meters so three meters long and two meters wide and here's the exact same rectangle and instead of writing three meters I've written 300 centimeters because it's 100 centimeters in a meter so three meters would be 300 centimeters and instead of writing two meters I've written 200 centimeters because obviously 2 times 100 would be 200 centimeters now let's find the areas of these rectangles because they're identical the areas will be the same so to find the area of this rectangle we multiply the length from the width together so we're going to do three times two and three times two is equal to 6 meters squared so the area of this rectangle is six meters squared now if we look at this rectangle this rectangle is 300 centimeters long and 200 centimeters wide so to find this area we're going to multiply these together so 300 times 200 about 3 times 2 is 6 and then add one two three four zeros one two three four zeros and then that's measured in centimeters squared so we've got the same rectangle and its area will be six meters squared or sixty thousand centimeters squared so that means that six meters squared is equal to sixty thousand centimeters squared or if we divide both of these by six we get one meter squared is equal to ten thousand centimeters squared so in one square meter there'll be ten thousand square centimeters and that makes sense because if you get a square which was one meter by one meter that would be 100 centimeters by 100 meters that'll be 100 rows of 100 which would be ten thousands little smaller centimeter squares in there that's it so if we want to convert between meter squared and centimeters squared we can multiply by ten thousand or I like to multiply by 100 I multiply by 100 again because it's squared so I know to multiply by 100 twice and that's it okay let's have a look at our next one converting metric units for volume so here we've got two cubes one of them's got a side length of two meters and one's got a side length of 200 centimeters so these cubes are identical so if we know the side length was two meters we can write two meters for the length two meters for the width and two meters for the height so to get the volume of this Cube we're going to do the length times the width times the height so we're going to do two times two times two which is equal to eight meters cubed so the volume of this cube is eight meters cubed so this Cube's identical so it will have a length of 200 centimeters a width of 200 centimeters and a height of 200 centimeters so if we do 200 multiplied by 200 multiply by 200 that will give us the volume of this Cube and the volume of this cube is equal to 2 times 2 times 2 is equal to eight and then we're going to add on six zeros one two three four five six so eight million centimeters cubed so eight meters cubed would be equal to 8 million centimeters cubed and if we divide both of these by eight we get one meter cubed is equal to one million centimeters cubed and that's it so that's going to be really useful for converting between meters cubed and centimeters so you can multiply by a million to convert between meters cubed and centimeters cubed and likewise convert back from centimeters Cube to meters cubed so you can divide by a median I like to multiply by 100 and by 100 and by 100 again and divide by 100 100 and 100 again if I want to convert between them and that will give me the same answer okay let's have a look at our next topic so the next topic is vectors now Vector is something that's got a direction and has got a size or another name for size is a magnitude so it's got a Direction so we've got a certain way and has got a magnitude or a size so it's got a certain length so these lines are all vectors they've all got they're all going in a certain direction shown by their arrows and they've all got a certain magnitude or size that's its length and we've already looked at vectors whenever we're dealing with translations because we move the ships we slid the ships and we use the column vector and it showed us how many squares to the left or the right and how many squares up or down and what we're going to do is we're going to represent these vectors as column vectors and we're going to use the same approach where the number on the top will be how many squares left or right and the number in the bottom will be how many squares up or down and if the top number is positive it's to the right and if it's negative it's to the left and the number beneath if it's positive it's up and if it's negative it's done so let's start off with looking at this Vector here so this Vector here we can see the starting point and the arrow shows the switch where we're going and we're starting here and we're going one square to the right so let's write that as a one and then we're going two squares up one two so that would be represented so this Vector will be represented by the column Vector one two one to the right and two up okay now let's have a look at this column Vector so this column Vector starts here and it goes in this direction and if we count the squares to the right it goes one two three four five to the right and then one two down so because it's five to the right we're gonna write five and then because it's two down we're gonna write negative two so this Vector would be represented by the column Vector 5 negative two okay our next Vector is this one and we start here and as you can see from the arrow we're going in this direction here and we're going three squares to the left one two three and one square up so because there's three squares to the left we're going to write negative three because if we go to the left it's negative and then it's one square up so we then write one so this Vector would be negative three one okay next column Vector is this one we're starting here and we're going up two squares one two so to the left or right well it doesn't go to the left or right so we're gonna write zero and then it goes two up one two so then at B2 so this is the vector 0 2 and finally our last Vector of this one we're starting here and we're going down to this point here so we're going one two to the left so it's going to be negative two and then one two three down so that's negative three so this would be the vector negative two negative three which means two to the left and three done okay so that's how you represent vectors as column vectors now let's have a look and see what happens whenever we add and subtract and multiply these column vectors so here are some column vectors we've got a the vector a is 3 negative one so the vector a is three to the right and one down and the vector B is 4 6 which means four to the right and six up now these letters A and B they're the vectors and they're in bold which means that they've been typed in a thicker ink obviously now whenever we're writing we can write a and bold we can just sort of write it over and over and over what we could do is to show that a is both show us that to show that something is a vector we write it and then put a line underneath it so instead of writing a bold a which is very difficult we can just write a with a line underneath it so if first question says find the vector 2A so we've got the vector a which is 3 to the right and one down so the vector 2A will be double that so instead of being 3 to the right it'll be just right 2A is equal to and instead of being 3 to the right it's going to be 6 to the right instead of being one down we're going to double it it's going to be two down so the vector 2A will be double the vector a so it would be 6 negative 2. if we're asked to find the vector 3A we would multiply both these numbers by three if we're asked to find the vector 10A we'll multiply both of these numbers by 10. if we're asked to find the vector negative 4A we would multiply both these numbers by negative 4 and so on okay let's have a look at our next one so this time we've been asked to find the vector a plus b so here's the vector a which is 3 to the right and one down and here's the vector B which is 4 to the right and six down so to find the vector a b we'll just add these vectors together so the vector a plus b would be equal to and if we add them together 3 plus 4 is equal to 7 and then negative 1 plus 6 well negative 1 plus 6 is equal to 5. so the vector a plus b would be seven five and that makes sense because if we start off at a certain point and we go three to the right and one down so three two and one down and then we go four to the right and six up all together we have gone seven to the right and five up and that's and that's what the vector a plus b is okay and finally we've been asked to find the vector B subtract 2A so we've got the vector B which is four six and the vector 2A is 6 negative two so we've got the vector B and we've got the vector 2A so now we just need to do B subtract 2A so that's going to be equal to B subtract 2 a would be equal to well 4 subtract 6 would be negative two and then six subtract negative two well six subtract negative two would be six plus two which would be it so the vector B subtract 2A if you get B and 2A and you subtract 2A from B would give you negative two here and that's it okay so that's column vectors now this video 353 a and corporate Maps right so we've looked at column vectors now let's look at vectors on diagrams so here we've got a diagram and as you can see it is a large parallelogram and there are nine smaller congruent parallelograms and we're told that the vector go from o to a is equal to little a so from o to a is equal to little letters so let's show that so here we've got o and a so to go from o to a that's that way that's equal to little a I'm going from o to D so you're going from o to D is Little B so from o to D that way is equal to Little B and we've been asked to find the vector OG so we go from o to J now on this diagram we've got nine smaller congruent parallelograms so that means from A to B would be that layer as well because each one of these horizontal lines is little light so going from B to C with B little a as well and then each one of these diagonal lines going upwards from here to here from here to here from here to here and from here to here each one of them is Little B each one of these diagonal lines would be little B so we want to find the vector from o to G so to get from o to G we're going to do an A and another a and another a that would be three a and then to go from C to G well that would be a little B so to go from o to G it would be three a plus b so three a plus b and that's it okay so we're now going to look at our algebra topic so that's the blue topics on your revision checklist and we're going to start off by looking at some algebraic notation so that's video 19 on corporate Maps so let's have a look and make sure we know what each of these algebraic expressions mean so we've got X plus three that means a number plus three if we've got X take away four that means a number take away four or four less than a number we've got ten subtract X that means that it's ten take away whatever X is or X less than 10. we've got 4X now that means four times x because in algebra we don't write the multiplication sign so 4X means 4 times x so if we knew what x is you just multiply by 4. next we've got this X and then the line two so remember in a fraction the lamb means divided by so we've got x divided by 2 and we sometimes call this X over 2 or x divided by 2. so this is X over 2 which means x divided by 2. and finally we've got x squared and that just means x squared so if we know X's we'd multiply it by itself because it's a square something means to multiply it by itself okay let's have a look at our next topic so our next topic is writing expressions or formal expressions and that's video 16 on corporate Maps so we've been told that Jake is X years old so he's X years old we don't know how old he is he could be 10 he could be 50 he could be 100 it could be 67 we don't know and there's nine years older than Jake so whatever age he is she's nine years older and Beth is twice as old as Jake so she's twice his age and the first question says write an expression for Anna's age so if we knew how old J equals we would add 9 onto it to find Anna's age but we don't know it so we're just going to write down he's X and we would add on nine so an expression for her age is X plus nine the next question to write an expression for births age now she's twice as old as Jake so whatever age he is you'd multiply it by two so we want to times his age X by two now remember in algebra we don't write the multiplication signs we just write 2 x that means two times x two times his age okay our next topic okay our next topic is on collecting like terms and so that means we're going to simplify expressions such as this one such as 7x plus y take away X Plus 5y and that's video 9 in corporate map so if you do want to recap this watch video Nano called Maps Okay so we've got 7x plus y take away X Plus 5y so when we're collecting like terms we collect the like terms so let's start off with our X's we've got seven X's so here we've got seven x's and we've got subtract and X so if you get seven x's and you take away an X or 1X you'll be left with six X's so let's write that down six x now let's deal with our y's we've got plus y so that's positive y or one y and we're going to add a number five wise so if we added one y and over five y's all together that's adding six wise so it'd be six X Plus 6y and that's it so we've simplified an expression by collecting the like terms okay let's have a look at our next topic so our next topic is multiplying terms which is video 18. and here we've been given nine times y now remember in algebra we don't write the multiplication sign so if we do nine times y we would just write nine y here we've got 4X multiplied by 5. so that's four lots of X but we've got five lots of it so we've got a 4X a 4X a 4X a 4X and a 4X if you added all those four X's up you would have five lots of four that'd be 20 X's all together now if I was multiplying 4X by 5 I would just take the 4 and Times by five so that's 20 and then it's 20 lots of X just 20 x like so okay let's have a look at our next topic so our next topic is laws of indices and last video 174 on corporate maps and then what we're going to do is we're going to look at laws of indices whenever we're using algebra as opposed to using numbers but the same rules apply so for instance if we had m to the power of 3 multiplied by m to the power 4 we just add the power so be m to the power of seven if we had m to the power of 8 divided by m to the power of two we take away the powers if we're dividing so be m to the power of six and then finally if we had a power to a power so we've got so a power and then a bracket and then a power outside we'll multiply the powers together so if we had M cubed squared we then have m to the power of 6 by multiplying the powers together okay and this is the corporate revision card on laws of indices so this might be quite useful if you've got the revision cards as well so let's have a look at our examples we've got y to the power of 8 multiplied by y to the power of three so we've got y times y times y times y eight times and then we're multiplying that by another y times y times y so that also go over to be 11 y's multiplied together so to be y to the power of 11 and of all words adding the powers together eight plus three is eleven so be y to the power of 11. next we've got y to the power of 15 divided by y to the power of 5 well if we're dividing and we've got the same base so they've both got a base of Y we just take away the powers so we've got 15 take away 5 is 10 so be y to the power of 10. and finally if we had y to the power of 6 squared we've got a power over power so we multiply the powers together 6 times 2 is 12. so it'll be y to the power of 12. or another way to look at this one is we've got y to the power of 6 multiplied by itself so we could then add the powers and that would give us 12 as well that's it okay now we're going to look at expand and brackets so sometimes we're asked to expand or to multiply out a bracket and that's video 13 on corporate Maps so here we've got our first question we've been told to expand six bracket y plus seven that means we've got six lots of the bracket y plus seven so we've got y plus seven y plus seven y plus seven y plus seven you know we've got six of them and we want to find out what we've got all together now if you had y plus seven micro seven or six times you would have six y's all together and a quick way today is just to do six times y so six times Y is six Y and then if you had y plus seven y plus seven y plus seven y plus seven you had six of them that'll be you'd be adding seven six times and six times seven is forty two so we'll be adding 42 so be plus 42. and a quick way to know this is just multiply whatever's in the bracket by the number outside so you just do 6 times Y is 6y and then you do 6 times 7 and 6 times 7 is 42. so it's a multiply a bracket or to expand the bracket use multiply what's inside by the number outside okay the next one we've got two bracket one minus three x so we're going to multiply what's inside by two so one times two is two and then we've got a minus answer minus and then we're going to do 2 times 3x is 6X so answer would be 2 minus 6X just like that is expand X bracket 3x plus 4. so remember to expand our brackets we multiply what's inside by the term outside so we're going to do 3x multiplied by X and we're going to do 4 multiply by X so 3x multiplied by X but that would be 3 x squared because we're multiplying the X by X would give us the x squared and we've got a 3. so 3x multiplied by X is 3x squared then we've got our plus sign and then we've got 4 multiplied by X so that's just going to be 4X and that's it okay our next question so here we've got our term outside 2y and we're going to expand our brackets y minus 3. so we're going to multiply what's inside by the term outside so we're going to do y multiply by 2y well that would be 2 y squared because we've got y times Y is y squared and then we've got the 2 as well so that's 2y squared then we've got our minus sign and then we're going to do 3 multiply by 2y of 3 times 2i would be 6y so answer would be 2y squared minus 6y and if you want more practice in this remember you can go to video 13 on corporate maps you can watch that video beside that this is the practice questions and also the textbook exercises and then also remember there's that bumper booklet of questions and there'll be questions there that involve expanding brackets okay our next topic okay so let's have a look at our next topic so our next topic is to expand in two brackets so as you can see here we've got two brackets X plus six and x minus two and we've been asked to expand and simplify them so to expand two brackets I use this approach I take my first term my X and I do x times x so x times x is x squared then I do my x times minus two and x times minus 2 is equal to minus 2x so 6 times x is 6X so plus 6X and then we've got 6 times minus two and six times minus two is equal to minus 12. so whenever we expand these brackets we get x squared minus 2x plus 6X minus 12. as you can see we've been asking expand and simplify and if you look at these two middle terms we've got minus 2x plus 6X now minus 2x plus 6X is 4X so our final answer would be x squared plus 4X minus 12 and that's it and another approach to expand on two brackets is to use a grid so here we've got X plus 6 we write X and then plus six and then we'll get X subtract two so we write X and negative 2. and what we're going to do is we're going to multiply each of these terms and put the answers inside of our grid so we've got x times x well that's x squared we've got x times six so x times positive 6 that's going to be plus 6x then we've got negative 2 times x so it's going to be negative 2X and finally we've got negative 2 times 6 well negative 2 times positive six well negative times a positive is a negative and 6 times 2 is equal to 12. so we've got x squared plus 6X subtract 2x subtract 12. so let's write that down x squared plus 6 x subtract 2x subtract 12 and then we need to simplify so let's collect our like terms we've got 6X take away 2x well 6X take away 2x is 4X so that would be x squared plus 4X subtract 12. so that's another approach we can use to expand on two brackets is the user grid okay let's have a look at our next question so our next question says expand and simplify 2x plus 1 X plus four so again we're going to multiply both of these terms by the 2X and then we're going to multiply both of these terms by the one so 2x times x well 2x times x would be 2x squared so 2x squared then we've got 2x times 4 well 2x times 4 would be 8X so plus 8x now we're going to multiply both of these terms by the one so 1 times x is equal to X or 1X so that's X Plus X and then we've got 1 times 4 well 1 times 4 is equal to 4. so what we're going to do now is we're going to simplify these two middle terms ax plus X is equal to 9x so our final answer would be 2x squared plus 9x Plus 4. that's it and again we could use the grid to expand these brackets so we've got 2x plus 1 and X plus four well x times 2x well x times x is x squared and then we've got Times by 2 so that's two x squared then we've got x times one that's gonna be Plus One X or plus X we've then got 2x times 4 or 2x times 4 would be 8X or plus 8X and then we've got four times one but four times one is equal to four so plus four and then we can just write it all out to x squared plus X plus eight X plus four and then collecting our like terms X plus eight X is nine X so we've got two x squared plus nine X plus four and that's it okay let's have a look at our next topic which is factorizing which is video 117. so to factorize an expression what we do is we want to put the brackets back in so we wanted to figure out what was expanded or what we had multiplied out is the opposite of expanding so to factorize if you've got 15x plus 20 here to factorize it you think what's the biggest thing that you can divide both by what's the highest common factor of 15x and 20. so if I've got 15 and 20 but the biggest thing I can divide these by is five so I'm going to put five outside the brackets and then open the brackets and then I'm going to divide both of these by five so 15x divided by 5 well that would be 3x and then 20 divided by 5 is 4. so B4 and it's Plus next we've got factorize 12y minus 36. so if I've got 12 Y and 36 the biggest thing you can divide these by is 12 because 12y is obviously divisible by 12 and 36 is divisible by 12 as well so we put 12 and then brackets now 12y divided by 12 well that should be one y or we would just write Y and then we've got minus and then we've got 36 divided by 12 and 36 divided by 12 is 3. and let's just check it 12 times Y is 12y and 12 times 3 is 36 and we had a minus sign so 12 times minus 3 is minus 46. that's it factorize W squared plus 8w as you can see both terms have W's so let's take W out so W squared divided by W that would just leave U of w and then we've got our plus sign and then 8w divided by W would just be 8. so our answer would be W bracket W plus 8 and you can test it by expanding the brackets so W Times W is W Squared and W Times a is a w okay next one we've been asked to factorize 4y squared plus 6y so we're looking for common factors of 4y squared and 6y in terms of the numbers I can see I can divide four and six by two so I'm going to take a 2 out as a common factor and also in terms of the letters we've got y squared and Y so we can divide both of those by y so we're going to factorize this by taking out 2y so we're going to divide both of these terms by 2y and see what we get well 4y squared divided by 2y that will leave me with 2y because 2y times 2A is 4y squared and then plus and if we had 6y divided by 2i that would just leave us with 3. so answer would be 2y bracket 2y Plus 3. okay our next topic okay so our next topic is factorizing quadratic so that's video 118 in corporate Maps so we're going to be factorizing quadratics that's expressions of an x squared term where it is just an x squared term so it's not going to be 2x squared of 3x where it would just be x squared or 1X squared and whenever we factorize these quadratics what we need to do is figure out what the two brackets were that we expanded to get that so we want to figure out what two brackets we have multiplied together are expanded to get x squared plus 8X plus 15. what I'm actually going to do is I'm actually going to just pick two brackets to begin with okay and I'm going to call it X plus 4 and X Plus 3. and I'm going to expand these brackets so x times x would be x squared x times 3 would be plus 3x 4 times x would be plus 4X and 4 times 3 is equal to 12. so whenever we expand these brackets we would get x squared and then we'd have 3x plus 4X would be plus 7x and then we've got our plus 12 on the end so if we had the pair of brackets X plus 4 and X plus 3 and we expanded them we would get x squared plus 7x plus 12. now one thing that I not address is that the 4 multiplied by the 3 gives us the number on the end the 12. so the two numbers in the brackets at the end the three and the 4 will multiply together to give you the number at the end of the quadratic so here where we've got our x squared plus 8X plus 15. the number here and the number here well times together to give us 15. another thing to notice about these two numbers the four and the three four plus three is equal to seven they add together to give you the term in the middle so whenever we're factorizing this quadratic x squared plus eight X plus 15 well because it's x squared we know there's an X at the front above brackets because x times x is x squared and in terms of the two numbers that follow the X's they will times together to give you the 15 on the end and they're going to add together to give you the eight in the middle so we want to find two numbers that we'll multiply together to give you 15 and add together to give you here now first of all I'm thinking here is going to be three and five because if you had three and five three times five is equal to 15 and 3 plus 5 is equal to n so let's just check if we had X plus 3 and X plus five if we expanded these brackets we would get x squared plus 5x plus 3x plus 15. and the 5x and the 3x would add together to give you the 8X so that's it so to factorize quadratically where it's just an x squared term at the front you're going to factorize and put your two brackets down you're going to put x's in the front of them and you're going to think of the two numbers that will times together to give you the number at the end and then we'll add together to give you the number in front of the X the coefficient of x okay let's have a look at another example so this time we've been asked to factorize x squared plus x minus 6. so we're going to have two brackets and we're going to put X at the front of both of them and we want to find two numbers in these brackets that will time us together to give you minus six and then we'll add together to give you the number in front of the X and as you can see here it's just X that means it's one so they're going to add together to give you one so we want to find two numbers so we'll times together to give you minus six and add together to give you one so let's think of numbers that times together to give you minus six so minus six we could have minus six times one we could have six times minus one we get a three times minus two or we could have minus three times two so these are all options so which were times together to give you minus six we want the one that we'll add together to give you one because it was plus X or Plus One X so as you can see here we've got minus six plus one well minus six plus one's minus five that's not going to be a right answer we've got six plus minus one that's going to be five that's not going to work you've got minus two plus three well minus two plus three is one so that's going to be our correct option we're gonna have X plus three and x minus two so that means our answer must be X plus three and x minus two because the two numbers are three and R minus two well times together to give us a minus six and then we'll add together to give us one okay let's have a look at our next question so our next question says factorize x squared minus eight X plus seven so we've got our quadratic and we're going to start off by putting our brackets down because we know that we're going to have brackets with x's at the front of both of them and we want to find two numbers that will times together to give us seven and then we'll add together to give us negative eight so let's start off by thinking of numbers there were times together to give a seven so it could be one and seven because one times seven is equal to seven but it could also be negative one and negative seven because negative one times negative seven is seven now we want the option where they will add together to give us negative eight so first of all I know it's not going to be the top option because they will add together to give us a year let's check our other option upon negative one plus negative seven well negative one going down another seven would be negative eight so this will be our correct option so in their brackets we will have x minus one and x minus seven and that's it okay so our next topic we're going to look at solving quadratic equations in this video 266 in corporate Maps so to solve this equation we're going to use the technique we just looked at that factorization so if you have a quadratic equation y equals zero you can try to factorize the left-hand side and if it factorizes then you can solve it really quickly and easily so let's factorize x squared plus 14x plus 45 so we'll put our brackets down bracket bracket bracket bracket and we're going to put X's at the front of both of them and we're going to look for two numbers that were multiplied together to give us 45 and we'll add together to give us 14. you could do five times nine five times nine is 45 and they add together to be 14. fantastic so we've got X plus five and then we've got X Plus 9 equal zero so we've factorize this quadratic and we've got in Brackets X plus 5 and then X plus nine equals zero now whenever you get two things that multiply together to give you zero one of them has to be equal to zero so that means that either this bracket's equal to zero or this bracket's equal to zero so let's look at this bracket so you have a number plus five X plus five equals zero so X plus five equals zero or X plus nine is equal to zero either that bracket's equal to zero so let's solve this equation if you had X plus five equals zero well you could subtract five from both sides and you'd get x equals negative five and in this case you had X plus nine is equal to zero so if you subtracted nine and subtracted nine you'd get x equals negative nine so if you had this equation x squared plus fourteen X Plus 45 equals zero we've got two possible answers and they are x equals negative five or x equals negative nine and one other thing to note is sometimes I take a little bit of a shortcut whenever I'm doing these questions so if I had X plus five and I'm trying to find when that bracket is equal to zero I just know it's a negative five because negative five plus five is zero so I would often take a bit of a shortcut so whenever I'd be doing these questions I would often just go straight to this point here so I would say well X plus five zero so x equals negative five or and in this case I know that it'd be negative 9 plus 9 is 0. so I just jump to x equals negative nine and that's it okay so let's have a look at our next question so next question is to solve x squared minus 4X minus 12 equals zero and that's great because the equation the quadratic equals zero if it's not equal to zero you'd want to make it equal to zero and then let's try and factorize this left hand side so let's put our brackets down and put equals zero so we put our X's at the front of both brackets and we're trying to find two numbers so we'll multiply together to give us negative 12 and I'll add together to give us negative four and two minus six when negative six plus two would be negative four so that's going to be our option so our brackets will be X plus two and x minus six now we're trying to find out when each of these brackets could be zero we know this bracket would be equal to zero whenever X is equal to negative two because negative two plus two is zero and for this bracket we know it'll be whenever X is equal to six so it would be x equals six and that's it so x equals negative two or x equals six okay let's have a look at our next topic now this actually isn't really a different topic this is really a special case so this is the difference between two squares and this is video 120 in corporate maths so what I'm going to do is I'm going to expand these brackets where I've got x minus seven and X plus seven now I've just used these brackets where we've got our x minus and an X Plus and then the same number and I just want to show you what happens whenever you have this this is x times x is x squared x times positive 7 will be plus seven X then we've got minus 7 times x that's minus 7x and then we've got minus seven times seven well a negative times a positive is a negative and 7 times 7 is 49 so it's going to be minus 49. and whenever we look at our two middle terms here we've got seven x minus seven X now they will give you zero whenever you do 7x take away 7x that's zero so we would be left here with x squared this and this cancels out and we're just left with minus 49. so if you expanding brackets where you've got an x and x and then the same number but then we'll move a plus and a one with a minus sign the two middle terms will cancel out now that's very useful because this we can factorize expressions like this and it's called difference between two squares because whenever we expand these brackets the answer will always have a squared term so x squared and a square number and then you'd have a takeaway in the middle so that's your difference you're taking away the difference between two squares okay let's have a look at some factorizing using this then okay so our first question says factorize x squared minus 49. Okay so we've just done that one actually so we know whenever we expanded x minus seven X plus seven we got x squared minus 49. so we were asked to factorize this so our answer would be x minus seven and X plus seven and the shortcut would be well we had x squared so the square root of x squared is just X so that goes at the front of both of them and then we've got 49 so you're square root 49 that's seven so you know that's going to be 7 at the end of both brackets and then you put one with a plus sign and one with a minus sign and that's it so if I wanted to factorize x squared minus 9x I would put two brackets down I'd square root both terms because they're both squares so square root X where just x and x the square root of nine is three and three and we put one with a plus sign and one with a minus sign and it doesn't matter which way around they go and that's it so if we were to factorize x squared minus 9 the answer would be X plus three x minus three okay next we've been asked to factorize x squared minus 36 so again bracket bracket bracket bracket x x because the square root of x squared is X and then 36 well the square root of 36 would be six so six and six one for a minus animal over plus sign that's it okay now the last one our last one is the factorize 64 minus x squared and this is just to show you that it's the difference between 2 squared so it doesn't have to be x squared at the beginning it could be the other way around so let's factorize and we'd get the square root of 64 is 8 so 8 is going to be at the front of both brackets we've got our two squares so one's going to have a plus sign and One's Gonna Have A minus sign and the square root of x squared is X so it's going to be 8 plus X and 8 minus X and that's it so our next topic is substitution so that's video 20 and equivalent miles and we've been given the question given that W equals 9 and Y equals five find the value of 8w minus 3y so remember in algebra we don't read the multiplication sign so 8w means 8 times W or eight times whatever value W is in this case it's 9 and then subtract 3 times y so it's three times whatever value Y is and that's 5. so let's work out what 8w is so that's 8 times W that's eight times nine and that's equal to 72. so 8w is 72 and we're going to take away whatever three times wise so 3 times 5 is equal to 15. so 3y is fifteen and then we're going to do 72 take away 15 that's equal to 57. so 8w subtract 3y if W is equal to 9 and Y is equal to 5 then that would be equal to 57 and that's it okay our next question now with substitution sometimes it's just substituting the values into an algebraic expression such as this sometimes we've got a formula which involves words and we've got to substitute values into that so here we've got Erin used as formula to work out how long it should take to cook a turkey now I've just made this up so please don't be a Kentucky using this formula so you've got the cooking time in minutes is equal to 90 plus the weight of the turkey in kilograms times 20. and the question says how long should it take to cook a seven kilogram turkey so it's the weight of the turkey in kilograms that's seven times twenty and then we're going to do 90 plus whatever that is so remember our order of operations we've got to do the multiplication before we do the addition so we're going to do 7 times 20 so 7 times 20 is equal to 140 and then with 90 plus 140 so 90 plus 140 is equal to 213 minutes and that's it so we've looked at how to substitute into expressions and we've looked at dealing with wordy substitution questions sometimes you give them substitution questions that involve odd and even numbers are positive and negative numbers and I want to look at one of these now so here we've got X is a positive number and Y is a negative number and we've been asked to state if the following expressions are positive or negative so we've got 5x so that means 5 times x and x is a positive number so it could be numbers such as 10 and 5 times 10 would be 50. but remember X is a positive number any positive number so 5 is a positive so positive times a positive would always be a positive so 5x would be positive third on X expression X Y that means x times y now X is a positive and Y is a negative well a positive times a negative is a negative for instance if we had 10 and negative 2 10 times negative 2 is negative 20. so this would be a negative and finally we've got y squared so y squared means multiply by itself so Y is a negative number so we're going to multiply the negative number by itself and a negative times a negative is a positive so that means that y squared would be a positive and that's it so sometimes you're asked to do substitution questions where you're asked to find if things are positive or negative or odd uneven and if you have a look at that bumper pack of questions there's another one there for you to practice now so our next topic is solve an equation so that's video 110 on corporate Maps so we've got some equations here we're going to solve we've got W plus 9 is equal to 24. now whenever we're solving an equation what we want to do is we want to work out what value W is so we want to get the W on its own we want W equals and then a number so what we're going to do is we're going to get rid of this plus 9. so what we're going to do is we're going to do the inverse so the opposite of add 9 is take away 9 from both sides of this equation so W plus 9 take away 9. so that gets rid of the plus sign so we're just left with w and on the right hand side we've got 24 take away 9 that's equal to 15. so that means our answer is W equals 15. to our next equation our next equation is x minus 7 equals 8. so we want to get the X on a zone so you want to get rid of this take away 7. the opposite of take away 7 is ADD seven so add 7 to both sides of this equation so we've added 7 to get rid of the minus seven so we're just going to be left with X and on the right hand side of this equation we've got 8 plus 7 and 8 plus 7 is equal to 15. so X is equal to 15. okay next equation so we've got three x equals 24. so that's 3 times x equals 24. so we want to get the X on its own so we want to get rid of this multiply by 3. and the opposite of multiplying by 3 is divided by three so we're going to divide both sides of this equation by 3. so we've got 3x and we're going to divide that by 3 so that's One X over just X and then on the right hand side we've got 24 divided by 3 that's equal to 8. and finally we've got C divided by 2 is equal to 7. now we want to get rid of this divided by two so we're going to do the opposite which is Times Square 2 and Times by 2. we're timesing by 2 to get rid of the divided by 2 so we're just left with C so C equals and 7 times 2 is 14. so that would be 14. that's how we solve these equations okay next okay now we've got some more equations and these ones have got two steps so we've got four W minus seven equals nine so we want to get the W on its own so what we're going to do is we're going to get rid of this minus 7 to begin with so we're going to do the opposite of -7 which is ADD 7 to this side and to the other side so we've added seven to get rid of the minor seven that's just going to leave us with 4w so we've got 4 W equals and on the right hand side of the equation we've got nine plus seven and nine plus seven is equal to 16. so we've got four W equals 16. now we want the W on its own so this is 4 times W so we're going to divide by 4 and divide by 4. we've divided by 4 to get rid of the Times by 4 so we're just left with w and on the right hand side of the equation because 16 divided by 4 and 16 divided by 4 is 4. that's very important to practice this topic of solving equations so please make sure you're doing the questions in the practice booklet the booklet that accompanies this video but also have a look at video 110 and then this practice questions and textbook exercises beside that and it's a good idea to practice this okay our next equation is C divided by 2 plus 1 equals six so we want to do is we want to get the C on its own so let's get rid of this plus one so we're going to minus one a minus one from both sides of the equation so on the left hand side we had C divided by 2 plus 1 we'll have to have taken away one so this is going to leave us with C divided by 2. and on the right hand side of the equation we had six we take where one that's five now we've got C divided by two so we want to get rid of this divided by two so we're going to Times by two and Times Square two so on the left hand side we had C divided by 2 we Times by 2 to get rid of the divide by two so it's going to leave us with c and on the right hand side we had five times two and five times two is equal to ten so C is equal to ten and that's it okay we're now going to look at equations and we're going to deal with equations with letters on both sides so here's an equation with letters on both sides and we've got 5x plus 1 equals 3x Plus 19. now whenever I'm solving equations with letters on both sides I want to get rid of the lowest number of x's now if you look at this the lowest number of x's would be the 3x so I'm going to get rid of this 3x remember we do the inverse to both sides so if I want to get rid of 3x I'm going to take away 3x so minus 3x and minus 3x so if we had 5x plus 1 and we take our f3x that leaves us with 2x we've still got our plus one and on the right hand side of the equation we had 3x plus 19 we're taking away the three X's so that just leaves us with 19. so we can solve this by taking away one and taking away one so that would leave us with two x equals 18 and then we can divide by two and divide by two and see that X is equal to nine and again we can check our answer 5 times 9 is 45 plus 1 is 46 and 3 times 9 is equal to 27 plus 19 is 46 so that's right okay next question so our equation this time says 4X plus 5 equals twenty subtract X now whenever I'm solving equations with letters on both sides I always want to get rid of the lowest number of x's now minus X is lower than 4X so I'm going to get rid of the minus X now if I want to get rid of minus X I'm going to do the opposite which is ADD X to both sides of the equation so I'm going to add an X and add an X so on the left hand side I had four x's and I add an X or one more X it's going to be 5x plus 5 the numbers will stay the same equals and on the right hand side of the equation I had 20 minus X I added X so the minus x and x will add together to give you zero so they're just going to disappear she's just gonna be left with 20. so we've got 5x plus 5 equals twenty and this is just like the equations you will have seen in the M1 video so we're going to minus five minus five so that will give us five x equals fifteen and then finally we're going to divide by 5 and we're going to divide by 5 and that gives us x equals three and that's it okay let's have a look at our next topic so our next topic is forming equations so here we've got some information we're going to form an equation out of it so with Daniel is X years old Daisy is five years older than Daniel so she's five years older than him so he's X so to find her age you would do X plus five so we'd write X plus five that's Daisy's age and Chris is twice Daniel's age So Daniel has X and we're going to times it by two to get Chris's age that would be 2X and the sum of their ages is 53 and our first question says a form an equation using the information given so we've got the sum of their ages is 53. remember the word sum means to add up so if we add up our X our X plus 5 and our 2x we will get the sum of their ages and we know that's equal to 53. so X plus X plus 5 plus 2X the sum of their edges equals 53. so let's collect our like terms let's do X Plus X plus two x's so X Plus X is 2x plus another two x's is four x's and we've still got our plus five and that's equal to 53. so that is our equation we've got an equation 4X plus 5 is equal to 53. and part basis solve the equation to find Daniel's age so we're going to solve this equation so let's then get rid of this plus 5 by take away 5 and take away five so we'll leave us on the left hand side with 4X because we took away 5 to get rid of the plus five on the right hand side of the equation we had 53 take away 5 is equal to 48. now we've got 4 times x is equal to 48 well we don't want this multiplied by 4 so let's divide by 4 and divide by 4. so 4X divided by 4 is just X and on the right hand side we had 48 divided by 4 that's equal to 12. So Daniel was X years old so we know that X is equal to 12 so Daniel is 12. so the next topic is formal equations and that's videos 114 and 115 on corporate Maps so here we've got an isosceles triangle and we've been told the angle at the top is 2X and we've been told the angle at the bottom left is X Plus 20. now from where the sides of the mark the same on the left hand side on the right hand side I can see this angle is equal to this angle so this is going to be X plus 20 as well now the angles in a triangle add up to 180 degrees so if we add up all these terms for the angles the 2x the X plus 20 and the X plus 20 we'll find that equal to 180 degrees so let's do that so let's let's take our 2X and add our X plus 20 and add another X plus 20 and that will be equal to 180 degrees so let's add up our X's we've got 2x plus X Plus X that's four x's and then we'll put 20 plus 20 so that's going to be plus 40. and that equals 180. now we've got 4X plus 40 equals 180 so let's take away 40 and take away 40 so take away 40 and take away 40 leaves us with 4X equals 140 and then we want to get X on its own so we're going to divide by 4 and divide by 4 and that gives us x equals and 140 divided by 4 would be 35. if the question asks us to find the size of the angles then you would substitute the 35 into these Expressions so 2 times x would be 2 times 35 so this angle would be 70 and if we took our 35 and added 20 this angle would be 55 and this angle be 55 so our angles would be 70 55 and 55. but the question didn't ask is that it just asked us to find X and that's what we've done okay let's have a look at our next topic so our next topic is solving inequality so that's video 178 in corporate maths so our first question says solve 5x is bigger than 30. now whenever I'm solving inequalities I use a similar approach to I would use for solving equations so here we've got 5x is bigger than 30. now we don't want 5 5x where you want just X so we want to get rid of this multiplied by 5. so to get rid of this 5 we're going to do the opposite which is divide by 5. so we're going to divide both sides by 5. so if we divide both sides by 5 well 5x divided by 5 would be 1X or X and that would be bigger than and if we done 30 divided by 5 that would be six so answer would be X is bigger than 6. okay this time we've been asked to solve 3x plus 4 is smaller than or equal to 31. so if this was an equation the first thing I would want to do is get rid of this plus 4 so let's take away 4 and take away 4 from both sides of the inequality so we'll go 3x plus 4 we're taking away four so that leaves us with 3x and then we've got our less than or equal to sound and then we're taken away 4 31 take away 4 is 27 so we've got 3x is smaller than or equal to 27. now we don't want 3 times x we just want X so let's divide both sides by three so dividing by three and dividing by three would give us well on our left hand side we're dividing by three to get rid of this Times by three so we're just left with X and then we've got a smaller than or equal to and then if we do 27 divided by 3 well 27 divided by 3 is 9. so if we've been given the question 3x plus 4 is smaller than or equal to 31 well we know that X will have to be smaller than or equal to 9. that's it okay let's have a look at our next question so this time we've got an inequality with letters on both sides so remember whenever we're solving equations with letters on both sides we want to get rid of the lowest number of x's so here we've got 8X plus 1 is smaller than 10x subtract 6. so with this inequality I'm going to get rid of the lowest number of x's which is our 8X so let's take away 8X from both sides of this inequality so when we do that we had 8X plus 1 we took away the e at X so we're just going to be left with plus one or one and that's smaller than well we have 10x take away 6 and we're taking away e at X so that's going to leave us with 2x so we've got 2x take away 6. okay now I like an equation I would want to get rid of this take away 6 so let's add 6 to both sides of this inequality so 1 plus 6 is 7. so we've got 7 is less than 2X and we added 6 to get rid of the minus six so we're just left with 2X and finally we want the X on its own so we want to get rid of this multiply by 2 so let's divide by 2 and divide by two and whenever we divide 7 by 2 we get 3.5 so that's 3.5 is less than x and that's it so we've got 3.5 is less than x now we might want to write this the way around so we've got the X at the front so if we write this the way around just be aware whenever we read it the way around we've got X now we've got 3.5 is less than X or if we read the way around we'll get X is bigger than 3.5 so we need to flip that inequality sound round whenever we're origin the way around because X is bigger than 3.5 that's it so whenever solvent inequalities are similar to solving equations but you just need to be aware of whenever you are wanting to flip it around so in other words whenever you want the X at the front row from the number at the front then whenever we flip around that you need to be aware of that inequality sign okay let's have a look at our next example so our next question says solve 9 minus X is bigger than 11. and the first thing I notice whenever I look at this inequality is we've got a minus x what I'm actually going to do is I'm going to add X to both sides it's inequality so when I add X and add X that would give me well on the left hand side I had nine take away X and then I added X well I added X to get rid of the minus X I would just leave me if 9 on the left hand side then we've got our big events and our greater than 10 and then we've got 11 plus X so we'll just be 11 plus X like so next well we want X on its own we don't want this 11 here so let's take away 11 from both sides of the inequality sign so let's take away 11 and take away 11. well 9 take away 11's minus 2 so we're gonna have minus two on the left hand side and on the right hand side we had 11 plus X we took away 11 well 11 take away 11 is 0. so we would just be left with the plus X or the X so we've got here negative 2 is greater than x we wanted the relative way around we can turn it around but we have to turn that inequality sound around as well so we would write X and I said right now because I'm going to write a smaller than and then we've got negative 2. so that means our answer is X is smaller than negative 2 and that's the 0 is negative 2 is bigger than x so that's our answer okay let's have a look at our next topic so next topic is inequalities but this time we're dealing with number lines so this is represented inequalities on a number line and it's also looking at how to solve inequalities and then represent the answer on a number line and this would be video 177 on corporate Maps so let's look at how we'd represent some inequalities on a number line so here's a number line and our first inequality is X is greater than negative one so X is bigger than negative one so first of all what we do is we go to negative one and because X is greater than negative one it can't actually be negative one so what we do is we put a Hollow Circle at negative one so you put a Hollow Circle negative one and because X can be bigger than negative one what we do is we put an arrow to the right like so and we just did an arrow and that tells us the X can be any value that is bigger than negative one and because the circle's Hollow it shows us that it can't actually be negative one so it's Any number greater than negative one so if we want to show that X was greater than negative one on our number line we do a Hollow Circle negative one and then an arrow pointing to the right the numbers that are bigger than negative one if our inequality was X is greater than two we would do a Hollow Circle at two and then an arrow to the right and that's it as it says X is bigger than 2. okay let's have a look at our next inequality okay this time our inequality says X is greater than or equal to two now this time we've got X is greater than or equal to two so that means it can actually be two and any number that's bigger than two so instead of doing a Hollow Circle what we do is we actually do a circle and color in so she added in circle and then a narrow to the right like so and what that means is the X can be 2 or any number are bigger than two if our inequality was X is greater than or equal to zero we would just sheared it in circle of zero and then we would do an arrow to the right and that would mean that X is bigger than or equal to zero that's it let's have a look at next inequality so next inequality is X is smaller than negative two so this time what we're going to do is because it's smaller than so it's not equal to we do a Hollow Circle and the arrow points to the left because it's smaller than negative two and that's it so if we had an inequality such as X is smaller than four what we would do is we'd do a Hollow Circle of four and arrow to all the numbers that are smaller than four and an hour just says that it goes on forever okay and our next inequality so our next inequality is X is smaller than equal to three so because the smaller than are equal to instead of doing a Hollow Circle we do shaded in circle at three and we do an arrow to the left like so and that's it so that's how we represent inequalities where we've just got X is bigger than a number or bigger than equal to a number or less than a number or less than or equal to a number and if we're in an inequality such as this where X is bigger than one but less than or equal to three and I wanted to represent the inequality of that on the number line because we know that X is bigger than one but we know it's bigger than one so it can be equal to one so we do a Hollow Circle at one and because X is smaller than or equal to 3 because it can be three with a shaded in circle of three and because X can take the values in between those numbers We join them up like so so if we wanted the representative inequality such as X is bigger than one but less than or equal to three we would do a Hollow Circle at one a shaded in circle of three and that represents that inequality on a number line okay let's look at our next question so this time we've been asked to solve 5x minus six is less than four and we've been asked a short answer on a number line so let's first of all start off by solving this inequality so we have five x minus six is less than four so like an equation we want to solve this and get X on its own so let's add 6 to both sides so add six and add six so on the left hand side we've had minus six we added six to get rid of it so we're just going to be left with 5x and on the right hand side of our inequality so we have our less than symbol and then we've got four plus six and four plus six is ten so we've got five X's less than 10. we don't want 5x we actually just want X on its own so let's divide by 5 to get rid of this multiplied by five so dividing by five and dividing by 5 gives us X is less than and 10 divided by 5 is 2. so solving our inequality our answer would be X is less than two and if we wanted to show that on a number line because this is less than two we do a Hollow Circle and then because it's less than two with an arrow to the left for all the numbers that are less than two and that's it okay so let's have a look at another type of inequalities question so we've been asked to list all the integers that satisfy the inequality 2N is greater than 7 but less than or equal to 14. so we're looking for integers so numbers such as negative two negative one zero one two three and so on to satisfy this inequality now if we have a look at this inequality we've got 7 is less than 2N this is less than or equal to 14. now we'll put 2N here what I'm going to do is I'm actually going to divide this inequality by 2. I'm going to divide everything by two and that will give me n instead of 2m so if I divide everything by 2 7 divided by 2 is equal to 3.5 is going to be less than and then 2N divided by 2 would be n and then we've got our less than or equal to symbol and then we've got 14 divided by two that's seven so we know that n is greater than 3.5 but less than or equal to seven so we need to list all the integers that are greater than 3.5 but less than or equal to seven so let's list our integers so our first integer that is greater than 3.5 would be four and then we've got five and then we've got six and then it says less than or equal to seven so seven is going to be an integer that works so I integers to satisfy this inequality would be four five six and seven and that's it okay next topic is changing the subject so here we've got a formula we've got T is equal to Aw take away C and at the minute the capital t is the subject because it's on its own the t's on it so in there so T is the subject but we want to make W the subject so what that means is we want to get the W on its own so we don't want this takeaway C and we don't want this to multiply by a so what we're going to do is like an equation we're going to get rid of this take away C and multiply by a so let's write out our formula we've got T equals Aw take away C now we want to forget the W on its own so let's get rid of this take away C to begin with so if I wanted to get rid of take away C I'm going to add C to both sides so add C and add C well t plus C we're is just t plus C we just write it out and then the right hand side of our formula but we had a w minus C we added C to get rid of the minus C because minus c plus c is zero so we would just be left with aw now we want to make W the subject which means we want to have the W there on its own this is a times W we're multiplying the W by a we don't want to multiply the W by a so let's divide both sides by a so let's divide by a and divide it by a on our right hand side we had a w we divided by a to get rid of it so we'd just be left with W on the left hand side of this formula we had t plus C and we're dividing that by a so what we're going to do is write t plus C now remember in algebra we don't write the divided by sound and we do over a and that means we've got t plus C divided by a and that's it so we've got WR so w as a subject so our answer would be W equals t plus C over a and that's it and changing this object is video seven on corporate map so if you want to watch more examples of continue the subjects starting off with some simple questions and building up to questions like this video seven will be really really useful and also remember there's the practice questions in the textbook exercise there as well if you do have the revision cards changing the subjects is one of the revision codes and as you can see here we've got a typical change in the subject question where it says make W the subject and it goes through step by step explaining how you would make W the subject of that formula so change instruction is a very important topic it's video seven or corporate maps and if you do have the revision cards that revision card will be really useful for it so we've been given a question that says Circle the identity so here first of all we've got an equation with a 4X plus 1 equals 21 and we can solve this we can take one away from both sides and we'll get an answer and if you solve this you would find x equals five now because that's just got an answer that's an equation so it's not an identity now next we've got 3x minus 7 is less than or equal to 50. that's what we call it inequality because we've got this inequality time we've got the less than or equal to sign so if it's got a greater than sign a greater than or equal to a less than or less than or equal to sign they're called inequalities that's an inequality it's not an identity next we've got 6X minus 8 and that's what we call expression we've got these terms 6X and minus eight so that's just what we call an expression and finally we've got an identity and an identity is something that's always equal to each other so we've got a 5 bracket X plus one close brackets and then we've got this symbol here of three lines that's the equivalent to symbol and then we've got five X plus five and if you expand this set of brackets you get five X plus five so five bracket X plus one is always equal to five X plus five so that is an identity it's always equal to it so an identity is something that's always true and that's it so okay our next topic our next topic is function machines so here's a function machine and we've got input multiplied by three subtract in and that gives us our output so part A says work out the output when the input is equal to seven so we've got the inputs equal to seven we're going to multiply that by three seven times three is twenty one and then we're going to subtract it so 21 take away at this 13. so if the input is seven the output is 13. okay our next part says work out the input if the output is 22. so we've got the output well let's get rid of this working out from our other part so we've got the output is 22. and then we've got subtract it well we're going to do the inverse so we're going the opposite way so we're going to do 22 plus it and 22 plus 8 is 30. and then instead of multiplying by 3 we're going the opposite way so we're going to divide by 3 and 30 divided by 3 is 10. so our input would be 10 whenever our output is 22 and we can check that if we had 10 times 3 is 30 subtract 8 is 22. so we've done part A and Part B and part C says find an expression for the output when the input is X so if the input is X we're going to multiply it by 3 well that's 3x and then we're going to subtract it so that's three x subtract it and that's it so an expression for the output will be 3x subtract it okay let's have a look at our next topic so our next topic is coordinates so that's video 84 and corporate Maps so here we've got a set of axes and we've got some coordinates we've got a b c and d and we're going to write down the coordinates of these four points and whenever random coordinates some people remember the center along on the chord or up the stairs so I'm going to use that just to help us find these coordinates so the coordinates of point a so we've got a and as you can see here we've sorted the origin so that's this point here the origin and to get the a we'll go to along the corridor and four up the stairs so the coordinates would be 2 4. next according to the point B so we'd start again at the origin we're going to go two along the corridor so that's going to be 2 and then we go down to -2 so we're going to go down to so the coordinates at this point here is 2 minus 2. next the coordinates of this point C so again we started the origin we're going to cross the minus five and up one so it's going to be minus five one and finally the coordinates of the point D well we're not going left or right we're not going to cross anywhere and we're just going to go straight down four so the coordinates would be 0 minus four so that's it okay and our next topic our next topic is drawn linear graphs so that has drawn a straight line graph and we've been asked to draw the graph of y equals two X plus one so we're going to plot the coordinates where the Y value of the coordinate is twice the x coordinate plus one so to do that sometimes in the question they give you a table but occasionally they haven't given a table so I'm going to show you how to do this without a table but if they give you a table it should look something like this so it starts off with X and Y and then you've got some values of X at the top so it could be something like minus two minus one zero one two and if they haven't run a table for you you just do something like this and to find what Y is well Y is equal to 2 times X plus one so we're going to substitute in these values of X into this and we're going to find what Y is so we've got 2x plus 1 so we're going to times all the x coordinates all these numbers by two and add one so let's start off here with our two two times two is four added one would be five this point is one so two times one is two plus one is three next if x is equal to zero two times zero is zero plus one is one now we've got some negative numbers two times minus one well two times minus one or negative one remember a positive times a negative is a negative so two times one is two so it's going to be negative two add one means going back up towards zero so it'd be minus one and finally we're going to do two times negative two now two times negative two would be negative four add one come back up towards zero would be minus three and and that's it so we've got our coordinates now we just need to plot them this is set of coordinates here two five one three zero one negative one negative one and negative two negative three so we'll plot these five points and then draw a nice straight line through them so let's start off with two five so two across five up so it's going to be here one three one across three up is going to be here zero one a zero across and then just one up that's there minus 1 minus one so that's negative one negative one so it's going to be here and finally negative 2 negative 3 so it's going to be negative 2 negative 3 down there and that's that point and then get a ruler and a pencil and draw a nice straight line through those points and it looks something like this and that's it so that's how you draw a straight line graph our next topic is to find the midpoint of a line now the midpoint of the line it can be done in two different ways if it's drawn on a grid for you sometimes you can do it just by inspection just by having a look and seeing if you can spot where the midpoint is so here we've got the line a b and if I wanted to find the midpoint of this line I can see straight away it's here that means that the coordinates of the midpoint would be one minus two because it's one across two down so it's one minus two so that would be the midpoint of the line so if you've got on a grid you can just sometimes spot it but sometimes you're given the coordinates so the coordinates of B would be the point three zero the coordinates of the point a here would be minus one minus four minus one minus four and if you were given the coordinates and you're asked to find the midpoint what you do is just add the coordinates together and then divide by two and that will tell you the coordinates of the midpoint so if we have minus one and three well minus one plus three is two and divide by two is one so fantastic and then in terms of the y coordinate if we add the Y coordinates together the minus four and zero or minus four plus zero is minus four divided by two is minus two so if you add the coordinates together and divide by two you'll find the coordinates of the midpoint okay our next topic okay our next topic is to find the length of the line now if you have a look at this line you can see if we just go across and then go up we can turn it into a right angle triangle and the great thing is then we can use Pythagoras's Theorem to find the length of this line so this is a right angle triangle so let's mark on our length so we've got this point here which is the point one one in terms of horizontally you can see we're going across one two three so it's going to be three and in terms of the heights you can see here we're at one and we're going up to five so that means the head of this triangle would be one two three four so we've got the lengths three and four now they're not centimeters they're just three and four units and we want to find the length for the line so remember Pythagoras Theorem is a squared plus b squared equals c squared where A and B are the shorter sides and C is the longest side so A and B well so a is going to be three so three squared plus and B let's let b equal four so four squared equals and then we've got c squared that's going to be a b so we could call it X if we wanted to and just write x squared then let's work out 3 squared plus 4 squared of 3 squared is 9. so we've got 9 plus and four squared is 16. so 16 equals x squared so we've just squared 3 and squared 4. then we're going to add our 9 and 16 together so 9 plus 16 is 25 and that equals x squared so we've got 25 equals x squared or x squared equals 25. now we want to find the length of this line so what we're going to do is we're going to square it the opposite of square square root Define the length of this line so we're going to the square root of 25 and the square root of 25 is 5. so that means the length of this line is 5 units and that's it our next topic is to work out the gradient or the steepness of a line so to work out how steeper line is we use the formula gradient equals rise over run so if we've got a straight line drawn on a grid if we do the rise of the line divided by the run of the line that will tell us the gradient of the steepness of the line and this is video 189 in corporate Maps so let's have a look and see what rise and run are so if you've been given a line drawn on grid such as this one and you've been asked to work out the grain of it choose two points on the line to begin with now in terms of the points here we could choose this point here minus 1 minus two or zero zero or one two or two four or three six or four eight they would be good points to choose I'm just going to choose this one here one two and I'm going to choose this point here three six and I'm going to turn it into a little right angle triangle so I'm going to score across and then I'm going to go up so the rise is the rise here so it's how far up you've gone so whenever you draw your little right angle triangle it's the Rises how much the line's gone up by so if you can see we've gone up and make sure you know what you're going up in so we're going up once here so we've got about one two three four so the rise here would be four and the run and again check what we're going to cross in we're going to cross them ones so the Run would be well we're going to cross one two so the Run would be two so we've got our rise of four and we've got a run of two so let's work out the gradient the gradient is the rise 4 divided by the run two and that's equal to two so the gradient of this line is two and what that changes for every one unit you go across the line's gone up two and as you see if we started at zero zero if we go across one it goes up two if you go across enough one to two you go up another two if you go across another one to three you go up another two and so on so the gradient of this line is two and the gradient is found by doing the rise divided by the run and you just draw a little right angle triangle you just work out what the Run would be and the rise would be and then you do the rise divided by the run so if you had a line going downwards like this and you wanted to find the gradient of this line um again you would draw a right angle triangle you choose two points so I'm going to choose this point and I'm going to choose this point and as you can see we're going across one so I'm going from five to six so run is one and in terms of our rise when we're going downwards this time so a rise is negative four because we're going down and you would do negative four the rise divided by one and that would be negative four and that means for every one you go across the line goes down four if you go across one it goes down four and so on okay let's have a look at another gradient question this time I've changed the scale so the numbers are a bit bigger and we've been asked to work out the gradient of this line so here we've got a line and we're going to choose two points on them I'm going to choose this point which is the point zero fifty I'm going to choose another point and another point I think looks quite good would be 25 250 here and let's draw our little right angle triangle so we're going to go across and then we're going to go up now in terms of the run we're going from zero here all the way across to 25 so the run here would be 25 we'll run across 25 and the rise well we're going from here up to here so we're going from 50 up to 250 so our rise would be 200. and then to find the gradient we're going to do the rise divided by the run so we're going to do the rise 200 divided by 25 and that's equal to 8. so the gradient of this line is it and that what that means is for every one you go across one unit you will cross you're going to be going up here so the green of this line would be it so the next topic is looking at the equation of a line so the equation of a line is in the form y equals MX plus C so for instance something like this y equals 4X plus 3. now we're going to do is we're going to look at this equation so whenever we got in the form y equals MX plus C the number in front of the x is the gradient so if I had the line Y equals 4X plus 3 the gradient of that line is four and you'll remember that means that for every one square you go across a one unit you go across the graph goes up four units so you go across one it goes up four and so on and then R plus C is the Y intercept this shows you where the graph crosses the y-axis so for this graph here this graph would have y equals 4X plus 3 has a gradient of four and it crosses the y axis at zero three it has a y intercept of three now here's part of the chord modular vision card on the equation of a line and you've got y equals MX plus C where m is the gradient so this number is the gradient and C is the Y intercept and if we had this line here if we wanted to find this equation well first of all we would find the Y intercept and crosses the y axis or meets the y axis at one so that means it's going to be plus one on the end in terms of its gradient well we can make a little triangle and we can do rise divided by run and if the rise was 4 and the Run was two but four divided by 2 is 2 that means the grid of this line is two for every one you go across it goes up two when you go across goes up two so the Grid in this line is 2 so the equation of this time would be y equals two X plus one so let's find the equation of this line that's been drawn on the grid so we know it's going to be in the format y equals m x plus C we know the M's degree in so let's find the gradient to begin with so let's make a little triangle so let's choose two good points I think that a good point would be this one here at minus one one and another good point would be this one here I you could have also chosen zero four so I'm going to make a little triangle it's going to cross and up and for my triangle it's got a run of two it goes across two units it goes from minus one to one it's going to run a two and the height it goes from one up to seven so that's got a rise of six so for the gradient we've got rise divided by run so m equals rise over run so it's going to be equal to the right 6 divided by the Run 2 and 6 divided by 2 is 3. so the gradient of this line is three so we know it's going to be y equals three x now in terms of the Y intercept it crosses a 4 positive four so it's going to be plus four if we cross down here minus one you would write minus one and so on if we cross through the origin zero you wouldn't write anything you would just write y equals three x but this graphic cross I had a one step to four so we wrote y equals three X plus four and that's it okay next question says a straight line with gradient minus two passes through the point one five find the equation of the line so we know first of all it's a straight line so we notice in the form y equals MX plus C so we know that the answer is going to be y equals something X plus or minus something and it says it's a straight line with Grant negative two so we know it's going to be y equals minus two x and then something after it and it passes through the point one five and it says find the equation of this line so we need to find this y-intercept and that's where the straight line meets the y-axis there's two ways which I would typically do this question one is by looking at the gradient the gradient's minus two so that means that this graph is actually going down so we've got a x and y axis it's got a gradient of minus two so it's coming downwards like so and we now goes through the point one five so it goes through the point one five so because we know the gradient is minus two that means for every one you go across it comes down two you go across one goes down two so we for this point here if we want to find a way acrosses the Y ax is here so the y axis we know that it's gone across one and it's come down two and it's now going to height of five that means it must have had a height of seven to begin with because it must have had a higher seven to go across one and come down two until you now have the height of five so that means that a y intercept will be seven so the equation of our line is y equals minus two X plus seven so one approach is to know what the gradient means and to look at the point you're being given and to work out where it would have crossed the waxes now there is another approach and this is actually the person I would typically use and that is whenever we had the whenever we knew it was y equals minus two X plus something I would just write plus C which stands for the Y intercept and I know the point that they've given me is one five so I know that it has an x coordinate of one and a y coordinate of five and we can substitute these in to our equation so we can instead of writing y equals we could write 5 equals so 5 equals and then next we know we've got minus 2x so that's minus 2 times whatever X is and X is equal to one so minus two times one is equal to minus two and then we've got our plus C and then we want to find interval C is our Y intercept well we want the C on its own so we want to get rid of this minus two so we'd add 2 to both sides of this equation and you get 7 equals c so C is equal to seven so you get y equals minus two X plus seven okay let's have a look at our next question so this time we've been given a straight line passes through the points 3A and 5 20. and we've been asked to find the equation of the line that passes through those points so again we know it's going to be in the form y equals MX plus C so we want to find the gradient of the line that passes through these points and we want to find where it crosses the y-axis so first of all let's find the gradients okay so let's do a little sketch so we've got our x and y axis and we've got our first point which is 3 8 so 3 across eight up somewhere like that and then we've got 520 so a little bit across a much further up like that so we've got 5 20 and we've got three it and we want to find the gradient of that line so the coordinate of that straight line and so what I do is as before I would draw a little right angle triangle like so and get our rise and our run so our run well we'll go from three across the five so our Runners two and our rise we're going from 8 up to 20. so rise would be 12. so we've got a rise of 12 and a run of two so the m equals rise of a run so rise over run so that's going to be equal to rise 12 over 2 is equal to six so the gradient of this line is six so for every one we go across we go up six and let's just check it if we went across one we would get to four and then if we go up six we'd get to 14 and then if we went across another one we get to five and up and over six we get to 20. so yeah that's right okay so we know it's y equals six x plus something so plus C let's just choose one of our points because we know the line passes through both of these points three eight and five twenty so let's choose one of these points I'm going to choose 520. I could have chosen three here and I know X is equal to 5 and Y is equal to twenty the x coordinate is 5 and the y coordinate is 20. so let's substitute those into our equation so y that's 20 equals 6 times x so that's six times five six times five is thirty plus C so to get the other two and I'm going to want to take away F ready from both sides so I will leave me with 20 take away 30 is negative 10 and then we'll go 30 plus C take with ready just leaves us with C so C is equal to negative 10. so our Y intercept is equal to negative 10. so our answer would be y equals six x minus ten and again let's just check it we knew the gradient of the line was equal to six so I meant that so for every one we go across we got six so if we're at this point if we go across one we go down six that would then bring us to 2 two two if we go across one and down six that would bring us to one minus four and across one and down six would bring us to zero negative ten so yeah the Y intercepts would be negative ten and that's it okay so let's have a look at our next topic and it's part of the lines which is video 196 on corporate maps and parallel lines their lines that go in the same direction and they never meet each other and because they go in the same direction they have the same gradient of the CM steepness so if we had y equals two X plus one and Y equals two x minus three those lines would be parallel to each other because they've got the same gradients and this is the couple miles revision card on that so if you do have the chord marriage revision cards make sure you know this one and stick it up on your wall or bring it with you on the bus in the morning or whatever make sure you remember the parallel lines have the same gradient and that's it that's a very useful bit of information that parallel lines have the same gradient and you can use that in questions as well we've got an example and the question says the graph below shows the cost of hair in a hot tub and this is video 171 and Corbin Mavs so here we've got a graph this uses the cost of hiring a hot tub and you've got the number of days is going along horizontally and we've got the cost going up vertically so going from zero up to 350. so the question says the graph intersects the vertical axis at 100. so as you can see here it starts at 100. what does this represent so as you can see here this would be on zero day so this is like a set fee it's like a set charge that you're charged for higher in the hot tub so no matter what there's a hundred pound to begin with and then you're going to be charged so much per day so what this represents so this represents a fixed fee a fixed cost of 100 pound so that means no matter how long you hire this hot tub for is a hundred pound fee to begin with if this was the cost of a plumber that might be a call out fee or if this is a taxi that whenever you go into a taxi sometimes you see the meter starts at a particular number like a set fee that you have prepared to begin with so this is a fixed cost this value of 100 pound the next question says find the gradient of the graph so we've got this graph and we want to find this gradient so let's choose two points I'm going to choose the point here 0 100 I'm going to choose this point up here which is 10 350. and let's make our little right angle triangle so we're going across and going up and let's work out the rise divided by the runs so our gradient M that's the letter for gradient m is Rise divided by run so for our right angle triangle the rise well it's gone from 100 up to 350 so that's a rise of 250 so we'll be 250 divided by and the run if you look at how much we've gone across by we've gone from 0 to 10 so the Run would be 10. so we're going to do 250 divided by 10 and that's equal to 25. so that means the gradient of this line is equal to 25 and the question says explain what the gradient represents so remember the gradient it means for every one you go across how much the line goes up by now in terms of this context of hiring a hot tub what that would mean is for every one more day you hire the hot tub how much the price increases by so that's what it means it means the hot tub means the cost per day of having the hot top is 25 pounds so that'd be the cost per day 25 pound so our next topic is to look at conversion graphs and here we've got a conversion graph and conversion graphs can be used to maybe compare distances like in this graph here where we've got miles and kilometers conversion graphs can be used to convert to money perhaps you'll get them from currencies and you want to convert between one and the other and the conversion graph can be quite useful for that so here we've got a conversion graph and horizontally we've got miles going naught one two three all the way up to 10 miles and vertically we've got kilometers going naught one two three and so on and let's convert four miles into kilometers using our conversion graph so it's important people to know how to use a conversion graph so we get a learner pencil and we want to convert four miles so we're going to 4 miles on the horizontal axis here for and we'll get our learner pencil and we'll go up to the line and it's just there and then we go across from the line and just with return a pencil on the test paper like so so as you see we've got the second box above the six so let's find out where each one of those boxes is worth so from naught to one there's one two three four five boxes that must mean we're going up another 0.2 is 0.2 0.4 0.6 0.81 so that would be 6.2 6.4 so four miles would be 6.4 kilometers using this conversion graph now we've been asked to change eight kilometers into miles so again getting a Rolanda pencil so getting your pencil Ritter and going from eight kilometers across to the line and then down from the line we get to exactly five miles so using this graph eight kilometers would be equal to five miles and that's it now if you were asked to maybe change something like 80 kilometers into miles I would go from eight across and get five and then say well eight kilometers is five miles eighty kilometers would be 50 miles and so on so you can use conversion graphs to convert what's on the scales but you can also use that information to work out other conversions as well that go beyond the skills okay let's have a look at our next topic and our next topic is quadratic graphs in this video 264 on corporate Maps so what we're going to do is we're going to draw this quadratic graph and whenever we draw quadratic graphs what you'll find is instead of being straight lines they'll either be this u-ship or a parabola like so this u-ship Parabola or they could be an n-shaped Parabola like so and so whenever you draw a chronolographic it will either have this U shift or this end chip and it depends on whether this x squared is positive or negative so because this is a positive x squared I can tell it's going to be a u ship if it was minus x squared it would actually be an N chip okay so the question asks us to complete the table for y equals x squared plus x minus 4. so we've got this X Y table and we've got lots of points the reason we've got lots of points is it's not a straight line any water curve so we're going to need more points so we could draw a nice curve through them and we've got our X values of minus 2 minus one zero one two and three and what we're going to do is we're going to work out our y values and to find our y values we're going to square X add X and take away 4. so let's start off with three so we're going to do 3 squared plus 3 and then take away 4. so if 3 squared is 9 plus 3 is then going to be 12. take away 4 is it so that means that would be it now we've got two so we're going to do 2 squared plus 2 take away 4. so 2 squared is 4 plus 2 is equal to 6 take away 4 is 2 so that would be 2. now we've got y so we're going to do 1 squared plus 1 take away four so one squared is one plus one well one plus one is two take away 4 is negative two so it's going to be negative 2. now we've got zero so we're going to do 0 squared plus zero take away four or zero squared is zero plus zero is zero take away four is minus four so that'd be minus four next we've got negative one so we're going to do negative 1 squared plus negative 1 and then take away 4. so remember when we Square negative we're going to negative times a negative which is a positive so we're going to do negative 1 squared so that's negative 1 times negative 1 which is one we're then going to add on negative one and then take away four so then we've got 1 plus negative one well if it was one add one it would be two but we're adding negative one which means it's going to go down one so one add negative one is actually zero and then we're going to take away four which is minus four so that'd be minus four and then finally negative two so we're going to do negative 2 squared plus negative 2 and then take away four and when we do negative 2 squared that's a negative times a negative so it's negative two since negative two so that's a positive so that's gonna be four and then we're going to add negative two that means it's going to go down two so it'll be two and then take over 4 will be negative two so we've now found our coordinates now what we're going to do is we're going to plot them so we're going to go three across and eight up so three across and eight up we've got two across and two up so two across and two up one across and two down so one across and two down zero across from Florida to zero minus four we've got negative one and negative four so negative one negative four and finally negative two negative two would be negative two and negative two would be there so we've now got our points and as you can see they make a curve so what we're going to do is we're going to draw a nice curve through them so we're going to do this on the computer and actually can actually be quite tricky on the computer so um please bear with me whenever I'm doing it so it might not be absolutely fantastic but I'll do the best I can yeah that's pretty good so that's it so we've drawn a quadratic graph and we've drawn a nice curve for our points and if you want more practice video 264 in corporate Maps will give you more practice on that and remember you've got that bumper pack of questions so um ultimate video practice question booklet and if you go to that in the description below and click on it there'll be questions there on the quadratic graphs I'll give you a chance to draw one yourself okay so let's have a look at our next topic so our next topic is solving quadratics graphically so it could be something like this where using your graph estimate the values of X whenever Y is equal to negative three so if we go back to our graph we need to now draw the graph of y equals negative three because we're told to find the values of X whenever Y is equal to negative 3. so if we draw that graph of y equals negative that's going to be all the coordinates with a height of negative 3 such as 0 negative three one negative three two negative three three negative three and so on the graph y equals negative three is a horizontal line that passes through three on the y axis so y equals a number graph will be a horizontal line and it'll be horizontal going through whatever the number is so if it was negative y equals negative three it'll be horizontal line going through negative three on the y axis because all the coordinates on that line have a height of negative three if it was y equals eight it would be horizontal line going through eight and so on okay but in our question we were asked to find an estimate of the values of X whenever Y is equal to negative three so if you have a look at our Parabola and our straight line our y equals negative three you can see they actually meet each over twice they meet here and they meet here and what we're actually going to do is we're going to go up to the x-axis so we're just going up to here there I'm going up from this intersection point the second intersection here we're going to go up again to the x-axis there and we're going to read off what those values would be and there would be our two solutions and it says estimate because remember you've drawn a curve and we've drawn it freehand so it's not going to be a perfect discretion it's an estimate to our answers so let's have a look and see what we've got so let's start off with this one we've got zero we've got one and then the middle is not 0.5 I think that's about 0.6 so I'm going to say x equals 0.6 would be one of our Solutions and this one over here well we've got zero we've got negative one we've got negative two that's negative 1.5 I think that's about negative 1.6 so negative 1.6 so our two estimates would be x equals 0.6 or x equals negative 1.6 so it will be our two estimates so x equals 0.6 or x equals negative 1.6 and just to also point out that the question could be written in a slightly different way rather than saying find estimates of the value of x whenever Y is equal to negative three it could have had the equation of the graph that we drew in the first question this x squared plus x minus for and it could have just written equals negative three and then what you do is you look at that value you just draw the graph of y equals negative 3 like we did so we just draw a horizontal line going through negative three and go up to the x axis and find those values and another thing you could be asked whenever you're dealing with quadratic graphs is to find the roots of the quadratic so here we've got a quadratic graph and it's y equals x squared minus x minus two and we've been asked to use this graph to find the roots of x squared minus x minus 2 equals zero so when the quadratic equals zero and that means where the quadratic has a height of zero and that means where it crosses the x-axis so if you have a look at where this quadratic crosses the x-axis it crosses the x-axis at negative one and two semi's of this equation will be x equals negative one or x equals two and that's it so to find the roots of the quadratic if you've been given the graph and then they've written the graph equals zero just find where that graph crosses the x-axis and another thing that they sometimes ask you is to write down the coordinates of the turning point and that would be the coordinates of this point here so you just right down the coordinates of that point and that's it okay let's have a look at our next topic so our next topic is reciprocal graphs and this video 346 in corporate Maps so a reciprocal graph is something in the form of y equals a number and then over X so something like y equals 2 over X and you will have heard of the term reciprocal before whenever you're dividing by fractions because whenever you're dividing by fractions instead of dividing by a fraction you multiply by the reciprocal that's what you get when you flip it over and that's what we mean whenever we've got a graph in this format y equals 2 over X so we're going to draw a graph and we've got an X Y table and we've got quite a lot of points because it's there's curves involves not nice straight lines so we're going to substitute in these values of X into our equation to get our values for y so let's start off with the Positive values so let's use x equals five so two divided by 5 is 0.4 then we've got 2 well 2 divided by 2 is 1. then we've got one two divided by one is two then we've got 0.5 so we've got two divided by 0.5 well there's four halves in two so that would be four now let's move on to our negative numbers so we've got 2 divided by negative 5 well a positive divided by a negative is a negative so it's going to be a negative answer and 2 divided by 5 is 0.4 is going to be negative 0.4 over the angle 2 divided by negative two well that's going to be well positive divided by negative would be a negative so negative one now we've got 2 divided by negative one that'll be negative two and finally 2 divided by negative 0.5 would be negative 4. okay so we're not going to points now let's plot them so we've got five across and 0.4 up so five across would be here 0.4 up but we've got ten little squares for two two divided by 10 is not 0.2 so it'll be two of them up which is or two of them up so be like that somewhere like that then we've got two one so two across one up would be there then we've got one across two up so one across two up and then we've got 0.5 across four up so 0.5 across would be halfway in between and four up would be there so we've now got our point now what we're going to do is do a nice curve through them this is quite tricky on the computer so just bear with me okay this won't be perfect but it's just gonna be a nice curve but looks something like so that's not too bad and as you can see the curve it never actually reaches the y axis because if you've done two divided by zero well that's on the find you can't divide two by zero so this curve this Curve will never actually reach the y axis and as you divide by smaller smaller and smaller decimals so so 2 divided by 0.1 would actually be twenty so that's really how 2 divided by 0. not not one would be equal to 200. so what happens is the graph will just shoot off to infinity and it would never actually reach the y axis and similarly whenever we're going across horizontally to the right here and because we're doing two divided by something even if we divided by a million there'll still be a tiny tiny number there would be 0.99 and so on so the graph will approach the x-axis but never actually reach it they're what we call asymptotes it's quite a fancy web but that's just what they're called okay and these points on our left hand side of the table we've got negative 0.5 and negative four so that would be there then we've got negative one negative two so negative 1 negative 2 would be there negative 2 negative 1 would be there negative 5 negative 0.4 would be there so we've got our points and we're going to draw a nice curve through them so that would be the graph of y equals two over X and if you had something like y equals 4 over X it would have the same shape but it would just be slightly further out so it'd be slightly higher up and slightly further down and so on okay let's have a look at our next topic so the next topic is cubic graphs so here we've got the graph y equals X cubed so cubic is whenever we've got this cubed and here we've got Y equals X cubed so here we've got our X Y table so let's Cube all these values to get our y values two times two times two is eight one times one times one is one zero times zero times zero is zero negative one times negative 1 times negative one well negative 1 times negative one is one and times that by negative one is negative one because a negative times a negative times a negative is a negative then we've got negative 2 so negative 2 times negative 2 is 4 times negative 2 is negative eight and finally negative or three times negative three times negative 3 would be negative 27. and now let's pop these points on our grid let's draw our curve so it'll come upwards we'll go through the points and then what will happen is it'll flatten out and come through zero and then curve back upwards and look something like that and excuse my sketch it is free hand on the computer is quite tricky and now let's have a look at our y equals negative X cubed so what this means is we've got a cube our value of x and then we're going to make it negative so we're going to do two cubed so 2 times 2 times 2 is it but we're going to make it negative so it's going to be negative 8. then we've got 1 times 1 times 1 which is one but we're going to make a negative so negative one then we've got zero times zero times zero zero well that's just zero then we've got negative one times negative one times negative one that's negative one but we're gonna make it negative well that means then because there's already a negative means we're going to make a positive it's going to be one and then we've got negative two so negative two times negative two times negative two is negative eight but then we're going to make a negative but it's negative already so it makes it means make it positive we're going to change the sign so that's it and then finally we've got negative three times negative three times negative three which is negative 27. we're going to make a negative it's already negative so it's going to be twenty seven and another way to consider this negative signs remember if you have got something like X cubed that means One X cubed or One log of x cubed if you've got Negative X cubed that's the same as negative One X cubed so really what we're doing is we're finding our value for x cubed and then we're multiplying it by a negative one so here we had two cubed which is eight multiply by negative one is negative eight whenever we had our negative one cubed that's equal to negative one when we times it by negative one that's one and so on and just what it does is just it just changes the science whenever there's a negative sound in front of it now let's pull out points okay we now plot out of points now let's try and draw a curve and that's meant to be a curve and what you'll find is the same as the execute graph but it's just flipped around the other way so our next topic is sequences and we're going to start off by finding the next terms in some sequences so our first sequence goes 52 50 48 46. so as you can see the sequence is going down by two each time we're taking away two taken away two taken away two so that means if we take away another two that would be 44 and if we take away another 2 that would be 42. so we'll find the next two terms in that sequence okay let's have a look at our next sequence now you may recognize these numbers earlier on this video we looked for our square numbers one times one is one two times two is four three times three is nine and so on so four times four sixteen five times five would be 25 and 6 times 6 is 36 so there would be our two missing numbers because we've got one times one two times two three times three they are square numbers another way just in case you didn't spot there were your square numbers you can see that we added three so add three then we added five then we added seven so if we added nine we would then get 25 and if we added 11 we would get 36 but hopefully you would spot that they are your square numbers okay let's have a look at our next sequence but this time we've got 8 9 11 14 well we're adding one to go from eight to nine we're now adding two to go from 9 to 11. we're then adding three to go from 11 to 14. so if we add 4 so add 4 we would get 14 plus 4 is 18 and then if we added five well eighteen plus five would be 23. so the two missing numbers in this sequence would be 18 and 23. okay the next sequence of numbers well I can see again that these are our special numbers we've got our square numbers here these are R cubed numbers one times one times one is one two times two times two is eight three times three times three is twenty seven four times four times four sixty four five times five times five is one hundred and twenty five and six times six times six is two hundred and sixteen so it's very important to know those ear Cube numbers as well okay okay next so our next question says find the difference between the second and the seventh triangular number so here we've got a pattern of dots and these are triangular numbers where we start off with one dot then we add a row of two dots beneath it so one plus two is three then we add a row of three dots beneath that so three plus three is six then we add a row of four dots well six plus four is ten then we add a row of five dots which would be 15 and so on so to get our triangular numbers we start with one and then we add two add three add four add five add six and so on and there are triangular numbers and there are one three six ten fifteen twenty one twenty eight thirty six and so on that's it so they're your triangular numbers and let's just check we need our second and our seventh triangular number so our second one is three and our seventh one would be one two three four five six seven and the difference between those well we've got 28 take away three would be 25. so the difference between the second and the seventh triangular number would be twenty-five so the next topic is to generate a sequence and we're going to be given a rule to generate a sequencer to create a sequence and this is video 290 and corporate maps and we've been told a sequence is created by using this rule so we add 40 and then divide by two so that's the rule you just add 40 divided by two add 40 divided by 2 and so on and we've been asked to find the next three numbers in the sequence so we're given the first number which is 80. so we're going to take our 80 and we're going to add 40. so 80 add 40 is equal to 120 then we're going to divide by 2 so 120 divided by 2 is equal to 60. so that means our next number in the sequence is 60. now we're going to take our next number 60 and we're going to add 40 so add 40 and then that's equal to 100 and then we're going to take our 100 and we're going to divide it by 2. so 100 divided by 2 is equal to 50. so the next number in our sequence is 50. next we take our 50 and we follow the rule add 40 and divide by 2. so we're going to take our 50 and we're going to add 40 and that's equal to 90 and then we're going to divide by 2 90 divided by 2 is equal to 45 so that's equal to 45 and so on okay so carrying on with sequences our next Topic in sequences is pattern so we've got these patterns and it's video 219 corporate maps and we've been asked to draw a pattern for it so we've got pattern one which is this pattern adult then we've got pattern two where we've added an extra dot above an extra dot below and an extra dot to the left then we've got pattern three we've added another one on the top another one below another one to the left so just keeps on getting bigger by adding on another dot on the top below and to the left Okay so we've been asked to draw pattern four so we're just going to take pattern three and add another dot above below and to the left so that would mean that it would have one two three four five six seven eight nine so we'd have one two three four five six seven eight nine and with the middle one so one two two three four five this one and we then have instead of having one two three to the left of it we'll have another one which is four so one two three four so we just take what pattern three was and add another above below and to the left that's it and our next question says how many dots will there be in pattern five so in pattern one there was four dots one two three four and pattern two well there's one two three four five six seven we're adding three each time so this would then have ten this would have thirteen and on pattern five there'd be 16 dots because we're adding three each time adding one above one below and one to the left that's it okay and another type of sequence that we should know for our GCSE Foundation Maps is our Fibonacci Sequence so Fibonacci sequences we choose two numbers such as one and one we find the next term of Fibonacci sequence by adding the two previous terms so one plus one is equal to two now we do one plus two is equal to three now we do two plus three is equal to five then we do three plus five is equal to eight five plus eight is equal to thirteen and so on and that's it so if Fibonacci sequences to find the next number if we add the two previous numbers so here's an example it says find the next three terms of this Fibonacci style sequence so we've got two seven nine and sixteen so to get the next number up we need to add the nine and sixteen so nine plus sixteen is twenty-five to get the next number we're going to add 16 and 25 that's equal to 41 and finally to get our next number we're going to do 25 plus 41 and 25 plus 41 will be equal to 66 and that's it so Fibonacci sequence is where we find the next term by adding the two previous terms okay let's have a look at our next topic so our next topic is the nth term so whenever we've got a sequence there's often a rule or an M term for it which means that we can find out any term in the sequence by substituting which value we're looking for so here we've got our sequence 3 8 13 18 and so on as you can see we've gone up in fives because we're adding five adding five adding five and so on and we've been asked to find the nth term of the sequence in other words the riddle for the sequence so if we want to find the nth term our first step is to consider the sequence and see what is going up in or going down by and this as the sequence has gone up in fives we're adding five adding five adding five what we're going to do is we're going to write the five times tables beneath the sequence so the five times tables are five ten are the multiples of five five ten fifteen twenty and so on and because we're multiplying the numbers by five we're doing five times one is five five times two is ten five times three is fifteen and so on we're going to write five n because that means five multiplied by n because five times one five times two five times three and so on so if the sequence has gone up in files we write the five times table so the multiples of five benefit and we write 5 and I'm from now our sequence is not 5 10 15 20. our sequence is actually 3 8 13 and 18. and to get from five to three we take away two to get from ten to eight we take away two to go from 15 to 13 we take away two to get from 20 to 18 we take away two so if we took away two from all of these numbers we would get our sequence so that means if we had five n minus two that would be 3 8 13 and 18. that means our F term is five n minus two five n minus two and let's just check it well the first term in our sequence well if we do five times one is five take away two is three great the second term in our sequence is eight what if we do five times two that's ten take away two is it fantastic our third term five times three fifteen take away two is thirteen and so on let's work out another one so if we get the sequence 7 9 11 13 and so on as the sequence is getting bigger by two each time we write the multiples of two being the sequence so that's two four six eight and so on and that's 2N our sequence though isn't two four six eight our sequence is 7 9 11 and 13. so to get from the multiples of two to our secants we'd have to add five add five add five and add five so that means our nth term would be two n plus five so that means the nth term of the sequence would be two n plus five and let's just check it two times one is two plus five is seven two times two is four plus five is nine great two times three is six plus five is eleven and so on so to find the M term of a sequence you consider what the sequence is going up or down by you right there multiples beneath it and then you figure out what you would need to do those multiples to get to the sequence now because I said down by let's just have a look at an example where a sequence is going down instead of up so let's have a look at one of those now so if we had the sequence fifteen twelve nine six and so on and if we look at our sequence we can see we're going down by three each time we're taking away three taking away three take away three and so on so that means that what we're going to do is we're going to write down the negative three times tables so that's going to be minus three minus 6 minus 9 minus 12 and so on so that would be minus 3 n because minus 3 times 1 minus three times two and so on but our sequence is not minus three n it's not minus three minus six minus nine minus twelve our sequence was fifteen twelve nine six to get from -3 to 15 we would have to add eighteen to get from -6 to 12 we would have to add eighteen to get from minus nine to nine we'd have to add eighteen and so on so that means our nth term for our sequence would be minus three n plus eighteen and let's just check it our first term is 15 minus three times one is minus three plus eighteen is fifteen and so on so that is our term for our sequence it was going down now the F times really really useful because what we can actually do is we can use the M term to work out terms in the sequence if we have the sequence 3 8 13 18 and so on and someone asked me to find the hundredth term in the sequence rather than carrying on the sequence for 100 terms I can use this nth term so I can do if I want the hundredth term I can do 5 times 100 which is 500 take away 2 which would be 498 so the hundredth term in the sequence without having to list them all down would be 498 and that's great so it saves us a lot of time and effort another reason why the average time is quite useful is we can use it to work out if the terms in the sequence or not so for instance if somebody came along to me and said is 200 in this sequence well the first one I can tell it's not because our sequence goes 3 8 13 18 and so on and as you can see the numbers end in a three or an eight a three or an eight so because 200 ends in a zero it's not going to be in the sequence but we can actually use the nth term to show that we can actually write five n minus 2 the nth term and write equals 200. so that's an equation and if we solve this equation and we get a whole number so we get n equals a whole number it would mean that 200 is in the sequence and whatever whole number that is tells us which term in the sequence it is but if whenever we solve this equation we get a decimal number it would mean that the number 200 wouldn't be in the sequence so let's solve our equation and see what happens so if we had 5m minus 2 equals 200 we would add 2 and add two well we add 2 to get rid of the minus two so we're left with five n and on the right hand side we've got 200 add 2 is 202. now we've got 5n we don't want 5n so let's divide by 5 and divide by 5 so that would give us 5n divided by 5 is n and 202 divided by 5 would be 40.4 and as that's a whole number that means that 200 is not in the sequence and because it's 40.4 it means it's not the 40th term and it's not the 41st term in the sequence so that means that 200 is not a term in the sequence okay let's look at our next topic within sequences so we've now got arithmetic and geometric progressions and that's video 375 in corporate Maps so an arithmetic progression is where we've got a sequence of numbers where there's a common difference so in other words here we've gone from 6 to 8 we're adding two we're then going from 8 to 10 so we're adding two we're then going from 10 to 12 so we're adding 2 and so on so because of that common difference of adding 2 adding 2 adding two adding two that means that's an arithmetic progression here's an example of another arithmetic progression so we can have 50 60 70 80 and so on and as you can see there's a common difference we're adding 10 each time also we could have it going down so we could have 2017 14 11 and so on and as you see there's a common difference we're going down by three each time so these are arithmetic progressions where we're adding or subtracting the same amount each time there's a common difference and the geometric progression is where we've got a sequence of numbers where we're multiplying by the same amount each time there's a common ratio so you can see from three to six we're multiplying by 2 from 6 to 12 we're multiplying by 2 12 to 24 multiplied by 2 24 to 48 multiplied by two and so on so here are some more geometric progressions we could have 1 5 125 625 and so on and here the common ratio is five we're multiplying by five each time so that is a geometric progression now geometric progressions can get smaller so for instance if we started off with 100 and we multiply it by 0.5 and multiplied by 0.5 and multiply by 0.5 and so on if the common ratio was a note Point number the sequence of the geometric progression will get smaller rather than larger and that's it so these are arithmetic regressions in geometric progressions so an arithmetic progression is where we've got a sequence of numbers where we're adding the same amount each time if it's a positive number we're adding it's getting bigger if it's a negative number routing each time we're taken away it would get smaller and if we had a geometric progression that's where we're multiplying by the same number each time and that's it so next topic is simultaneous equations and here we've got a pair of simultaneous equations we've got 3x minus y equals 23 and 2x plus 3y is equal to it so whenever we could simultaneous equations what we want to do is cancel out one of the letters we want to add or subtract the equations so that one of the letters disappears so as you can see here we've got minus y and we've got 3y now if we had minus 3y and 3y we could add them together to get zero so what I'm going to do is I'm going to multiply this top equation by 3 to begin with so less than with them one and two so we've got equation one and equation two we're going to do three times equation one so whenever we do three times equation one we'll multiplying this by three gives us three times three x is nine X then three times minus y was going to be minus three Y and three times twenty three well three times twenty three is sixty nine then we've got equation two which is two x plus three y equals eight we've got one equation with minus three Y and one equation with three y so that's fantastic what we can do now is we can add these two equations together and our minus three Y and our three y will add together to give us 0 so we'll just be left with x's and that's fantastic okay so let's add the two equations together so let's write the word add I tend to avoid putting a plus or minus symbol just in case if it was a subtract I put a minus here in front of the two and make it confused and make it to minus two x so error add or sub now what I'm going to do is I'm going to add these two equations together so nine X plus two x well that would be eleven X minus 3y plus three why well that's zero so they cancel out and finally we've got 69 Plus 8 that's equal to 77. so we've got 11x equals 77. now we can divide by 11 and divide by 11 and we get the x equals seven so x equals seven so that's fantastic so we know X is equal to seven so let's now find y so let's substitute this x equals 7 into one of our equations and I'm going to substitute into equation two just because we have positive 3y and let's write that down to sub x equals 7 into 2. just to show what we're doing so whenever we substitute x equals 7 into this equation we get 2 times 7 so that's 14 plus 3y equals it now we're going to solve this equation so minus 14 and minus 14. well we had 14 plus 3y and we took away the 14 so we're just left with 3y and on the right hand side we've got 8 take away 14 that's going to be equal to negative 6. now we're going to divide both sides by three and whenever we divide both sides by three well three y divided by 3 is just Y and negative 6 divided by 3 is equal to negative 2. so that means we've got our answers we've got x equals seven and Y equals negative 2. and let's just check our answer now we substitute X into equation two to find our value of y let's now substitute both of them into equation one so let's check in one so check in one well if we put these numbers if we substitute these numbers into equation one we've got three times x so that's 21 3 times 7 is 21 minus y is minus two so we're going to do minus minus two and that's meant to be equal to 23. so let's check it 21 minus minus 2 that's 21 plus 2 and that's 23 so that's right so we know we've got this question right and that's it okay let's have a look at our next question so next question says solve the simultaneous equations 2x plus 3y equals 1 and 7x plus 2A equals negative 22. and I've chosen these numbers on purpose because what we're going to have to do is multiply both equations by certain numbers to get uh the same number in front of e for the X's or y's so let's look at our equations equation one and equation two now we could calculate our X's so we could multiply the top equation by seven to get 14x and we could double equation two to get 14x and we could cancel those or we could cancel our y's so we could multiply let's see if we've got three and two let's get six so we can multiply our top equation by two and our bottom equation by three and that would give us 6y and six way and because they're both six ways we would then take the equations away from each other to get zero let's actually do that let's cancel our y's so let's multiply equation one by two because we want to get six y so we're going to do two times one and when we do two times equation one we get we'll multiplying this whole equation by two would be four X plus six y equals two now we've got equation two but we want to get six y so let's now multiply this whole equation by three so let's do three times equation two and when we do that we get multiplying these all by three we get three times seven X is Twenty One X three times two I must be plus 6y and three times minus 22 that's going to be minus 66. so this is great because we've now got two equations both with six y's in them now what we can do is we can now take away these equations from each other the only thing is that whenever we take these away we're going to have 4X take away 21x which would be negative 17x and so on so what I'm actually going to do is I'm going to write this top equation again just beneath this other one so I'm going to write 4X plus 6 x y equals two just so that whenever I take these equations away from each other I'm going to get 17x rather than minus 17x so now let's put a line beneath them and let's say subtract so we're going to subtract these equations from each other so 21x take away 4X that's 17x 6 YT equals 6 5 well that's zero they cancel out and then we've got minus 66 take away two so when we've got minus 66 take away two we're going to go down another two so it's going to be minus 68. so we've now got 17x equals minus 68. so let's divide both sides of this equation by 17 so divide by 17 and divide by 17 and we're going to get that x equals negative 4. so we now know that X is equal to negative 4 we can substitute that into either equation one or equation two to get our y let's substitute it into equation one so equation one so sub x equals negative 4 into 1. and when we do that we go well 2 times x well that's going to be 2 times negative 4 and 2 times negative 4 is negative 8 plus 3y equals one now if we want to solve this we want to get rid of our negative 8 so let's add it to both sides so add it and add it that will give us three y equals and one plus it's nine and divide by 3 and divide by 3 and we get Y is equal to 3. so our answers are x equals negative four and Y equals three and again we can check our answer in equation two check in two and when we do that we get well 7 times X was seven times -4 would be minus 28 then we've got plus 2y well Y is equal to 3 so 2 times 3 6 we're going to add 6 and that should be equal to negative 22. and let's just check it negative 28 plus 6 would be negative 22. so that is equal to negative 22. so we're right we know that our answers would have to be X is equal to negative 4 and Y is equal to 3. okay let's have a look at our next question a wordy context maybe where you're buying so many of some fun and so many of something else and you're given the price and so on and you've got to create your own simultaneous equations and solve them so let's have a look at a typical question so we've got five adult tickets and three child tickets cost 58 pound and two adult tickets and eight child tickets cost forty seven pounds find the cost of each type of ticket so we've got our two equations we've got five adult tickets but let's let an adult ticket be X pounds unless that a child take would be y pounds so let's make our equations we've got five x plus 3y is equal to 58 pound so equals 58 and then we've got 2X plus 8y and that's equal to 47 pound so we've got our two equations now what we want to do is either get the same numbers in front of our x's and y's or the same but one being positive one being negative because if they're both the same we can take them away from each other or if they're both the same but once plus and one's negative we can add the two equations to get zero we could cancel out our X's or we can cancel out our Y's just for fun we're going to calculate our X's this time so I'm going to actually cancel out our X's so I'm going to multiply our top equation let's equation one by two to get 10x I'm going to multiply the second equation equation two by five to get 10x as well and then what I'm going to do is take the two equations away from each other so let's do 2 times equation one and when we do two times equation one we get well multiplying these all by 2 would get 10x plus multiplying three y by 2 would be 6y equals and multiplying 58 by 2 would be 116. and then multiplying equation two by five because we want to get 10x so 5 times equation 2 would be multiplying these all by five would give us 5 times 2x is 10x 5 times 8y well it's going to be 4ty and then multiplying 47 by 5 would be 235. now we want to cancel out our 10 X's so if we've got 10x and 10x we want to take them away to get 0. now because we're taken away rather than 6 y t equal to 40y I'm actually going to write this top equation just beneath the other one again so let's scroll down and let's write 10 X plus 6y equals 116. and let's write sub because we're going to subtract them from each other so 10x take away 10x is 0. 40y take away 6y well that's 34y and then we've got 235 take away 116 and when we do that we get an answer of 119 pounds or 119. so we now know that 34y equals 119 we can divide both sides by 34 to get y equals 119 divided by 34 is equal to 3.5 or 3.50 so we now know what Y is and that's the cost of a child ticket which is 3.5 or 3.50 let's find the cost of an adult ticket so let's sub y equals 3.5 and we can choose either equation one or equation two I'm going to choose equation two so n 2 2. when we substitute 3.5 in for the value of y into equation 2 we're going to get we've got 2x plus and then we've got 8 times 3.5 that's equal to 28 and that's equal to 47. so we've got 2x plus 28 is equal to 47 so let's take away 28 from both sides so that gives us 2X equals and 47 take away 28 was equal to 19 and then if we divide that by 2 we get x equals and dividing 19 by 2 is 9.5 so as the question says find the cost of each type of ticket so we've got an adult ticket which is X and that's going to be equal to 9.5 or 9.50 so let's write it as a price 9.50 and the child ticket which was equal to Y is equal to 3.50 and that's it okay let's have a look at our next topic and our next topic is solving simultaneous equation by using graphical approaches in other words drawing the graphs and finding out where the graphs meet and using that to find the answer to simultaneous equations so here we've got a pair of simultaneous equations so that's two equations at the same time we've got y equals three minus x and y equals 2x minus three so to solve this what we're going to do is we're going to draw suitable lines in other words we're going to draw the graph for y equals 3 minus X and we're going to draw the graph of y equals 2x minus 3 and where those two straight lines meet will be our answer so let's first of all draw the graph for y equals 3 minus X so to draw this graph I'm going to draw an X Y table so it's X and Y and I'm going to do a little table like so and if you need to recap on this if you go to corporatemath.com and you go to linear graphs there'll be a topic called Draw and graphs using X Y tables and what I'm going to do is I'm going to choose some values for X with 0 1 2 3 and I'm going to work out the values for y so y equals 3 take away X so if I take away X from 3 I can find the value for y so if x is equal to 0 well 3 take away 0 is 3 so that means the Y is 3. 3 minus X well if x is one it would be three minus one that's two if x is equal to 2 or 3 minus two is one so that'll be one and if x is equal to 3 I'd have 3 mandatory which is zero so the points in this graph will be zero three so zero three there'll be one two so one across two up there'll be two one so two across one up and there'll be three zeros so three across and zero up like so and if I get my ruler and pencil I can draw a nice straight line through those points and I would look something like this and that's it that's the graph for y equals three minus X now in a typical question I would imagine what they would is draw one of these two lines for you and you just have to draw the other one and then find out where they meet okay let's have a look at our next one so the next graph is y equals two x minus three so we'll do an X Y table X and Y and we'll choose some points and again I just tend to use zero one two three and I choose a few of them or in this case I choose four of them just so whenever I plot them I can see that all four they're on a nice straight line if they're not in a straight line then I know I've done something wrong so what I'm going to do is to find y we do 2 times x and we take away three so we're going to times all these numbers by two and then take away three so let's start off with zero two times zero is zero take away three would be minus three then we've got one well two times one is two take away three would be minus one now we've got two two times two is four take away three is one and finally we've got three two times three is six t equal three is three so we've got our coordinates zero three one minus one two one and three three so let's plot them zero minus three one minus one two one and three three and get our ruler and pencil we'll draw a nice straight line through those points and as you can see here the two straight lines meet at the point two one this is the point of intersection here two one there so that means to me that the point two one now when we're solving some multinos equations what we're trying to do is find the value for x and we're trying to find the value for y so if we look at this point two one we're going two across that means the x value is 2 because the coordinates two one the first number is the answer for x and the second number is the answer for y so X is equal to two and Y is equal to one and let's just check that 2 across one up the x value is two and the Y value is one and that's it okay let's have a look at our next topic so our next topic is tally charts and whenever you're doing the tally chart it's important to know that whenever you go to Five People You go one two three four and the fifth person is across and then you just carry on so here we've got a tally chart and we've got Monday There's 5 10 12 and that's the frequency 12. on Tuesday there's one two three so let's complete this three on Wednesday the frequency seven so we need to do seven so it'll be one two three four five and then we've got two more six seven on Thursday let's count up in fives five ten fifteen and then another four that's going to be 19 people on Thursday and finally Friday there's 10 so that's gonna be one two three four five one two three four five and that's it so tally charts if you want to revise tally charge this video 321 on corporate Maps our next topic is frequency tree so here's a frequency tree and we've got more an appointment even an appointment so we've got this 40 appointments all together we know there's 23 in the morning and there's some in the evening and then we've got on time late on time layer so we've got 40 appointments all together 23 in the morning so if we do 40 take away 23 we'll find out how many evening appointments there were so 40 take away 23 is 17. so there must have been 17 even appointments because they need to add together to be 40. if we focus on the 23 morning appointments 21 were on time so that means there must have been two that were lit because 21 plus this number must be 25 that must be two and in the evening appointments where there was 17 altogether five were laid so if we do 17 take away five that's 12. so 12 must have been on time and that's it so we've completed the frequency tree and video of 376 and corporate miles we'll give you more information on that okay let's have a look at our next topic so our next topic is two-way tables and if you want to revise two-way tables it's video 319 on corporate Maps so here we've got a two-way table we've got some subjects at the top English and art and then total and then we've got our information about whether they pass the course or failed the course and again we've got total at the bottom so we've been asked to complete this two-way table well we haven't but I'm saying it now we're going to complete this two-way table and so we've got um first of all let's look at Art we've got 12 students that are failed art of 19 in total so 19 shouldn't study that and 12 of them failed so it must be quite a hard test and we would we can find out how many students passed because we know there's 19 all together and 12 Fields so if we take that 12 away from 19 that leaves us with seven so seven students must have passed the r course next well I'm looking at this top row here for how many students passed English and art and we can find this total we know 25 students passed English and seven students passed art so if we add those two together we can get the total 25 plus 7 is 32. so 32 students passed their courses next I'm looking at these total so we know 32 students passed the course and 13 failed their course so if we add together the 32 and the 13 we can see how many students are in total so 32 add 13 is 45 so let's put that in next let's look at the students that failed their courses we'll also give our 13 students filled their course and 12 of them were from art so that means that only one student failed English so that means that we've got that number of one and then finally how many students studied English in total well we've got 25 that passed and one that failed to all together that b26 and let's just check our answers 26 plus 19 is 45. so that's a two-way table so next topic is pictograms so you can find videos on pictograms on video 161 and 162 in corporate Maps so here's an example on a pictogram so a pictogram uses symbols to represent a number and here we've got a pictogram shows the information about the number of hours of sunshine in four cities during the day in June so we've got Paris cork London and Swansea and we've got each circle here and we've got a key and it's very important that if you are drawing a pictogram you include a key we've got this key a circle represents four hours so that means that the whole circle represents four half a circle would represent a semicircle would represent two a quarter of a circle would represent one and three quarters of a circle to represent three so let's have a look at Paris we've got four and then we've got another four so that's it and then another two so that's going to be ten all together then you've got corks that's four eight twelve so there's 12 hours of sunshine and cork London so you've got four and then another three so that's seven all together over there and then you've got Swansea and you've got four eight and then another one so that's nine all together there and the question says how many more hours of sunshine did Paris have than London so Paris hit 10 and London had seven to find out how many more hours of sunshine we just need to take those away to find the difference 10 take away seven is three so the answer would be three hours okay and our next topic our next topic is bar charts and that's videos 147 and 148 in corporate Mars and here's a typical question we've got a bar chart we've been asked some questions about it so here's a bar chart showing the number of ice creams sold gone up vertically from zero up to 400 and we've got day of the week going across the horizontal axis Monday Tuesday Wednesday Thursday Friday and the first question says on which two days with the same number of ice cream sold so we're looking for the same number of ice cream sold we're looking for the bars being the same height so if you look here we've got weddings day and Friday where we've got the bars the same holidays each other so that means a Wednesday and Friday would be the two days of the week where the same number of ice creams were sold so Wednesday on Friday okay the next question says how many more ice creams were sold on Thursday than Friday so if we look at Thursday we've got our bar going up here and we've gone past 300 so let's see what this number would be so we've gone up from zero up to 100 so let's see how many squares there are going up from zero to 100 so we've got one two three four five so if we divide 100 by 5 we get that's equal to 20. that must mean we've gone up in 20 so let's check 20 40 60 80 100 yeah so if we're at 300 that's going to be 320 and 340. so all together there are 340 ice cream sold on Thursday and on Friday well we've got 100 then it's going to be 120 40 60 and 80. and then everyone will be 200 so yes that's 180. so it says how many more ice creams were sold on Thursday than Friday so if we take away 180 from 340 that would tell us how many more ice creams were sold and that would be 160. okay let's have a look at our next topic so our next topic is a bar chart sounds video 148b on corporate Maps so here we've got a dual bar chart we've got this key that shows us in Blues last year and in yellow we've got this year and we're told the Dual bar chart shows the number of goals scored in the cut by three ice hockey team so we've got the teams at the bottom the flames and these two bars are for the Flames last year and this year we've got the Steelers last year and this year and we've got the blizz last year and this year and Catherine says that the three teams scored more goals in the Cup last year than they did this year is she right so let's have a look at the Flames so the Flames scored six goals last year so let's write six above that bar and this year the flame scored 14 goals so the flame scored a lot more goals this year than last year the Steelers they scored seven goals last year you can see it's seven because in between six and eight so seven last year and seven this year they've scored the same number of goals last year and this year and the blaze they scored 11 goals last year you can see it's 11 because in between 10 and 12 so it's 11 last year and this year they scored four so Catherine says the three teams or more goals last year than this year so let's add up what they got last year so let's write last year and last year they scored six for the Flames seven for the Steelers and 11 for the Blazers so let's add those up and see what we get six plus sevens for 13 plus 11 is 24. so 24 goals were scored last year let's have a look at this year so this year the flame scored 14. the Steelers scored seven and the players scored four so we're gonna do 14 plus 7 plus 4. and 14 plus 7 is 21 plus 4 is 25. so we've got one more goal this year than last year now the Catherine said the three teams scored more goals in the Cup last year than this year well no they didn't score more last year they scored more this year so is she right now they scored more this year okay let's have a look at our next topic so our next topic is composite bar charts and that's video 148 a on corporate Maps so here's a composite bar chart and it's a bar chart but yeah as you can see each of the bars is split up into different sections so we've got this bar for January and in if we have a look at our K in pink we've got hot drinks and in Gray we've got cool drinks we've then got up for February the hot drinks and the cool drinks and for March we've got the hot drinks and the cool drinks and we're told this composite bar chart shows us information about the number of drinks sold over three months so if we look at the bars and we look at the totals in January we can see the 600 drinks sold in February we can see the 650 drinks sold and in March is 750 drinks sold but also because it's a composite bar chart we can actually see the breakdown of cold drinks and hot drinks sold as well by you looking at each of these regions so for instance for January there's 450 hot drinks sold and there's 150 cold drinks sold and we can see the same for February and March and the question says in which two months were the same number of hot drinks sold so let's have a look at our key in the question we wanted to find the two months that got the same number of hot drinks sold so that's going to be the pink sections of each bar so if we look at the bar for January and the bar for February they both go up to 450 so that means that in January there's 450 hot drinks sold and in February there's 450 hot drinks sold whereas in March there's not as many hot drinks could sold maybe it's a bit warmer and a March is only actually 300 hot drinks sold so in which two months with the same number of hot drinks sold that will be January and February our next topic is line graphs so here we've got a line graph and we've got frequency gone up vertically and we've got the time of the day going across horizontally so we'll go 0 20 40 60 80 100 gone up vertically and then we've got the time 9 o'clock eleven one o'clock three o'clock five o'clock and seven p.m so we're going up in two hours there as we go across and the question says Sally records the number of cars in a car park every two hours so this is the number of cars in the car park at each of the times she began at 9am and finished at 7 pm the line graft shows her results when were the most cars in the car park that'd be whenever the line graphs at its highest point and as you can see here it had 90 cars at 1 pm and the question said when were most cars in the car parks that would be 1pm the next question said how many less cars were in the car park of 3 P.M the 1pm so if we look at 1pm well we know there's 90 cars then and if we look at 3pm we've got us just above the 60 let's see what we're going up in here so we've got we've got 10 squares and that's 20 cars so 20 divided by 10 is 2 so we went up in twos so that would be if we counted up the twos that would be 20. so we've got 60 62. so if we do 90 take away 62. that will tell us how many less cars were in the car park at three o'clock than one o'clock and that would be 28. next topic is pie charts so here we've got our table we've got rugby team and we've got frequency so 90 rugby fans were asked which team they supported England France Ireland Scotland and Wales and we've got 20 supported England five supported France 15 supported Ireland 25 support Scotland and 25 supported Wheels okay and we're going to draw a pie chart for this information so if you run a pie chart typically they'll be Circle drawn for you the line there going from the center up to top for you and we're going to draw a pie chart for this so to draw a pie chart the first thing you need to do is add up the frequencies well we know there's 90 Ruby fans so if we add them all up we'll find this 90. then we need to divide the whole circle 360 by that number of people so 360 divided by 90 is equal to 4. so that means that each person is given four degrees in the circle so each person is 4 degrees now 20 people supported England now each of those people get four degrees so if we multiply that by four we'll see how what sides of angle they should have in the pie chart 20 times 4 is equal to 80. so the angle we're going to draw for England fans would be 80 degrees next we're going to multiply 5 by 4 and we're going to multiply all these numbers by 4 because each person gets 4 degrees so we're going to multiply by 4 multiply by 4 multiply by 4 and multiply by 4. so 5 times 4 is 20 degrees so we'll draw 20 degrees for the French fans 15 times 4 is 60 degrees for the Irish fans that should be much bigger then we've got 25 times 4 25 times 4 is 100 so 100 degrees for the Scottish fans and 25 times 4 25 times 4 is 100 and then we've got all our angles the Highlight to add these up to make sure I get 360. 80 plus 20 is 100 plus 60 is 160 plus the number 100 is 260 plus another 100 is 360 Degrees that's it so whenever you're on a pie chart you need to add up the frequencies then divide 360 by the total frequency to find the number of degrees per person or per item or whatever you're looking at and then multiply all the frequencies by that number to find the angle you should draw okay so we're going to draw 80 degrees for the English fan so let's start off by going to our pie chart and we're going to get our protractor and we're going to line it up like so so the cross goes in the in the center of the circle like so and that the zero goes at the top and we're drawing an 80 degree angle so we're going to go around to our 80 degrees is so that's here and we'll do a little Dot now you'll move your protractor and you will draw a nice straight line so let's move our protractor and you draw a nice straight line from the center through that point to the edge of the pie chart like so so that's for the English fans so let's label it for English fans for England so like so and that's that sector done and it's an 80 degree angle so we can put that in as well if we want to next we're going to draw an angle of 20 degrees for the French fans so we're going to get our protractor we're going to have to turn it so that we put our zero along the line we've just drawn you always put the zero on the line you've just drawn or if there's no lines There Yet the one at the top so we started off with the zero at the Top Line because I was only line now we've drawn this line we're going to put 0 there and we're going to draw an angle of 20 degrees so we go to zero we go around to where 20 degrees is that's here so put a dot and then move our protractor into a nice straight line through there with a pencil so it looks something like that so that's 20 degrees you don't necessarily need to put the angles in but we do need to put it in we do need to label that it's France okay next was Ireland and if we go back and look that was 60 degrees so we're going to get our protractor we're going to turn it so the zero is along the line we've just drawn so we're going to rotate it slightly again the cross has to go in the center of the circle and zero is on the line like so and then we're going to go from zero around to 60 degrees for the Irish fans so to there 6 see move our protractor so move our protractor and draw a nice straight line from the center of the circle through that point and to the edge and again what we're going to do is we're going to label it for Ireland and we can put the angle in if we want to it's at 60 degrees so you don't need to do that bit but again label it for Ireland next was the Scottish fan so that was 100 degrees so again take up retractor and rotate it so that the zero is on that new line like so so start off with zero and we're going to go around to 100 degrees so we're going to sign that out go around 10 20 all the way around to 80 90 and 100. so remember we are dealing with the outside numbers here so don't look at this in inside 100 we're dealing with the outside number so we're going zero all the way around to 100 there again move our protractor and draw a nice straight line from the center of the pie chart through that point to the edge and that will be for the Scottish fans so Scotland to 100 degrees and finally the last sector will be for wheels that should be already drawn Force for 100 degrees because that's all that's left but let's check it so let's take our protract and rotate it and when you do that you can clearly see that is 100 degrees and that's it Okay so we've had a look at drum pie charts now let's look at reading them or interpreting pie charts and that's video 164 on corporate Maps so we've been told 180 years students were asked how they travel to school and the pay chart shows their responses so we've got bus some travel to School by bus some walk some travel by car and some cycle and we're asked what fraction the students travel by bus so sometimes when we're given a pie chart we're asked to find a fraction or a percentage so we could be asked what fraction the students walk to school or what percentage of students travel by bus to school and to do that what we're going to do is we're going to look at the whole circle the whole pie chart and figure out what percentage or what fraction is that region so if we have a look at this sector for a bus we can see it's got a right angle that means it's a 90 degree angle and a 90 degree angle will be a quarter of a full circle somebody said this is a quarter of the pie chart so what fractions students travel by bus a quarter off them if we were asked what percentage students traveled by bus that would be 25 because a quarter is 25 and think back to our fractions decimals and percentages if we were asked what fraction students walk to school because that's 120 degrees to find what fraction students walk to school we would write that as a fraction we would write down 120 out of the whole circle which is 360. and we would cancel that down we could divide both of these by 10 to get 12 over 36 and then we could divide both of them by 12 to get 1 over 3. so if we were asked what fraction students walk to school the answer would be one third if we're asked what fraction students traveled by car we would write 80 out of 360 or 80 over 360 and then just cancel it down it can be useful to remember some key angles whenever we're dealing with pie charts so for instance if you've got 180 degrees a straight line 180 degrees would be a half if we had 120 degrees because three 120 degrees is 360 Degrees that means 120 degrees would be a third if you get 90 degrees that would be a quarter as we've seen a quarter if you've got 60 degrees that's a 6 because 660s is 360 degrees another useful one to remember would be 36 degrees that would be a 10th because 10 times 36 is 360. so those are some useful ones to remember but if you don't remember it or if you give it an angle such as 80 degrees what you can just do is write that out of 360 or over 360 and cancel it down so our next question says how many students walk to school so we've got 120 degrees for walk so that's going to be a third so that means that a third of the students who walk to school and all together there were 180 students so we need to work out a third of 180 and to get a third of 180 we're just going to do 180 divided by three to get a third of a number you just divided by three so 180 divided by 3 is equal to 60. so that means that 60 students walk to school if we wanted to find out how many students traveled by bus to school because there's a quarter we would find a quarter of 180 and so on and we're going to look at the probability skill so here we've got the probability scale where we've got zero is impossible so if something's got a probability of zero that means it's impossible it cannot happen for instance rolling is seven on an ordinary dice if with an ordinary Dice and we roll it we we the highest number we can get to six so we can't get a seven so that's impossible so it's got us probability of zero then we've got in the middle here we've got even chance so in other words if you flip a coin getting a head or a tail the probability of each one would be 0.5 an even chance anything in between those so anything bigger than zero or less than 0.5 is unlikely to happen now some of them are very unlikely perhaps winning the lottery is going to be almost zero it's very very very very very very unlikely whereas maybe you know rolling a number or two or three and a dice because there's two out of six and that's a third so it'd be around here somewhere so that's quite unlikely so you've got n of in between zero and 0.5 is unlikely then you've got the probably of one well that's certain to happen so for instance if we chose the day at random and an ending in the letter Y well that's certain because every day ends in the letter Y so it has to happen so then that's certain so that's one an elephant in between 0.5 and 1 and of an in between them is likely happening so in other words a day choosing at random in July Being Sunny well it's likely so to be in there somewhere that's it so this is the probability skill and if you want to do some questions on those if you look at video 251 on corporate Maps there's practice questions beside those but also remember you've got that question that bumper booklet in the description below next let's look at some probability questions okay let's have a look at our next topic so our next topic is probability and that's what we're going to express with the probabilities as fractions decimals or percentages and that's video 245 in corporate Maps so here we've got a question and it says Thomas's 12 tiles each with a letter on it so we've got these 12 tiles they spell out Corbin Mavs c-o-r-b-e-t-t-m-a-t-h-s and we've got these 12 tiles and each of them has a letter on them and he's going to pick one of them at random and we've been asked to find what's the probability that it is the letter s so what we're going to do is we're going to write this as a fraction and first of all let's consider how many different possible answers he could have well altogether there's 12 possible tiles he could pick so we're going to put 12 on the bottom of our fraction on the denominator and then we're going to put how many s's there are on the top so there's one Tau with an s on it so the probability that he could get the letter s is one out of 12 or 112 like so because there's one s and there's 12 Tides all together next we've been asked to find the probability that it is the letter T so again he could pick any of these 12 letters so it's out of 12. and in terms of the ones with t on it there's one two and three tiles with the letter t on it so that means he could choose any of those so it could be three out of 12. so the probability that the tile he picked at random has a t on it would be 3 out of 12 or 3 12. now this fraction could be simplified they're three and twelve are both divisible by three they're both in the three times tables so if we divided both of those by three we would get one quarter but unless you're told in a probability question to simplify a fraction you can write it like so you could write it as three twelfths and only simplify a few tool to so the probability of getting the letter T would be three twelfths okay let's have a look at our last question so our last question says what's the probability that is not the letter c so all together there's 12 tiles one of them's got a c on it and eleven don't so the question says also probably there's not the letter c we'll all together there's 12 tiles and 11 do not have the letter c on it so they're probably for not being the letter c would be 11 out of 12. another way to do that is to say well the probability for C is 112 and they take that away from one and that would leave you of 11 12 and that's how to find and the probably of something not happening to do one take away the probably if it happened but that's it sort of red sometimes it probably is a fraction you put down the total number of possible outcomes on the bottom of the fraction so here you could have picked any of the 12 tiles that goes on the bottom and how many of the ones that you're looking for on the numerator so for the probably if an S well there's one s so it's one out of 12 or 112. the probability for T well there's three tiles with the letter t on it so it's 3 out of 12 or 3 12 or family they're probably if not the letter c well there's 11 without the letter c on it so it's 11 out of 12 or 11 12. that's it okay let's have a look at another probability question so here's our question it says there are only pink yellow green and blue countries in a bag so there's four different colors of countries in the bag and the table shows the probability of picking each color so the probability of picking a pink counter is 0.5 the probability of picking a green counter is 0.1 the probability of picking a blue counter is 0.2 and we don't actually know the probability of picking a yellow counter and the question says find the missing probability so that's the problem probability of picking a yellow counter because there's only pink yellow green and blue counters in the bag there's no other colors that means it's certain that we're going to pick one of these colors so I mean to these probabilities will add together to give us one so that means that if we add together a 0.5 the missing probability 0.1 and 0.2 they will add together to give us one so if we add up the properties that we know so if we add those properties together we get 0.5 plus 0.1 it's 0.6 plus 0.2 is 0.8 so that means the probability of picking a pink green or blue counter is not 0.8 so to find the property I found in the yellow we're going to take this away from one to see what's left so one take away not point eight is equal to 0.2 tell me it's the probability of picking a yellow counter would be 0.2 and that's it so the next topic is actually the property of something not happening and that's video 250 in corporate Maps so as we looked at previously the probability of something not happening is one take away the probability of it happening and that will tell you the probability of something not happening so here we've got the problem with Hannah when in the competition is 0.28 so the chance of her winning the competition is not a point to it and we've been asked to find out the probability that Hannah does not win the competition to do that we just need to take away not point to it from once we just do one take away 0.28 and that would be equal to 0.72 so if the probability of Hannah winning the competition is 0.28 the probably if we're not winning the competition would be 0.72 because one of these two things will have to happen that means they have to add together to give us one okay let's have a look at our next topic so our next topic is expectation which is video 248 and corporate Maps now if you know this probably if something happened and you know how many times an experiment or something has taken place if you multiply those two things together you'll find out how many times you should expect something to happen so let's have a look at an example we've got a factory make 6 000 plates each day so the factory makes 6 000 plates each day and the probability that a plate is faulty is 0.035 so it's quite unlikely it's very unlikely that one's faulty and it's got a probability of 0.035 and we've been asked to find out how many faulty plates will be expected in one day so if we multiply how many plates that are made by the probability of one of them being faulty we'll find out how many faulty plates we'd expect so if we do six thousand multiply by 0.035 that will tell us how many faulty plates we would expect and that is 21. so we would expect 21 faulty plates if you're asked to find out how many plates were in Fault are you then you could just take the 21 away from 6 000 and that'll tell you how many players would be not faulty okay let's have a look at our next topic okay let's have a look at our next topic so our next topic is called relative frequency or sometimes called experimental probability and that's video 248 in corporate Maps so here we've got some letters and I've done an experiment where we've picked a letter at random from the letters a b or c and the letters picked at random word and I've done it seven times where a a b a c b a so a Letter's been picked a random from the letters a b or c and we've done experiments seven times one two three four five six seven times and these are our results so if we're asked to find the relative frequency or experimental probability of an a we just write down what fraction of our results were is so as we're seven all together it'd be out of seven and there's one two three four A's so the relative frequency of an a would be four sevens if we're asked to find the relative frequency of a b well we've done the experiment seven times so will be out of seven and there's two B so the relative frequency of a b would be two sevens and if we're asked to find the relative frequency of a c it'll be 1 7 because there's one C out of the seven results okay so relative frequency or experimental probability is just fine by writing the number of successes over the total number of Trials okay let's have a look at another type of question now so this time we've been given a table and with two people Susan and Helen and they've done an experiment and Susan's done the experiment 20 times she's span a spinner she's spanner spinner 20 times and on that spinner there's different letters and we're told the number of bees that she got whenever she smile at 20 times was it so if we're asked to find the relative frequency of Spin and a b so because you span the spinner 20 times it would be out of 20 and because she got eight B's the number of successes is eight we write 8 out of 20 or 8 20th like so we could write this as our decimal or we could even simplify our fraction simplify our fraction we'll divide in both of these by four would give us two two fifths so we could write an answer of eight twentieths or two-fifths if the question doesn't tell you to simplify we don't have to so we could just write eight twentieths alternatively because Helen's result is a decimal number we could also give it as a decimal number we could write well two-fifths as a decimal will be no point four okay let's look at our next person Helen and Helen spun the spinner more times she spent 120 times and this time we don't know how many beats she's got but we know the relative frequency if we're getting a b we know the relative frequency is 0.35 now to find the number of bees that Helen got all we need to do is multiply the relative frequency by her number of spins thinking back to we had a topic called expectation where to find how many times you expect something to happen you just multiply the number of Trials by the probability so if you multiply the relative frequency of the experimental probability by the number of Trials it'll tell you how many successes you had so if we multiply 120 by 0.35 that will tell you how many times Helen spinner landed on a b whenever she spell it so that would be 42. okay so our next topic is listing outcomes and I really like this topic and this is part of the Corp mileage revision card and we've got Emily is making a pizza with two toppings so she's gonna make a pizza with two toppings and she can choose from Ham chicken olives and pepperoni so you can choose any two toppings and they've got to be different toppings you can't go for like double ham or double olives so she's going to choose two different toppings and we've been asked to list down all the possible combinations so whenever you're listing outcomes it's very important that you don't just work in a random manner so you don't just go like ham and pepperoni Olive and chicken chicken and pepperoni you work for it in like a systematic way so what I mean by that is you start off with choosing ham and you choose down what can she have we have so she could have Hammer chicken so THC so you can have ham and olives h o and she could have ham and pepperoni so that's HP next we've done all the possible combinations with ham now we're going to move the chicken so she can have chicken and olives so chicken and olives or she could have chicken and pepperoni so it's chicken and pepperoni CP so we've done all the combinations with chicken now we moved to olives now with olives we can only have olives in pepperoni so she can have all of some pepperoni and that's it so in terms of if she had these possible toppings these four possible toppings that she had to pick two different ones the possible combinations would be ham and chicken ham and olives ham and pepperoni so you can have chicken and olives chicken and pepperoni or she could have olives and pepperoni so they would be the possible combinations okay and that's it and this is part of the corbination card and listen outcome so if you have a look at that revision card that'll be useful for you as well okay so our next topic is sample spaces and here we've got a sample space question and it says the counters picked at random from each bag so we've got two bags bag one and bag two in bag one the counters have the numbers two one and five on it and in bag two the countries have the numbers four three and two on them and it says they're kind of just picked at random from each bag and the numbers are multiplied together and we've got this table and this will help us find all the results so we've got bag one and in bag two we've got two three and four and the numbers have been times together so we could get a one and a two well one times two is two we could get a two and a two a two times two is four we could get it we could get a five and a two well five times two is ten we could get a one and a three we'll multiplying those together it's three a two and a three well that'll be six multiplying them five times three would be fifteen one times four would be four two times four would be eight and finally we could get a four and a five or a five and a four and that would be twenty when we multiply them and the question says what's the probability of getting a multiple of four okay so let's look at our outcomes and see which ones are multiples of four well two is not a multiple of four four is a multiple of four tens and up freezing up six isn't fifteen's not four is it is and twentie is so out of our nine possible outcomes four of them will be multiple to four so a probability of picking a multi or four would be four out of nine or four ninths and that's it okay let's have a look at our next topic and our next topic is scatter graphs and they have videos 165 up to 168 on corporate Maps so here's a scatter graph and it shows the cost of ten plumbing jobs so we've got these ten plumbing jobs you've got the duration from naught up to five hours and the cost from not up to three hundred pounds and you've got all the points there and our first question says what type of correlation is showing so because the points are going upwards in other words as the hours are going up the cost is going up that is a positive correlation so positive so if the points were coming downwards in this direction it would be a negative correlation and if the points were just scattered around everywhere that would be no correlation okay our next question our next question says draw a line of best fit and I've just shown the liner best fit like so I've got my real learning pencil I've drawn a line going straight for as many of the points as possible or as close to the points as possible this point here is an outlier so I've just ignored that one and I've drawn my line the best fit so it's as close to possible to these points so I've tried to sort of minimize the distance between my line and all the points that I've been given um I normally go for roughly half an either side but because some are a bit lower down here I've sort of put it around there okay now the good thing is with lines that best fit the examiner has a template that they put over and they've got sort of a range of areas where you can put your liner best fit so um you know you don't need to sort of worry too much about it as long as you think it looks good okay so our first question says draw Lana best of it I've done that our next question says estimate the cost of a job lasts in two and a half hours so if we go under horizontal axis here for hours two and a half hours would be here and if we get our ruler and pencil and so using ruler and pencil whenever you're doing this and go up to the line of best fit and then go across you can clearly see that the cost would be 150 pound so our estimate for the cost of a job lasts in two and a half hours would be 150 pounds next the question says estimate the duration of a job costs them 180 pound so if we got a 180 pound on the vertical axis here and we get our learn pencil and we draw a cross to our line of sfit and then we drill down we can see that's three and a half hours so ours would be 3.5 hours that's it Okay so we've drawn a line of best fit and we've used that to make some estimations okay let's have a look at the next part the next part says Circle the outlier so here we've got a scatter graph and you can see this point just stands out it just stands out from all the rest so it's what we call a night layer okay let's have a look at our next topic so our next topic is stem Leaf diagrams so stem Leaf diagrams is one of two topics which differ really between the exam boards at Foundation level where edxl cover stem Leaf diagrams and frequency polygons whereas OCR and AQA don't so if you study for Ed Excel Foundation you need to watch this and learn this and do the practice questions and write notes and so on if you're studying for OCR or AQA it's only two minutes I would probably watch it because you could practice for the mode and the range in the median anyway uh but feel free to skip ahead if you do OCR AQA but it's up to you okay our next topic is stem and leaf so stem Leaf is a great way to represent information in order so represent numbers in order and we've got here this would be videos 169 and 117 corporate Mouse and the question says the following stem Leaf diagram shows the times taken for 15 people to complete the jigsaw so we've got the key and that's very important for a stem and leave so three line one means 31 minutes so we're dealing with minutes here and we've got 31 39 40 43 46 51 57 57 58 59 60 63 64 66 and 75 minutes and the question said what is the modal time taken so the motor is the most common so we've got 31 39 48 and so on so if you look here we've got 57 and 57 so the modal time taken would be 57 minutes the next question said find the range of the times taken so the range is the largest take away the smallest so the largest amount of time taken was 75 minutes and we're going to take away the shortest time which was 31 minutes and 75 take away 31 would be equal to 44 minutes so the range the difference between the largest and the smallest is 44 minutes and the last question says find the median time taken so the median is the middle value so I like to do this by Crossing off the values and pencil of course across off the smallest and then the largest the next smallest which would be 39 then the next largest 66 the next smallest 40 64. the next smallest 43 the next largest 63 the next smallest 46 the next largest 60. the next smallest 51 next smallest 59 and then cross off cross off and we're left with 57. so the median time taken is 57 minutes and make sure that with this seven you remember it's 57 so it's 57 minutes that's it so the system beliefs it's very important whenever you're doing a stem Leaf diagram to read the key and if you've drawn one yourself make sure you include a key so next topic is the mode and here's part of the chords revision cards so the mode is the value that appears most often so here we've got the number of goals scored in eight football matches and they are two four one zero three two four and two and as you can see two appears most often it appears three times so the mode is two so the most common result okay our next one our next one is the median and the median is found by arranging the numbers in order and then selecting the middle value so here we've got the number of goals scored in seven netball matches so we've got 47 41 51 58 32 55 and 49. now we're looking for the median so that's the middle value so let's arrange them in order so 32 and then we've got 44 and then 47 and then 49 and then 51 55 and 58. now we're looking for the median which is the middle value so I tend to cross them off in pencil okay in case I need to rub them out so cross off the smallest cross off the biggest cross off the next smallest and the next biggest the next one and the next one we're left with 49 so 49 is the median is the middle value once they've been arranged in order that question was quite nice because we had an odd number of numbers so there was definitely one in the middle so what happens if there's an even number of values so let's have a look we've got nine five four ten seven and one so let's arrange these in order so one's the smallest and four five seven nine and ten cross off the smallest cross off the biggest cross off the next two and then we've got five and seven in the middle well to find the median that's in the middle of those two values so in the middle of five and seven will be six so the medium for this one will be six if the two numbers in the middle are the same well then there would just be that value because for instance if it was here eight and here in the middle of it and it is it okay our next average our next average is the mean so the mean is fine by adding up all the values and dividing by the number of values so here we've got the ages of five basketball players and they are 23 30 20 27 and 30. so let's start by adding these values up so 23 plus 30 plus 20 plus 27 plus 30. and the nice thing is M1 is a calculator paper so let's add them up on our calculator and that gives us 130 so the tool is 130. now we need to divide by the number of values so there's one two three four five values so if we take 130 and divide it by 5 that will tell us the mean so dividing that by 5 gives us an answer of 26 so the mean is 26 and the mean is fired by adding up all the values and dividing by the number of values okay in the next topic the next topic is the range and the range tells us how spread out our data is our data is so we've got our data our data and we've got the range is the largest subtract the smallest and the number of shots taken in Crazy Golf are five eight four three five two and seven and we want to find the range we need to find our largest value so our largest value is 8 and our smallest value is 2 so take away two and eight to equal 2 is equal to 6. so the range for this information and avoiding the word data is equal to the range for this information is six it's the difference between the largest and the smallest value okay next topic okay sometimes we're asked to find the mode from a table so here we've got some information and we've got the age of some people and there are five or people are animals or whatever it is with age of something and the ages are five six seven and eight and we've got the frequency so there's two five-year-olds two six-year-olds five seven-year-olds and one eight-year-old now whenever you find in the mode from a table it changes the word from often the mode age to the modal age and we're trying to find the Moodle age that just means the most common age and remember we had five seven-year-olds so that means that was the most common age so the mode here would be the one with the highest frequency which is seven so the mode of the modal age is seven our next topic is finding the mean from a table so we want to find the mean from this table and remember to find the mean we add up all the values and divide by the number of values so we've got two five-year-olds two six-year-olds five seven-year-olds and one eight-year-old so we could write down two five rules of five five two six year olds six six five seven year olds right those all lights you know so seven seven seven seven seven I'm one eight year old and then you can just add them all up manually but these numbers might be quite it might be a bit larger so let's find it an easier way to do that so we want to find the ground tool now this if there's two five year olds I add on this column called the FX column and that stands for the frequency multiplied by whatever column this is so if there's two five-year-olds well five plus five is ten another way to do that is just two times five and two times five is ten if there's two six-year-olds we could do six plus six but two times six is twelve so 12. if there's five seven year olds well 5 times 7 is 35 and one eight year old well that's gonna be one times eight is it so we'll find this column called the FX column and it's fired by multiplying the frequency by whatever the values are in the table and then if we add that up we will get the grand total because we know that if there's two five-year-olds that's ten years if it's two six year olds that's another 12 years if this is five seven years if you add those up you'll get the grand total and that's equal to 65. so if you added up all the ages that would be 65 now we need to divide that by how many people there were well if we look at the frequencies and add those up there's two two five and one so it's two five-year-olds two six year olds and so on so two plus two is four plus five is nine plus one is ten so all together though is 10 people so if we do the total which is 65 divided by 10 that would tell us the mean age so 65 divided by 10 would be 6.5 so the mean age is 6.5 okay our next topic the next topic is to find the medium from a table so the median is the middle value so if you look at this table we're trying to find the middle value so one way to do it is to arrange them all in order so to write down two 18 year olds we could write down to 18 year olds three 19 year olds we could write down 13 20 rules and one 21 year old so there we've got all the edges and then we want to find the median and the middle one so we can then just work out the middle one so and as you can see the median age is 20. so the medium would be 20. so that's one way today is to list down all the ages in one list and then just find the middle one so another way to do it though is we can consider the frequencies now all together we've got two people three people another 13 people and one person so if you add those up you'll find there's 19 people all together and if we were to line up 19 people well the median person would be the 10th person let's see why that would be well if you had three people one two three the median is the middle one which is the second person and then you can find that by doing three plus one which is four and half is the second one if you had five people one two three four five you can then add one which is six people and divide by two which is then the third person if you had for instance six people one two three four five six you can add one which would be seven and divide by two is three point five and if you look one two three point five that would be the median so if you had 19 people you could add one which is 20 and divide by two which would be the tenth person and the 10 10th person if you line these people up in order of age the 10th person wouldn't be in here there's only two 18 year olds the 10th person wouldn't be here because there's only five so far and the tenth person would definitely be in this group of people so the temperature would be 20 years old so you can choose which way I wanted it you could list out all the ages or so if they're values such as 18 19 20 21 and so on you could take the frequency add one and divide by two and that'll take you the position of the median and then you could find it and it's up to you which approach you use okay next topic is to look at the combined mean so to work out the compound mean let's just make sure we know what the mean is so the mean is found by adding up all the values and dividing by the number of values so it's going to be very useful and what's also useful is if we know the means found by adding up all the numbers and dividing by the number of numbers if we have the mean and multiply it by the number of values that will give us the grand total so that's very useful as well okay so our question says there's 20 students in class A and 10 students in class B and they sit the same test the main test score for class A is 60. and the main test score for class B is 14 and the question says work out or calculate the mean test score for all 30 students so we're going to do this question by considering if we know what the mean is for class A and we know how many students are we can find out the total number of marks received by class out by the students in class A so we know the mean score for class A is 60 and we know there's 20 students so if we do 60 times 20 that gives us 1 200. that means there are 1 200 marks obtained by the students in class they all together that's a total number Mark scored and if we divided that by 20 we get the mean score of 60. so that's the tool for class A now let's get the tool for class B so their class average was 40 and there's 10 students so if we multiply 40 by 10 that's 400 so that means in total the 10 students of Class B scored 400 points or 400 marks altogether so if we add together the 1 200 and the 400 that tells us how many marks were received by all 30 students all together in this test so 1 200 plus 400 is 1 600. now we're trying to work out the mean score for these 30 students so if we divide the grand total the 1 600 by 30 we will get the mean test score for these for any students so 1 600 divided by 30 is equal to that'll be 53.33333 so on or 53.3 reoccurring or let's just run our answer to two decimal places so 53.33 and that would be the mean test score to two decimal places and that's it so whenever we're working out the combined mean it's very useful to be able to work out the grand total and that can be found by taking the mean and multiplying it by how many people were involved with that mean and that will tell you the total and that'll be very useful and if you want to watch the video on this one called Mouse is 53a and just remember you do have that bumper pack of questions that M2 practice booklet which has loads of questions and there'll be questions on that on the combined mean also okay our next topic okay let's have a look at our next topic our next topic is called the estimated mean and that's video 55 in corporate Maps it's one of my favorite topics and we've been asked to work out estimate for the mean age so we've got a group of people and we know that there's nine people from zero up to ten we've got 13 people from 10 up to 20. we've got 16 people from 20 up to 30 and we've got two people from 30 up to 40. and we've been asked to work out the mean age and remember to work at the mean age we add up all the ages and we divide by the number of people let's actually start up a founding the number of people and to do that to allowed up 9 13 16 and 2 and that's equal to 40. so we know there's 40 people now we want to add up their ages and divide by 40 the number of people but unfortunately we're not able to do that in this question because we don't actually know all their ages we've made this group frequency table to make the data a bit more easier to interpret but unfortunately we're not going to be able to work out the exact mean that's why the question says an estimate for the mean age Okay so we've got nine people that have an age from 0 up to 10. we don't actually know their ages there could be four there could be eight there could be they could all be nine euros we don't actually know but what we're going to do is we're going to choose a fairy ager a representative age for these people because they're from not 10 we're going to use the midpoint which is five years old we're going to add a column on called midpoint and that will be the age that we're going to use for these people so for the not 10 year olds we're going to pretend that they're five years old each we're going to pretend there's nine and five-year-olds we don't actually know their ages that's the fairest thing to do now we've got 13 people with an age between 10 and 20. so let's pretend they're 15 years old so let's use the midpoint of 15 and for 20 to 30 the midpoint would be 25 and for 30 to 40 the midpoint will be 35. we can imagine there's 9 5 rules there's 13 15 year olds the 16 25 year olds and there's two to 35 year olds that's the the best estimate we can do in this situation so whenever you've done this estimated main question so you'll find the midpoint of each of these groups and you'll make a column for that so now what we're going to do is because we imagine there's nine and five year olds rather than doing five plus five plus five nine times what we're going to do is we're going to just do 9 times 5. so we're going to do 9 times 5. 13 times 15 16 times 25 and 2 times 35. so that'll tell us our estimate for the total ages of the nine people between naught and ten the 13 people between 10 and 20 and so on so we'll call this column FX and whenever you're doing the estimated mean question you'll add on a column for midpoint and you'll add on this FX column and your times the midpoint by the frequencies and that will tell you your FX column so 5 times 9 or 9 times 5 is 45 13 times 15 is 195 16 times 25 is 400 and 2 times 35 is 70. so we've completed our FX column now what we're going to do is we're going to add those up to find our estimate for the grand total so 45 plus 195 plus 400 plus 70 gives us a total of 710. so our estimate for the total of the ages would be 710 now what we're going to do is we're going to divide that by the total frequency so the ground total divided by the total frequency gives us our mean our estimated mean so 710 divided by 40 is equal to 17.75 so our estimated mean age is 17.75 years old and that's it so to find the estimated mean you add a column on for the midpoint you find the midpoint of all the categories whatever that is you put the midpoints down you then multiply those by the frequencies and that will tell us the FX column and then you divide the total of the f X column the grand total or the estimated grand total and you divide that by the total frequency and that will be your estimated mean sometimes we have grouped frequency tables what we need to do is find the modal class interval so remember the word modal is similar to mode and that just means the most common class interval so the most common class interval well we just look at the frequencies so we can see that this category with 16 is the frequency is the most common so that means that this is our modal class interval so D is larger than or equal to 20 but less than 30 and that's it so the modal class interval will be the class interval with the highest frequency and if you want to recap that in corporate Maps this video 56a okay let's have a look at our next topic so our next topic is found in the class interval that contains the median so here we've got a group frequency table with duration going from 0 up to 10 from 10 up to 20 20 up to 30 and from 30 up to 40. and we've got the frequencies and all together and if we add up these frequencies there's 40 altogether and we've been asked to find which class interval contains the median so if we knew these 40 durations and if we wrote them all out we're asked which of these categories from naught to 10 from 10 to 20 from 20 up to 30 from 30 up to 40 would the median ion would it be in this category this category this category and this or this category now straight away I can see there's not going to be in the first category because there's 40 all together and so because there's 40 altogether if we divide 40 by 2 that gives us a rough ideas to where the medium would be so 40 divided by 2 is the 20th so the medium would be roughly the 20th value and because this is grouped it and we don't actually know the actual numbers we may as well just use that 20th value now strictly speaking if we knew all the values and we could write them all out and there's 40 numbers we would do 40 plus 1 which is 41 and we would divide that by 2 41 divided by 2 is equal to the 20 point fifth value so if you had 40 people and you'd line them all up and we wanted to find the median the median would really be the 20.5th value but for grouped frequency tables where you don't actually know the values anywhere where it's just nine people in between these values 14 in here and so on we can just divide this frequency by two so 20th and find where the 20th one is so normally in a question like this the 20th and the 20 point fifth value will be in the same category anywhere just to make sure that you know whenever they're making the questions that you know you don't have a sort of a difference so that if students you went for this value or if students went for the 20th one they would design the questions that they'd be in the same category anyway and let's just see if that's the case so here we've got nine people in the first category now we're looking for the 20th one it's not going to be in there because that's a there's only nine people there now we've got another 13 well 9 plus 13 is 22. so up to the end of this category there's 22 people that means if you were to line up those people someone in this category would be the median or it would be either the 20th or the 20 point for value and as you can see that means that both of them will be in this category so the question says which class interval contains the median well it's going to be this class interval 10 Which is less than or equal to D Which is less than 20. and that's it okay so our next topic is frequency polygons and along with stem Leaf diagrams frequency polygon is the only other topic where is a bit of difference between the exam boards where at Excel says you should know frequency polygon whereas AQA and OCR don't mention frequency polygons so if you're studying for lxl you're going to have to watch this section it should take a minute or two and if you're studying for aqn OCR you may want to skip on but then again it's only a minute or two so I'd probably watch it anyway just to see what frequency polygons are and they're not that complicated so let's have a look at frequency polygon stats videos 155 and 156 and corporate Maps so here we've got a table and we've got time not the 20 minutes 20 to 40 minutes 40 to 60 Minutes 60 to 80 minutes and 80 to 100 minutes and we've been given some frequencies for them so there's five times that are between 0 and 20 minutes 11 times between 20 and 40 and so on so the digital frequency polygon what we're going to do is we're going to plot the frequencies in the midpoints of each of the categories so if we've got naught to 20 minutes we're going to 10 minutes and then we're going to go up to five so we're going to go 10 minutes and up to five next between 20 and 40 minutes well 30 minutes is in the middle that's the midpoint so we're going to go Freddy across and 11 up so 30 across and 11 up now whenever you're dealing with how far up to go make sure you know the scale so if we have a look there's 10 boxes for five that means each little box is not 0.5 here so we're going to go further across we're going to go up to 10 and then we're going to go up two more boxes to get to 11 so we're going to go to there now our next one we've got between 40 and 60. it's going to be 50 across and 20 up so 50 across and 20 up next we've got between 60 and 80 so it's going to be 70 across and 15 up so 70 across and 15 up and finally we've got between 80 and 100 that's going to be 90 across and nine up so 90 across and then 9 up would be well we've got 10 here each little box is 0.5 so we want to go down two and then that would be there so we've plotted our points now what we're going to do is going to learn our pencil and we're going to join them up so like so and that's it that's our frequency polygon one thing to note is not to join up the first point and the last Point you're only joining up the consecutive points so you join up the first one to the second the second to the third the third to the fourth and the fourth to the fifth so it's a frequency polygon like so don't join up these points and that's it another thing you can be asked to do with frequency polygons is to compare them so this could be the times for class A we could also have the same grid of number frequency polygon for class B and you can have a look at them and compare them and see you know which class was faster whenever you look at their frequency polygons okay let's have a look at our next topic so we're going to look at Venn diagrams now and that would be 380a and corporate maps and the question says 90 people were asked if they liked free drinks so we've got drink a drink B and drink C and 11 people liked all three drinks so let's put that on the Venn diagram they're like all three drinks so it's going to be 11 people in the middle it people liked drinks A and B but not C so they like a and b so they're going to be in the middle of A and B so here but not C so it's going to be here eight people like drinks A and B so then those circles but they're not in the C Circle so that's there next 17 people like drinks A and C so A and C would be here but not B so that means it's 17 people will go in there and nine people like drinks B and C so they like drinks B and C but not air so it means that's going to go there we're also told that four people only like drink base with four people only four people like drink B and we're told that 50 people like drinking a so that means that the whole circle here for a will be 50. so if we add up the numbers we know in this circle for a we've got 17 plus 11 that's 28 plus and over 8 is equal to 36. so that means that 14 people must go there and three people did not like any drink so they're gonna go outside and we've been asked to complete the Venn diagram so we've done most of it and there's just one missing number here now we're told that 90 people were surveyed so if we take all these numbers away from 90 we'll see what number must go here and when we take away all these numbers away from 90 we're left with 24 so that means there must be 24 people in here so we've completed our Venn diagram so in M2 or M3 or M4 you may encounter a Venn diagram with three circles Okay so we've looked at event diagram question whenever we're dealing with a worthy situation now let's have a look at some notation that you may encounter whenever you're looking at Venn diagrams so here we've got a Venn diagram so we've got a and b and as you can see here's a here's B this is the section where they overlap and this is the section that is neither in air nor B and then this is the universal set which is the whole rectangle and you've got a and a if you've been asked to shade a that would be inside of a if we're asked to shade B that'll be inside of B there next we've got this a with a little Dash above it that means the complement of air and the complement of a means not a it means anything that's not a so if you have a look here whenever we shaded in it's going to be anything that's not in a so a dash means not a or the complement of a and that means anything that's not a and then here if we've got B Dash that means the complement of b or not B so it's NFL outside of B so that's a b a dash which means the complement of a which is n of in this not a we've got B Dash which is the complement of B which is not B now we've got a and then this symbol B and this is a union B and that means A over B it means anything that's in a or b anything at all and that means it would be this region here and if it's in a or a B because it's a union B A or B now we've got a and then this symbol B and that means a intersect B and that means A and B so it's the region that's in a and it's in B so that's the overlap region in the middle so this is the core massive revision card it's very useful and it's very important to know this notation that a is inside of a b is inside a b a dash means the complement of it means not A and B Dash means the complements of B so it means anything that's not in b a union b means n if in an A or B anything tall it's in a or b and a intersect B Means A and B it means anything that's in that middle region that overlap and that's it okay let's have a look at the question now where we're dealing with that notation okay so here's a Venn diagram Square question and we've got the symbol as well and this symbol means a universal set and we've got the universal sets that's all the numbers that we're going to put in the Venn diagram are one two three four five six seven eight nine and ten and we're told that the numbers inside of a are two five seven and eight and the numbers inside of B are two four and nine and we've been asked to complete the Venn diagram so whenever I'm complete an event diagram like this the first thing I would do is look and see what numbers are going to be in the middle so in other words what numbers are in a and in B and if we have a look here two is in both of them A and B so we're going to put a 2 in the middle and then we've got no other numbers that means let's put the rest of the numbers in the a in a inside so it's going to be five seven and eight and if we look at B B has got two four and nine so that's two four and nine so if we look in a we've got 2 5 7 and 8 and then B we've got two four and nine now we're going to put the rest of the numbers outside of A and B because the rest of the numbers are in the Venn diagram but they're not in a or a B so it's going to be one we've done two we've then got three four five and we've got six seven eight nine and ten so we've completed a random grammar let's just check we should have ten numbers in this diagram one two three four five six seven eight nine ten so you've been asked complete so we've completed the Venn diagram we've done part A M and then we're told a number is chosen at random so one of these numbers is going to be chosen at random and we've been asked to find the probability of a intersect B so in other words that's a and b so there's any number that's in A and B so if we look that's going to be the region here so we want to find the probability of choosing a number that's in this region well there's ten numbers all together so we're going to put 10 on the denominator and there's only one number in a intersect B in this region in the middle so the probability of choosing this number would be one out of ten or one tenth okay next question part C part C says find the probability that the number choosing at random is in a union B in other words it's in a or b so that means there's anywhere inside of a or b or both any of these numbers at all so if we have a look we've got one two three four five six numbers in A or B so it's going to be six and then there's ten numbers all together so the probability of choosing a number in a or b in a union B would be six out of ten or six tenths so probably would be six tenths or if you wanted to simplify that would be three-fifths and that's it okay let's have a look at our next topic so our next topic is called tree diagrams and it's a probability topic and a tree diagram is a really useful way of showing what can happen Whenever two or more events take place and this video 252 according maths is a very important topic so I highly recommend you watch that video and try the practice questions as well so let's have a look at our questions our question says John and Dan are taken in terms of throw a ball at a Target so we've got two different events John's gonna throw the ball at the Target and then Dan's gonna throw the ball at the Target and what's great is this diagram will show us all the possible outcomes of what can happen and we've been given some probabilities so we've been given the probability that John hits the target is 0.6 and they're probably the down hits the target is 0.7 now before we get started let's actually label the probabilities on the diagrams now the problem is that John has the target is 0.6 so let's work out the probability that John misses the Target now it certain that he does one of these two things he's either going to hit it or he misses it that makes these two probabilities need to add together to be equal to one so if we want to work out this missing number we're going to do one takeaway 0.6 and that would be 0.4 so these two properties will need to add together to give us one okay now that we're down from the ball at the Target when he could hit the target which is not 0.7 so the probability of the dam misses the target is 0.3 and below for down again we've got 0.7 is the probably that he hits the target or not 0.3 is the property that he misses the target Okay so we've not got our tree diagram with our probabilities labeled on it now let's consider the outcomes of John through and the ball and then down from the ball well if we go along the branches we can find out all the possible outcomes and that's the great thing about a tree diagram so let's start here and John throws the ball and he can hit it and then dye can hit it so we've got a hit and a hit hit and a hit so that's a hit and then a hit then we could have John Hits the Target and Dan misses the target so it'll be a hit and a miss so that's a hit and a miss now if we go down this way John can actually missed the Target and then dank and hit the targets out of a miss and a hit and finally John could miss the Target and dank and Mr Target so it'll be a miss and a miss so we've got all the possible outcomes here we've got a hit hit Miss Miss hit and a miss miss and what's fantastic is we can use the properties we've been given in the question to work out the probabilities of a hit hit hit Miss Miss hit a miss miss really easily and all we do is multiply the numbers along the branches so if we wanted to work out the probability of John hitting the Target and Dan hitting the Target because it's and what we do is we multiply the properties together so we're going to do 0.6 multiplied by 0.7 so we go along the branches and we do 0.6 multiply by 0.7 so let's write that down 0.6 multiplied by 0.7 is equal to 0.42 so the probability of John hitting the Target and down hitting the target the hit hit will be 0.42 next let's work out the probability of John hitting and Dan missing so a hit Miss so I'll be 0.6 multiply by 0.3 so 0.6 multiplied by 0.3 will be 0.18 next we could have John Missin and Dan hitting so that would be 0.4 multiplied by 0.7 so 0.4 multiplied by 0.7 and that'll be equal to 0.28 and finally we've got a miss miss that's John missing and down missing so will be 0.4 multiplied by 0.3 so 0.4 multiply by 0.3 would be equal to 0.12 and what's great is if we add these properties together 0.42 0.18 that's 0.6 plus 0.28 that's not 0.88 and then plus 0.12 that gives us one because obviously one of these four things will have to happen over a hit hit Miss Miss hit or miss miss okay so let's go to our questions our question says what was the probability that at least one man hits a Target so at least one man hits the target means either one of them could hit it or both of them can hit it so hit hit that would work hit Miss yep one of them's hitting the Target that would work Miss hit yep one of them's hitting it that would work but miss miss wouldn't work some of these were interested in this problem ability this probability and this probability and we want to find the probability that either it's a hit hit or a hit miss or a Miss hip so if we add these three proper leads together we will find the probability that at least one man hits the target so we could do 0.42 plus 0.18 plus 0.28 and when we do that we get 0.88 so the probability that at least one man hits the target would be 0.88 and that's it okay let's have a look at our next topic so our next topic is reading tables and sometimes you might be given a table and you might be told to find some information from that table and if you want to practice reading tables it's 387 on corporate Maps so here we've got a table and we've got some Caravans a luxury one a basic one a family one and a comfort one and it tells you how many people they sleep so the luxury sleep six the basics four the family one six and the Comfort one four so it then tells us which Caravans have Cuts in them so they're basic and the family one it tells us which ones have decking so the luxury and the comfort and it tells us which ones have barbecues so the luxury the family and the comfort and then we'll give them the price to hire that Caravan and the first question says which Caravan sleeps four people and costs more than 300 pounds so let's look at the Caravans and sleep for people so we've got basic here and we've got Comfort here and the question says more than 300 pounds per view we've got basic or comfort and the basic one costs 290 pound and the Comfort one costs 325 pounds so which Caravan it sleeps exactly four people and costs more than 300 pound that's going to be the Comfort Chiron so Comfort okay the next question which Caravans do not have decking so the ones that do have decking are the luxury and the Comfort Caravan but the basic and the family Caravans do not have decking because they're not ticked so it would be basic so it'll be basic and family now samples can be very very useful rather than a surveying or interviewing everybody for instance in a school if you wanted to find information about what color Blazer you would like rather than asking every single student it can be very useful that's called a census asking everybody in a population rather than a census you can do a sample where you ask a smaller number of people but it's very important whenever you do samples they're fair so for instance if I was doing a survey of students about Blazers it's making sure they ask people from different year groups you ask a range of different classes so it's very important that you get a good sample of fair sample to make sure that your sample is representative of the whole population but here we've got a question and it says 480 students attend the school and the teacher asks 50 students which color Blazer they would like so we've got black 20 students with like a black blazer 15 students want a Navy Blazer nine students want a green Blazer and six students want a maroon Blazer and the question says estimate how many of the 480 students would like a Navy Blazer now altogether we knew there was 50 students all together and we want to work out an estimate with for how many students would like a Navy Blazer now if I look at the sample we know that 15 out of the 50 wanted Navy now if we work out 15 over 50 as a fraction that's equal to three tenths so that means that three tenths of the students asked would want a Navy Blazer now if this was a good sample a representative sample that means that it should be the same fraction of the whole school would want an AV Blazer so if we work out three temps of all the students in the school we'll find out how many should want a Navy Blazer so if we work out three tenths of 480 that would tell us a good estimate so we'll do 480 divided by 10 that's equal to 48 and then we'll do 48 multiply by 3 and that's equal to 144. so that's it if we're asked how many of the 480 students would like a black blazer we would have done 20 over 50 which is two-fifths and then work out two-fifths off the 488 and so on and that's it so that has been the whole of the GCSE Foundation Maps course for lxl aqn OCR so I've gone through every single topic I know I said I would spend two or three minutes going for each of them so some of them maybe have been a bit longer than that and some of them are a bit shorter but that's been a eight hours of going through all of those topics I hope you found it useful I hope you've watched it in chunks and you haven't just watched all eight hours in one go we went through the number of topics to begin with we then went through the shape space and measures topics or the geometry topics we then went through the algebra topics and we then went through the statistics or the probability topics so that has been the whole of the GCSE Foundation Maps course for Edexcel aqn OCR and I said there's only very subtle differences around stem only from frequency polygons and that's it so I hope there are any topics that you need a bit of extra help on so as you've been watching this video if you've identified a few that you need to go through if you go to corpmath.com you can watch the video tutorial on that topic so for instance if you needed some extra help on Venn diagrams you could go to video 388 and watch that video tutorial to help you on Venn diagrams and then there's the practice questions and the textbook exercises as well to help you also remember as you've been going through this video there has been that ultimate GCSE Foundation revision question booklet because you tell I know that booklet as you've been going through the video and watching the video there has been a question on every single topic as you've gone through and so that booklet may have been useful for you and you've got the the answers there so if you scan the QR code that bring you straight to the answers if you haven't tried that booklet it may be a good idea now I'd have a go at that booklet also in this video there's been lots of references to the chord Mars revision cards the GCSE Foundation revision College will be a fantastic tool to support you as you study for your GCSE Maps your foundation maps and so there's a link to them in the description below also there's the favorite booklet so that little unofficular approach to Eurovision be really useful and for foundation I'd highly recommend the orange the foundation books and the yellow Foundation plus books and this has been the corporate Maps ultimate GCSE Foundation revision video I really really hope you found this video useful it's taken a lot of time and effort to make this video so I really hope you found it useful if you have found it useful could you please like it could you please subscribe to YouTube channel as well and maybe could you share this video with your friends because then maybe they might find it useful as well um the video's been quite long it was about eight hours long but it's gone through every single topic of the GCSE Foundation course for AQA edxl and OCR the whole aim of the video has been to make sure you're familiar and confident with those topics so I really hope that it's that it's done that thank you so much for watching good luck with your studies good luck with your exams and all the very best cheers bye