Transcript for:
GCSE Maths Revision Guide

[Music] hello and welcome to mr tompkins ed tech this video covers pretty much the whole of gcse maps in just over two hours covering all the essential topics for both foundation and higher tier students aiming to get a good pass in their mocks and final exams so if that sounds like you then keep watching you'll find topic lists and time stamps in the video description as you go through i highly recommend you use these to make a list of topics that you're not sure about afterwards you can then look those topics up in your revision guides or simply follow the links provided to more focused revision videos on those topics higher tier students aiming for grades seven through nine will want to watch this next video too as it contains all those harder topics appearing towards the end of the higher tier paper if you find it helpful please do give it a like and subscribe and share the video with your friends you'll find video walkthroughs of pretty much every exam paper on my youtube channel along with revision resources and question compilations on all those key topics you struggle with as we run up to the exams i'll be posting important information predicted questions and general support and encouragement to help you achieve your target grade this summer good luck in your preparations and in your exams let's get into it write 120 as a product of its prime factors okay so if you've this is the calculated paper but if you didn't have a calculated paper or if you don't have a casio calculator you're gonna have to do this a long way by breaking the number down into factor pairs all you got to do is think of two numbers that multiply together to give you 120 so i could pick 2 and 60 or 3 and 40. it doesn't really matter i'm going to go for 3 and 40. uh now 3 is prime so that's a stopping point 40 isn't so we can continue to break that number down so we need to find two numbers that multiply together to give me 40. uh i could pick 2 and 20 perhaps so 20 and 2 2's prime so i'm going to stop on that one but 20 isn't so i'm going to keep breaking it down uh two numbers that multiplied together to give you 20 uh 10 and 2. so two's prime so i stop on that one and then finally i've got ten factor pairs of ten two and five now both two and five are prime so i've kind of come to the end of breaking my number down uh so then the product of my prime factor as i can say 120 then is all those um end points multiplied together so all of these multiplied together will give me 120. okay so what i've got i've got one two three twos of an article so that's two times two times two and then i've got a three and i've got five okay now you can rewrite that in index form if you like just write bit neater 2 cubed times 3 times 5. okay but as i say if you've got a casio calculator you can just type it into your calculator if you enter the number 120 and press equals and then press the fact key it will do this for you in one step so um yeah that kind of takes the hard work out of it and you just need to write the answer down for three marks amazing uh so if you've not got yourself one of these casio calculators uh i highly recommend them you can find a link to it on my um my amazon shop page i'll put a link in the in the video and you can jump there and order yourself one if you haven't already got one all right let's move on a is a common factor of 72 and 120 and b is a common factor of six and nine work out the highest possible value of a over b okay so this fraction is going to be its biggest when a is at its maximum value and b is at its minimum value isn't it okay you'll get a bigger number if you divide of a fraction by a smaller denominator uh so let's let's take b to start with then i think that one's going to be easier a common multiple of uh six and nine if we want b to be minimum we're actually asking for the lowest common multiple of six and nine okay so let's let's think about the the multiples of six and nine uh this multiples six are six twelve eighteen and so on and nine is nine eighteen twenty seven and so on so you can see we found our lowest common multiple already uh 18 is there 18 is that so our lcm then of those two numbers is 18. uh so that's going to be my value of b what about a then we've got a is a common factor of 72 and 120. so let's list out the factors of 72 and 120 i tend to write them down in factored pairs so one goes with 72 2 goes with a half of 72 is 36. uh i think three must go into it then as well so three and 24 is that three and 24 four four and what four and [Music] that's got to be 18 isn't it five no six yeah six goes into it six and twelve seven no eight yes eight goes with uh nine and i've met myself coming down the other way so that's as far as i'm going to get so quite a lot of factor pairs of 72. uh what about 120 are we gonna have so many so i've got one and one twenty uh two and sixty three goes with forty 4 yes goes with 30 5 goes with 24 6 goes with 20 also got a lot of factors seven no eight yes eight goes with fifteen nine nope ten yes ten goes with twelve and then i've got eleven in between that's not going to be a factor so that's all of them gosh lots of factors for both of those numbers so we want the hcfn of those two numbers so we want a to be as big as it can be so it's going to be the highest common factor of those two so looking through my list uh 72 no 36 is my next biggest that isn't on my list 24 yeah that's on my list so look 24 there 24 there so the hcf of these two numbers is 24. okay so what's the highest possible value of a over b then it's going to be so that was my a value wasn't it 24 and b is 18. so my highest value is going to be 24 over 18. okay uh which i could cancel down but i'm not sure i need to so it could it could simplify to four thirds also an acceptable answer eve has these coins so she's got two pound coin a pound coin 50 p two twenty's a five and a two pence ola has these coins she's got a pound a 50p or 2p and a 1p eve gives three of her coins to ola now ola has the same amount of money as eve which coins does eve give to ola okay let's see if we can work this out let's let's work out how much they've got each first off then so eve is starting with 2 plus 1 which is 3 plus 50 p plus 20 plus 24 supplies plus 2 that adds up to 3 50 90 1997 so ola starts with three pounds and 97 pence uh ola has a pound 50p 2p and a one piece that is one pound and 53. okay so then after eve gives ola three coins they should have the same amount okay so how many coins does she give him well if we can work out what the difference is between their two numbers and half it then we'll know how much eve gives to ola so what is 3 pounds 97 subtract 1 pound 53 that is um seven subtract three is four nine subtract five is four and three subtract one is two so the difference between their money is two pound forty so if i half that 2 pound 44 divided by 2 2's into 2 pound 4 4 2's and 2 go once 2's into 2 go twice 2's into 2 go twice so it's 1 pound 22 isn't it okay so i need to give well eve needs to give ola one pound 22 and that will mean that they have the same amount of money so if she gives the pound and the 20 and the 2p that's one pound 22 isn't it okay and then they'll both have the same amount of money eve will have two pounds left and the 50 and 20 and the 5p making two pounds and olo will now have two pounds and the 50 and the 20 and the two two pence and the one pence which is 275 as well okay so how am i going to write that down oh i need to write down in the answer box so she's giving her a pound a 20p and a 2p so the pound coin the 20p coin and the 2p coin question 19. kohler is sold in packs of six and packs of eight uh and we can see various prices here so one pack of six costs one pound ninety five or we can buy two packs of six 350 or we could buy one pack of eight for 264 and two packs of eight for five pounds what we need to do is work out the cheapest way of buying 48 cans of cola and you must show you're working okay so i think the easiest way to approach this is to work out the unit cost of one can from the four various deals we've got on offer there and then work out what the best deal is from that uh now we want to buy 40 cat 48 cans or red altogether six is a factor of 48 as is 12 as is eight and as is 16 they all go into 48 so we can just buy enough packs of the cheapest pack type to get us up to the 48 cans so just knowing the unit cost of each of the four deals should be sufficient for this question so let's take the one pack of six and so for my uh one pack of six so i've got one pound 95 which i need to divide by six so 195 divided by six gives me a unit cost of 0.325 so that's the cost of one can isn't it so then we've got two pack of six so that's three pounds fifty so two packs of six would be twelve cans and three pound fifty divided by twelve gives me a unit price of point two nine two rounded off to three significant figures next let's have a look at the one pack of eight now that costs two pounds 64 and i get eight cans for that so that gives me a unit cost of 2.64 divided by eight gives me 0.33 per can so interestingly the most expensive you think it'd be the cheaper of the two but no and then finally two pack of eight uh costs five pounds and all together there are 16 cans in two packs of eight so five divided by 16 gives me a unit cost of zero point three one let's round off to three seconds so freezing different figures as well three one three so now we've got our four different unit costs now it's clear now then that this one two pack of six is the cheapest unit cost okay so if it's cheapest to buy that per can then it's going to be the cheapest price all together so what is the cheapest way of buying 48 cans so we need to buy two packs of six how many times it's going to be 12 cans and 12 goes into 48 four times so the cheapest way of doing this is going to be to buy four lots of two times six pack beth sells hot dogs at a market each hot dog is a sausage in a bread roll hot dogs three pounds each the table shows her costs fees paid to the market 240 pounds bread rolls 42 peep a pack of six uh sausages two pound fifty per jar of ten other costs fifty seven pounds now it says that beth sells the hot dogs for three pounds each and she sells 300 hot dogs and we need to work out her total profit okay well our total profit is going to be profit is going to be total income minus total expenditure so your money coming in less your money coming out so we need to work out those two amounts don't we so let's work out income first how much money she's got coming in so we know how much money she's coming in from this part down here don't we okay so that's where we get our income from beth sells hot dogs for three pounds each and she sells 300 hot dogs so that's quite simple so income is gonna be her three pound per hot dog times her 300 hot dogs 3 times 300 is 900 pounds so that was nice and easy about expenditure so that was money coming in or about money money going out expenditure so what she got so a total expenditure then is going to be the 240 pounds from the fee pay to the market she's got 42 p per pack of six bread rolls so how many bread rolls she gonna need then if she's made 300 hot dogs and um bread rolls are in packs of six and what's 300 divided by six sixes into 300 go 50 times so she's gonna need 50 packs of those bread rolls so that's going to be 50 times 0.42 okay uh sausages 2 pound 50 per jar of 10 so 300 divided by 10 means i need 30 jars of sausages so i need 30 jars at 2 pound 50 each and i've got 57 other costs okay so then in total then my total expenditure is going to be 240 plus 50 lots of 42 pence so 50 times 0.42 that comes out to be 21 pounds i've got 30 lots of 2 pounds 50 30 times 2 pound 50. that comes out to be 75 pounds so my total expense is going to be those four amounts added up and isn't it so expenditure then is going to be 240 plus the 21 plus the 75 plus the 56 24 240 21 75 sorry 57 57 pounds in other expenditure which comes to 240 plus 21 plus 75 plus 57 that means a total expenditure is 393 pounds okay so what's her profit then so profit is going to be income less expenditure which is going to be her 900 pounds that she made less than 390 pounds that she spent uh that comes out to be 507 pounds so she's made a good profit on those hot dogs 507 pounds between january 2015 and january 2016 the price of diesel increased from one pound 12 per liter to one pound 19 per liter calculate the percentage increase in price okay now i tend to use this triangle when i do percentage questions uh f o m stick your thumb on it um f stands for final original and multiplier okay and it's arranged like this into a triangle f o m okay so what does the f stand for f stands for finals it's the last price it's going to be that one there one pound 19. it increases from one pound 12 to 1 pound 19. so 1 pound 19 is the end price uh or the final price o stands for the original okay so what was it to start with it was one pound 12 per liter wasn't it so that's our original price that's gonna be one pound twelve and then multiplier well that's what we're trying to find isn't it okay so then what we do is we take our little fom triangle and cover up the one we want to find i want to find the multiplier and that tells me what to do with the other two so multiplier that's going to be the final price divided by the original price okay so then subbing in the values i've got here final is one pound 19. original is or one pound twelve okay so then i just need to tap that into the calculator then uh one pound nineteen divided by one pound twelve uh that comes out to be a 17 16 or 1.0625 1.0625 okay now that's a decimal to convert it into a percentage need to times by a hundred or shift the decimal point two places along so that's 106.25 percent now remember we want to work out what the percentage increases okay so um when you're doing percentage increases you always start with 100 so if i started with 100 and now i've got 106.25 percent that is an increase then of 6.25 okay tom invests 2 000 pounds at 5 per year simple interest how much interest has been paid after six years okay i know this is simple interest not compound interest we are not putting the interest into the account and then in the second year calculating the interest on the original amount plus the previous interest is just going to be the same amount each time so if a hundred percent of something is 2 000 pounds then what's that 10 would be hundred five percent is gonna be a hundred okay so that's that's what five percent uh interest would be now we're going to apply 100 uh that five percent interest six times over because we've got six years to do it so our amount how much interest has been paid after six years it's just going to be six lots of a hundred pounds and isn't it which is 600 pounds part b says tom receives a 20 wage increase his new weekly wage is 360 pounds calculate his weekly wage before the increase okay so with percentage questions i often do these triangles uh f o m uh just to keep track of what i have to multiply or divide by what so f stands for final o stands for original m stands for multiplier you can apply to all sorts of percentage questions so then we just need to go through and work out which is which so uh he receives a 20 wage increase uh he's his new weekly wage is 360. uh we want to work out his weekly wage before okay so final is what it was at the end so that's going to be his new weekly wage isn't it so that's going to be 360. the original well that's what we're trying to find his weekly wage before the increase and then m the multiplier is going to be the 20 increase isn't it but remember we start with 100 so if we increase it by 20 uh we're now 120 which is a multiplier is 1.2 okay so now we cover up the thing we want to find so i want to find the original and then that tells me what to do with the other two so using my little triangle original equals final over multiplier in other words it is 360 divided by 1.2 okay okay to make things a little bit easier for ourselves i'm just going to multiply top and bottom by 10 and so that's 3600 divided by 12 and because it's a non-calculated papers now i haven't got a calculator to tap it in so 12's into 3 600. well 12s into 36 go three times and then i've got two extra zero so it's going to be 300 isn't it 300 pounds here are the interest rates for two bank accounts we've got northern saving bank which is 2.5 per year compound interest and central alliance bank which is 2.7 per year simple interest mia puts 6400 pounds in each account calculate the difference in value between the two accounts after eight years give your answer correct to the nearest penny okay so i'm going to work them out separately and then i'm going to compare them there so let's start off with the northern saving bank nsb uh so working out compound interest the final amount is going to be the original amount times the multiplier raised to the power of the number of years that's the formula for compound interest so what is the final is what we want to find isn't it original what's the original it is what she put in the bank account which is 6400 and the multiplier is going to be a hundred percent plus 2.5 which comes to a 102.5 which is a multiplier is going to be 1.022 five and isn't it so that's the multiplier we're going to use here so my final amount is going to be my original amount which is 6400 times by 1.025 raised to the power of 8. now tapping that into my calculator because i can't do that sort of maths in my head 6 4 0 0 times 1.025 to the power of 8 that comes out to be 7 797 pounds and 78 pence uh so i'm just going to write that down to the nearest penny seven seven nine seven seventy eight seven seven nine seven seventy eight okay uh now for the other bank account that was cab okay let's just scroll up a bit give myself a bit more space so let's work out the the second base now c a b c a b bank uh again the final is going to be original times multiplied but i need to find them a slightly different way here well the multiplier part anyway finals just the same the original is just the same still six thousand four hundred multiplier now i'm starting with a hundred percent and i'm gonna be adding on two point seven percent eight times so i'm just gonna multiply that by eight and work out one one multiplier so two point seven in my calculator times eight that comes out to be twenty one point six percent so add that to a hundred percent is a hundred and twenty one point six percent which is a multiplier is going to be 1.216 okay and then final amount it's just going to be my original amount times my multiplier don't have to compound it this time it's not raising it to a power and i've already applied it eight times so it's just going to be a straight forward multiplication so 6400 times 1.216 that comes out to be six thousand four hundred times one point two one six i made that seven thousand seven hundred eighty two pounds forty seven thousand seven hundred and eighty two pounds and forty pence okay what you'll notice is slightly lower than the compound interest amount which is i suppose the point of what we're looking at here isn't it so what is the the difference between the two work out the difference between the two so subtracting one from the other so then the difference is going to be 7797.78 subtract 7782.40 what's that then seven seven nine seven point seven eight subtract answer that comes out to be a difference of 15 pounds and 38 pence 15 38 okay that's my final answer then that's my difference to the nearest penny the height of zach is 1.86 meters he's a big boy and height of thread is 1.6 meters right the height of zach as a fraction of the height of fred give your answer in its simplest form right the height of the of zap as a fraction of the height of fred that means zach's height goes on the top and fred's height goes on the bottom so it's going to be 1.86 divided by 1.6 now again we've got decimals here which we can easily get rid of by multiplying top and bottom by let's do it by a hundred uh so what's that what that gives me that will give me 186 over 160 won't it so it's a top heavy fraction uh right so we're looking for common divisors now 160 and 186 well struggling to find a common divisor uh bigger than two let's just halve it and see have another look so that'll be 93 over 80 okay any common factors there again three no seven nope no i can't think of any more no i think that's about as far as i'm going to get with cancelling down now it's asking for the simplest form here so we cancel down as far as we can uh now we could write it as a mixed number we could write as 1 and 13 over 80. uh but i don't actually think that is any more simple than this i prefer to leave my answers as top heavy fractions i think they're more useful so i'm going to leave it like that calculate three fifths plus five eighths okay so we're adding fractions together here but you'll notice that we've got a different denominator it's got a five here and an eight here so to add them together we need to find a common denominator so what's a common multiple of five and eight then uh the smallest one i can think of is well it's just five times eight it's 40 isn't it so let's times the first fraction top and bottom by 8 and the second fraction top and bottom by 5. what's that going to give me so that's going to give me 3 times 8 is 8 16 24 40 plus 5 times 5 is 25 25 40. so now i've got two numbers with the same denominator i can simply add the numerators together then so the 24 plus the 25 that gives me 49 over 40. okay so that's an improper fraction and in a lot of cases that would be perfectly acceptable uh it's just that in this particular case it's asking me to express is a mixed number uh not as an improper fraction so i'm going to convert this into a mixed number by saying how many times does 40 go into 49 it goes in once and the remainder is 9 so it's 1 and nine fortieths just double check have we got any common divisors of nine and fourteen no don't think so so i think we're done so it's one and nine fortieths right the ratio six millimeters to 180 centimeters in the form one to n okay now when you're comparing things in a ratio you should really have the same units and these are not the same are they i've got millimeters and centimeters so i'm gonna have to convert them all into the same unit it's going to be easier to go from centimeters to millimeters i think uh you should know that one centimeter is the same thing as 10 millimeters so if i take the mil uh the centimeter one and i multiply by 10 i'm going to get millimeters then aren't i so the ratio is going to be 6 to 180 times 10 so that's 1 800 okay now i want to have a 1 here which means i need to divide through by six so doing the same thing this side dividing through by six eighteen hundred divided by six i can just do that in my head that's three hundred okay so write the ratio in the form one to n it's one two three hundred a deal kate and rebecca share 160 pounds in the ratio of two to five to three how much does rebecca receive okay so within these questions then we need to work out how many shares there are all together and we can do that by just adding the the ratio bits together so two plus five plus three that tells me that there are ten shares altogether okay now if there are 10 shares and we're sharing 160 pounds then each share is going to be 160 divided by the number of shares 10 so that's going to give me 16 pounds per share okay and then we've got in the ratio of two to five to three to start with so scaling that up then by a factor of 16. that means that a deal is going to get 32 pounds kate oh kt sorry it's going to get 80 pounds and rebecca is going to get three lots of 16 which is 48 now i've done a bit more work that i need to do here but it's good practice just to double check your three amounts actually add up to what you started with and 32 and 1848 does add up to 160. so i'm feeling pretty confident that i've got my answer right so 148 pounds for rebecca a box contains only red blue and green pens the ratio of red pens to blue pens is five to nine and the ratio of blue pens to green pens is one to four calculate the percentage of pens that are blue okay so i've been given two ratios here and they're kind of separate from each other aren't they so what i'm going to do first is i'm going to try and combine them into one ratio that has all three elements in it so i've got red and i've got blue and i've got green haven't i and i know from the first statement that the ratio of red to blue is five to nine so i can put a five in there and a nine in there and then from the second statement i know that blue pens to green pens is one to four so i can put a one in there and a four in there okay so notice i've got two blue numbers i've got a nine and a one now if i can take this second ratio one to four and scale it up so that one becomes a nine then i can kind of combine the two together so that's what i'm gonna do i'm gonna take the second ratio here and i'm gonna multiply up by a factor of nine and that's gonna give me a ratio of blue to green of 9 to 36 okay and you'll see now that that 9 there matches with that 9 there so that means i can bring down this five and write it in here so i've got a kind of a super ratio now that's got all three bits included so the ratio of red to blue to green is five to nine to thirty six okay now we know that we can work out the percentage of pens that are blue because we know that there are nine blue pens for every five plus nine plus 36 pens there are so it will be percentage blue it's going to be that 9 divided by 5 plus 9 plus 36 made into a percentage okay so what's 5 plus 9 plus 36 that is 50 isn't it so it's 9 50. uh well i might just make that into 1800 and say that's 18 percent advertisement break looking for a math tutor to help you through your gcc a-level or ib exams then why not check out mr tompkins tutors at mr.tompkins. forward slash tutors mr tompkins tutors offers personalized one-to-one quality tuition online via teams direct to your home tuition is delivered by myself and a team of first-rate teachers who i've personally known and worked with for years all have qualified teacher status in the uk and a recent full dbs disclosure so you can be confident that you're in good hands to find out more and request a trial lesson simply visit the website link below and complete the online form and we'll be in touch with you soon until then by writing each number correct to one significant figure find an estimate for this calculation now it's really important that you read these sentences above the calculation some people just see this and they'll start focusing on that straight away and they won't read the sentence that goes above it you really need to make sure that you do this one is giving you some very clear instructions uh they're asking you to estimate i've actually given you more information than they need they don't normally tell you this bit uh so we're gonna write each number correct to one significant figure even if they told me to estimate that's what i would have done anyway so a bit weird that they should tell you but anyway 606.3 what's that number correct to one significant figure uh well the first significant figure is the first non-zero digit which is that six there isn't it which is in the hundreds column so round in 606.3 to the nearest hundred is going to be 600 okay and the second number the first significant figure is the first non-zero digit it's that one there's the three that's in the tenths column so if i round off to one significant figure i'm going to round off that number to the nearest tenth so am i going to round up or down just look at the number next to it and to decide whether or not it's five or more or four or less in this case it's four or less i'm just going to round down to 0.3 finally uh 19.93 the first significant figure is that one which is in the tens column so round in 19.93 off to the nearest 10 that's going to be 20 then isn't it so my my uh calculation becomes 600 times 0.3 divided by 20. okay so let's let's do the math then what 600 times 0.3 uh that's the same thing as 60 times 3 isn't it so that is 60 120 180 180 over 20 just cancel those zeros out 18 divided by two that's going to give me an answer of nine okay so that's my estimate the length of the table is 110 centimeters to the nearest centimeter complete the error interval okay so i tend to draw a line when i do these and i mark the number i've been given now this is to the nearest centimeter so on my scale i'm gonna go up one centimeter so that'd be one one one and down one centimeter that would be one zero nine now the numbers that round to a hundred and ten are the numbers are going to be halfway or closer to a hundred and tens it's all these numbers in here these are the ones that round to 110 aren't they and halfway between 109 and 110 is 109.5 and halfway between 110 and 111 is 110.5 so they're going to be my boundaries i know it's really 110.5 would round up which is why you've got this strictly less than sign in here work out cube root of 4096. now this is a job for the calculator now your calculator if you've got a scientific calculator should have a cube root button of some sort sometimes it is written like this if you've got a cube root button some calculators don't have a cube root but they have an x root button that you can use or might have a little square like this so one of those symbols there now luckily my calculator does have a cube root button so i can just press shift and then that's cube root button and then type in 4096 and then press equals and it gives me the answer of 16 now clearly you need to know how your calculator functions to find cube roots uh similarly my next question is work out three to the power of seven now again different calculators have a power key that's written slightly differently some of them are written like x to the n some of them have a little x and a little square where the power should be uh some of them just have like a little up arrow like that again you need to work out which is your power button mine looks like this i'm going to press 3 to the power of 7 equals and my calculator gives the answer as 2187 2187 work out 512 divided by 5 to the power 3 times 5 squared is a power of 5. okay i'm just going to use the fact that when i multiply two numbers with the same base i subtract when i multiply them i add the indices and when i divide them i subtract the indices so this is going to be the same as 5 to the power 12 subtract 3 plus 2. so if 12 subtract 3 is 9 plus 2 is 11. so that's equal to 5 to the power of 11. it's that one there write the number 6 million 5 200 in standard form okay let's try and write it out as a number first off so 6 million is 6 followed by six zeros six million five thousand so that's gonna be zero zero five two hundred like that six million five thousand and two hundred okay now we've got it written as a number now we could try and write it in standard form uh so we're just going to write all the digits 6.0052 i can forget about the last two is it i'm not going to change it either way so times what 10 to the power of well if the decimal point starts here and we're going to end up one two three four five six gonna move it six jumps to get it to there aren't we so that's times ten to the power of six so my answer is six point zero zero five two times ten to the six match each sequence uh to its description one has been done for you you can see here we've got one two three four five six different sequences and the first one is mapped as a fibonacci sequence it is look because one plus one is two one plus two is three and with a fibonacci sequence you get the next number in the sequence by adding together the last two you see that's the case all the way through there about the next sequence i've got 1 2 4 8 16 32 now from one term to the next you can probably see that i'm multiplying through by two it's doubling each time isn't it times two so times two times two times two so anything that has a common ratio like that multiplied through by the same number that is a geometric progression uh next one one two three four five six that one is going up in even jumps it's plus one each time plus one plus one plus one uh so any number any sequences going up in a fixed number each time that's an arithmetic progression okay what we've got now one 3 6 10 15 and 21 now if you drew those as a sequence of dots i think you can probably see that this is the it's the triangle numbers it's that one there uh 149 16 25 36 i hope you can recognize is the square numbers and that leaves me with the cube numbers 1 8 27 64 125 216. yep that all looks good okay all done a linear sequence starts 11 21 31 41 and so on okay uh now if it's a linear sequence it's going to have uh gaps in between that are equal aren't they so this is a plus 10 sequence can you see is 10 between each numbers and that means the nth term is going to be something like uh 10 lots of n so where n is 1 2 3 4 then 10 n is 10 20 30 40 and so on but we've not got 10 20 30 40 we've got 11 21 31 and 41 now each of our numbers is one more than 10 n so we must have 10 n plus one and that is my nth term 10 n plus 1. the nth term of a sequence is 5 n subtract 2. work out the third term well if it's a third term then that means that n must be 3. as n stands for whichever term is in the whichever position it is in the sequence so if n is three then five n subtract two is going to be five lots of three subtract two five three is a and 15 subtract 2 that is equal to 13 so it's that one simplify 3a plus 7b minus a plus 4b okay so we need to collect like terms so we've got terms in a 3a minus a and we've got terms in b plus 7b and plus 4b now notice when i highlight those those i highlight the sign that goes in front of them because that is part of what we're doing isn't it so 3a subtract a is 2a and 7b plus 4b is 11b so we get 2a plus 11b expand four lots of 2d plus five so single brackets a number on the outside that gets multiplied by everything inside the bracket so 4 times 2d is 8d and 4 times 5 is 20. there's a plus sign in in the middle so it must be plus 8d plus 20. i've got a pair of brackets to remove so i don't know how you remember this it might be first outside inside last might be something like that or you might have a smiley face whatever way you can work out that you have to multiply everything in the first bracket with everything in the second bracket y times y is y squared y times negative four is minus four y five times y is plus five y and five times minus four is minus twenty at this stage here just uh look in the middle for your two uh like terms and add them up so i've got y squared minus four and plus five make y so my final answer is y squared plus y minus 20. we're asked to factorize 15 x plus 18y now that's pretty much the opposite of what we were just doing expanding removes brackets and factorizing puts them back in the thing that goes on the outside of the bracket is the thing that's common to both terms so we've got an x and one and a y and the other so they're not common but we've got a 15 and an 18. 15 and 18 are both in the three times table so i can put a 3 on the outside now to work out what goes on the inside of the bracket we just need to think about well what 3 times what is going to give me 15x well 3 times 5x gives me 15x i'll have a plus in between and 3 times what gives me 18y well 3 times 6y must give me 18y so i get 3 lots of 5x plus 6y factorize x squared minus 9x plus 20. so we've got a quadratic expression and we want to factorize it if it factorizes it will go into a double bracket like this won't it it starts with x squared so i'm going to have an x and an x now i'm looking for a pair of numbers that multiply together to give me plus 20 and add up to negative 9 okay so thinking about the factor pairs of 20 first off 20 and get by multiplying 1 and 20 together or 2 and 10 or 4 and 5. they're the factor pairs of 20. so we want one that adds up to negative 9 okay well because uh i've got a positive number at the end and a negative number in the middle i must have minus and minus so these must all be negative values negative negative negative negative negative negative and which pair adds up to negative nine it's this one here so it's going to be minus four minus five x minus four x minus five solve x squared minus nine x plus twenty is equal to zero now that's the same expression as we've got above so i can just factorize it the same way x minus four x minus five is equal to zero now when you have two things that multiply together and give you zero it means ma one of them must have been zero to start with so either x minus four is zero in which case x is four or x minus five is zero in which case x is five so my answer is x equals four x equals five two solutions solve the equations three x plus fourteen is equal to two uh right so subtract 14 from both sides or change sides change signs to move the 14 to the other side i'm going to get 3x those two things are going to cancel each other out and i'm going to get 3x is equal to 2 subtract 14 which is minus 12 then dividing both sides by 3 i'm going to get x is equal to minus 4 x is minus 4. solve the equation five x plus one is equal to two lots of x plus four uh so we wanna get all the x's on one side and numbers on the other but let's try multiplying out this bracket first um don't always have to multiply it sometimes you can divide by the number on the outside but here we've got x's on both sides so i think it's going to help to mark that out first so 5x plus 1 is equal to 2 lots of x which is 2x plus 2 lots of 4 which is 8 okay so you can see that we've got unknowns on both sides here so let's let's subtract 2x from both sides so that will leave me with a 3x plus 1 is equal to 8 and we've also got numbers on both sides so let's subtract 1 from this side and also one from here that's going to give me 3x is equal to 7 and then dividing both sides by three i'm going to get x equals seven over three now seven thirds a top every fraction or improper fraction and that is a perfectly acceptable answer uh so you can just leave it in that form there it's probably the easiest way of doing it uh so just leave it like that rearrange x equals 2y minus 6 to make y the subject okay so i've given this you can do this pretty similar to how you would um solve an equation just trying to get y by itself basically so we can start by adding 6 to both sides and that bit will cancel with that leaving you with 2y on the right and then x plus 6 on the left and then we're going to divide both sides by 2. okay so then that cancels with that leaving me just y y is equal to x plus 6 i'm just going to leave it like that x plus 6 all over 2. okay that's my answer y equals x plus 6 over 2. the rectangle has length 2x plus 5 centimeters and width x minus 2 centimeters the rectangle has an area of 35 centimeters squared use algebra to find the value of x okay so area of a rectangle it's just length times width isn't it okay now we know the area we're told it's 35 we know the length it's 2x plus 5. we know the width is x minus 2. okay so solve for x okay so we've got a quadratic coming on haven't we so we've got a pair of double brackets that we're going to need to multiply out so that times that times out that times that that times that so i'm going to get 35 is equal to 2x squared plus 5x minus 4x minus 10 okay then what i'm going to do is going to collect everything on one side so that's 35 is that 35 sitting there i'm going to move over to the other side so collecting like terms 2 x squared 5x take away 4x is x minus 10 with my other 35 coming over is going to be minus 10 minus 35 which is minus 45 and that's going to be equal to 0 because there's nothing left on the left okay so i've got 2x squared plus x minus 45. can i factorize that i hope so okay so common factor well common factor pairs of 45. i've got 1 and 45. uh two notes three note yeah three and fifteen um five's gotta be in there isn't it four note five and nine [Music] six no seven nope eight no so which one of those pairs is gonna work uh now remember i'm gonna have two x and x in here so the order i put them in is going to affect the middle one as well isn't it uh i'm thinking it's going to be this pair here okay because if i put the 5 at the end here plus 5 it's going to be multiplied by the 2x giving me 10 and then the 9 here which is negative 10 subtract 9 is going to give me the 1 in the middle so it's 2x minus 9x plus 5. okay that's equal to 0 so that's going to give me solutions of x equals uh nine halves and over two here which is four point five or x equals minus -5 okay now does that make sense in this question because if if x was minus 5 then up here i would get 2 lots of minus 5 minus 10 plus 5 which is a negative number and i can't have a rectangle with a negative value in reality so that one is not a solution okay it's a solution to my quadratic but it's not a solution to this problem okay so i'm just going to ignore that one i'm instead going to use the x equals 9 over 2 or 4.5 okay so x is 4.5 solve the simultaneous equations 3x plus 2y is equal to 10 and 3x minus y is equal to 13. now you notice that these two values of x the coefficients are the same so this question lends itself very well to solving by elimination so let's uh let's call this one equation one and this one equation two then equation one subtract two so taking the second equation away from the first so 3x so 3x take away 3x they're going to cancel each other out aren't they just take care when you're subtracting negative signs here so 2y take away negative y is a 3y and 10 take away 13 is minus 3. so that's going to give me y is equal to -1 okay so now substituting back into one of our equations substitute y equals minus one back into uh should we pick equation one so then i've got three x plus two lots of minus one is equal to ten uh two minus ones is negative two taking that to the other other side so taking that over there i'm going to get 3x equals 12 giving me x is equal to 4. so that would tell me that x is 4 y is minus 1. always a good idea to check these just by subbing both values into the second equation the other equation we didn't use so three x minus y equals thirteen so three lots of 4 minus negative 1 is that equal to 13 12 subtract negative 1 12 plus 1 is 13 so yes it does work always good idea to do checks in the exam you better to find out in the test that you're wrong rather than when your results come back solve the inequality 3x divided by 2 is less than or equal to 9. now as long as we don't divide through by a negative number we can treat these pretty much the same way as we solve any linear equation so the first step i'm going to do is to take that 2 and cross multiply it so move it across the equal sign or not equal sign the less than or equal to sine so that's going to give me 3x is less than or equal to 9 times 2 which is 18. 3x is less than or equal to 18 and then dividing both sides by 3 is going to give me x is less than or equal to 6. okay in the second part solve the inequality 4 lots of x plus 2 is greater than 12. okay the first thing i'm going to do is divide both sides by 4 to get rid of that 4 on the outside of the bracket so that's going to give me x plus 2 is greater than 3 and then moving this plus 2 to the other side i'm going to get x is greater than 3 subtract 2 x is greater than 1. finally represent the solution set that satisfies both answers to part a and b on the number line okay so from the first part we had x is less than or equal to six so x must be less than or equal to six and in the second part we had x is greater than one so x got to be between one and six it can include six but it cannot include one okay i'm going to show the range by drawing a line from 1 up to 6. so i'm going to draw a line from 1 to 6. now the ends of the line depend on the sign so if it's strictly greater than or strictly less than we do an open circle so at one i'm going to have an open circle and if it includes the end point like at the sixth end we're going to do a field circle okay so because this sign is strictly greater than we've got an open circle and because this one is a less than or equal to sine we've got a field circle on the grid below draw the graph of x plus y equals 2 for the values of x from -3 to 3. okay so we need to work out our coordinates so we can plot this graph okay now the easiest way to do that is to create a table of values so pulling up a ruler let's uh make a little table for our x values and our y values okay so we know that this range goes from -3 to 3. now you need a minimum of three points but there's no harm in doing it a few more uh so minus three minus two minus one zero one two three massively overkill probably only need three or four uh but anyway it's not a very difficult question so x plus y equals 2 that means y is going to be to subtract the value of x rearranging that for y uh so we can find the y value by taking at the x value away from two so if x is three then y is going to be two minus three two minus three is minus one uh if x is two then y is two minus two is zero if x is 2 then y is 2 subtract 1 which is 1. if y is so if x is 0 then y is 2 subtract 0 which is 2. if x is minus 1 then y is 2 subtract negative 1 2 subtract negative 1 is the same as 2 plus 1 which is 3. i think you can see the pattern going on here it's going to be 4 and 5 isn't it okay so i've got minus 3 five to plot minus three five minus two four minus one three zero goes with two one goes with one two goes with zero and three goes with minus one and that's they're all in a nice straight line it was a linear function so you'd expect it to be in a straight line if it wasn't a straight line you should be asking yourself some questions okay so now they are all in a nice straight line just pulling up a ruler and let's join them up now your line should stretch the full width of your graph and start and stop with the first point uh the like the the graph does continue in either directions so your graph should show that okay so there's my line complete the table of values for y equals x squared minus x minus two so they've given you some of the values and they want you to find some more uh now it's always it's usually a good idea to start with a positive end because that's a little bit easier to work out and you might want to just double check your understanding by picking firstly a pair of values that you do know both of just to make sure you understand what it is you're meant to be doing so if i did input 3 into this function i would get 3 squared minus 3 minus 2. 3 squared is 9. take away 2 is 6. take away 2 is 4. yes so that's right okay so that is what i'm going to be doing for each of these questions so then if i put 2 squared in and i take away 2 and then another 2 2 squared is 4. take away 2 take away 2 is 0. and then i've got minus 2 minus 2. you see it's going it's repeating itself so that's probably the point where it turns around and starts going up so i'm already thinking this is zero and four here but i'm just gonna double check um so put minus one in minus one squared minus negative one minus two negative one squared is positive one take away negative one is positive one so that's one plus one take away two is zero so yeah that is zero and then for minus two minus 2 squared minus negative 2 minus 2. negative 2 squared is positive 4 minus negative 2 is plus 2 so that gives me 6. take away 2 is 4. okay so is what i thought it was going to be right so draw the graph of y equals x squared minus x minus 2 for the values of x minus 2 to 3. so basically we just need to plot those points on minus 2 4 minus 2 4 is up here what's next minus 1 0 minus 1 0 0 minus 2 0 minus 2 1 minus 2. uh two zero and three four okay now this was a quadratic equation so we were expecting a parabola and that is what we've got so i'm feeling pretty confident here that we've got it right the last part is to draw a continuous curve passing through each of your points [Music] it doesn't you don't have to be leonardo da vinci but try and get it to be as smooth as possible what you don't want to be doing is put in like two marks on your on your paper that will get marked down okay and make sure you you're using a reasonably sharp pencil and make sure you go through each of the points i'm a bit off on that one actually uh it's good to go through the center of each one okay 6c says write down the x coordinate of the turning point of the graph now my turning point is this point here isn't it that's where the graph stops going down and starts going up again now we only want the x coordinate so just reading up to the line i can see that that's going to be at a half so the x coordinate is a half interestingly they don't want the y coordinate here okay a straight line has gradient 4 and passes through the point 5 23 work out the equation of the line give your answer in the form y equals mx plus c okay so uh we know that the equation of this line is of the form y equals m x plus c and we're told here that the gradient is equal to four and that means m equals four so i know the equation already is of the form uh y is equal to four x plus c then so all i really need to do is find c now the second bit of the information tells me that my line passes through the point 5 23 so i can use that information passes through 5 23 so that's the x coordinate there isn't it and that's the y coordinate so if we sub those into the equation uh they should work because it's a point on the line and then the only thing i don't know there is going to be c then so sub in those in so 23 is equal to 4 lots of 5 plus c what's that let's just multiply out those brackets 4 times 5 is 20 isn't it so i've got 23 is equal to 20 plus c let's just subtract 20 from both sides uh it's going to leave me with c equals three so now i know all the pieces i've got m i've got c i can write down the equation it is uh y is equal to four x plus three we're told that y is directly proportional to x and we've been asked to complete the table now if y is proportional directly to x then y is going to be equal to k lots of x okay some constant we can find now we've got a pair of values that go together here when x is 7 y is 63 so let's sub those in uh so when x is 7 y is 63 so 63 is equal to k lots of seven so that makes k equal to 63 divided by 7 which is 9. so our full equation is y equals 9x so very simple to to fill out the rest of the table uh if x is 0 then 9x is 0 if x is minus a then nine knots of minus eight is minus 72 all done y is inversely proportional to x complete the table okay so y is inversely proportional to x that means y is inversely proportional little fishy over one over x okay so that's what that is saying mathematically now we can replace the little fishy symbol with equals k so i'm going to rewrite that so y equals k over x where k is a constant that we can then find okay now you can find the constant k if you've got two values uh well your value of x and your value of y that go together now we've got one of those in the table if you look at this table i've got two numbers here that go together when x when x is six y is equal to four so sub in x equals six y is equal to four into my partial equation y equals k over x i should be able to find out what k is so uh that becomes four equals k over six replacing y for four and x six okay then just taking that six there cross multiplying up there or multiplying both sides by six if you'd rather i'm going to get k is equal to 24 aren't i okay so my complete equation then is going to be y equals k which we've now found to be 24 over x okay so now this is my complete equation which i can then go and find my other values so if x equals 12 then y is 24 over x 1a is 24 over 12. 24 divided by 12 is 2. so when x is 12 y is 2. uh and what about the other way around so if y is 8 what is x okay so if 8 is equal to 24 over x uh just cross multiplying these two just switching them around i'm gonna get x equals 24 over eight 24 divided by eight is three so x is three when y is eight once you've done a few of these uh proportional questions they get very very formulaic they always give you they always start off by giving you the rule that um links y and x you can then form your partial equation uh which involves the constant k and then you'll have a pair of values that you can sub in to find what k is to complete the equation and then you can go on and use your complete equation to find other values they're always the same very very formulated questions and after you've done a few you should be good on them an athlete runs 240 meters in 45 seconds work out her average speed in kilometers per hour okay well i've got another triangle handy triangle for um distance speed time this time i've got d on the top these are alphabetical dst distance speed time uh and again i'm going to do it pretty much the same way as i did the percentage problem going to look at what i've got what i haven't got and then use the other two to work out they're missing one so an athlete runs 240 meters so that's going to be my distance and isn't it 240 meters so that's 240 m uh speed well that is what i want to find and then time 45 seconds so that is 45 seconds okay so then my speed covering up the thing i want to find i want to find speed that tells me what i have to do with the other two so average speed equals distance divided by time now if i just go ahead and sub those numbers in i'm going to sub in meters and seconds into this into this calculation and it's going to give me an average speed in meters per second but this particular calculation and we have been asked to put it into kilometers per hour okay so before i sub these numbers in i'm going to convert them into uh the right unit so 240 meters what's that then in kilometers that's going to be uh well there are there are a thousand meters in a kilometer so i need to divide 240 by a thousand then don't i that's going to give me not 0.24 it's not 0.24 kilometers and then time 45 seconds what is that in hours well it's going to be there is a was a 60 seconds in a minute and 60 minutes in an hour so if i want to convert um seconds into hour i need to divide by 60 seconds and i need to divide by another 60 seconds so what's that equal to 45 divided by 60 divided by 60 1 80ths which is not point naught one two five okay so then subbing that in distance is not point two four time is 0.0125 so then 0.24 divided by 0.0125 that comes out to be 19.2 19.2 kilometers per hour okay so 19.2 an alloy is made from 28 centimeters cube of copper and 41 centimeters cube of gold so those two amounts are they're actually volumes aren't they in centimeters cubed uh so i know the volume of copper and the volume of gold now i told i'm told that the density of copper is nine grams per cubic centimeter and the density of gold is 19 grams per cubic centimeter work out the mass of copper used okay so we need to know the formula that links density mass and volume uh it's just density is equal to mass divided by volume that's the formula okay so in this case i've been given the density i've been given the volume i want the mass uh rearranging that mass is going to be density times volume isn't it so mass equals density multiplied by volume okay so my density of copper is nine grams per cubic centimeter my volume is 28 centimeters cubed it's a good idea just to check you've got consistent units when you're doing these sort of things so density is in grams per cubic centimeters and my alloy is measured in cubic centimeters so if i combine these together i'm going to get grams and aren't i for my mass so 9 lots of 28 then 280 minus 28 what's that that's 252 isn't it so 252 grams now part b says work out the density of the alloy okay now to work out the overall density of the alloy we need to know uh what the total mass is and what the total volume is okay so the total volume is going to be the two uh it's going to be the two volumes added together then isn't it okay so we had a volume of 28 and a volume of 41 didn't we so 28 plus 41 gives me a total volume of 69 now the total mass now we found the mass of um copper already in the last question it was 252. and so we need to add on the the mass of the gold okay so we need to do another mass calculation then don't we so to finish this off we need the mass of the of the gold don't we so mass of the gold is going to be remember it was density times volume wasn't it so the density of the of the gold times the volume of the gold so 19 lots of 41. so that came out that comes out to be 19 times 41. that's 779 so then the total mass is going to be 779 plus 252 comes out to be 100 okay so then the density of the alloy is going to be the total mass divided by the total volume say 1031 divided by 69. what's that 1031 divided by 69 14.94 i'm going to round it off to 14.9 so 14.9 grams per cubic centimeter a b c and d e f g are parallel lines that's already been marked on you can see by those arrows at the end at c and g and it tells me that b e h is a straight line okay so we've got a pair of parallel lines and a couple of lines crossing them so we need to work out the size of angle x okay so angle x is that one over there how can i find that one well when you've got a pair of parallel lines and a transversal or a line crossing them then it forms various angles and you should know about corresponding alternate and co-interior angles and i can see from here my x angle and my 25 angle are what we call co-interior angles so this one and the x should add up to 180 degrees co-interior angles add up to 180 degrees so 25 plus x is 180 so then taking 25 off both sides then should tell me what x is so x is equal to 180 subtract 25 which is 160 155 isn't it so x is 155 degrees and then b says work out the size of an angle y you must show your work in which may be on the diagram well it will be on the diagram because i always do that okay so where's where's why it's down there there's why look we need to find that one okay so again we've got a pair of parallel lines the yellow ones and we've got this line here that crosses them this purple one okay so we should know that this angle here the y angle corresponds to this angle up here now that angle up there is made up of the right angle which is 90 degrees plus the 25 angle which is equal to 115 degrees so y and 115 are corresponding so that is the answer so i'm going to say y corresponds to 90 plus 25 equals 115 degrees 115 degrees it shows a polygon abcde and that has been divided into three triangles and we need to use the triangles to work out the sum of the interior angles of a polygon abcde and we must show our working well this polygon has got five sizes it's actually a pentagon so a five-sided shape has got three triangles that go inside uh your maths teacher might have shown you how to derive the formulas for a sum of interior angles of a of a polygon and they do involve just dividing the shape up like this into triangles you know that the angles inside the triangle sum to 180 degrees so this triangle here adds up to 108 degrees now the triangle next to it also adds up to 180 degrees it's interior angles there and finally our last triangle adds up to 180 degrees too so you can see that those three angles there form that interior angle of my pentagon and then i've got another one there look and another one there another one there and another one there so the three sets of 180 degrees from my triangle add up to the interior angles of my pentagon so let's write that down below uh we're going to write angles in a triangle sum to 180 degrees so interior angle of abcde equals 3 times 180 which is 540 degrees okay here is a map of france and we can see we've got several cities marked on the map and we've got a scale down here it says one centimeters represents 80 kilometers okay what's the question asked over the page what is the free figure bearing of leon from bordeaux the three-figure bearing of leon from bordeaux so we're going to start in bordeaux and we want to go to lyon okay right so you'll notice at the at the top here we've got a direction line for north and three figure bearings are always measured clockwise from north so let's just get my ruler out so straight up in the direction of that north line that's north so if i want to measure a bearing i should really start by putting a north lone line on the point where i want to measure my bearing from so i want to go directly north from bordeaux so drawing an up arrow from there okay so that's my north line and from bordeaux i want to go in this direction don't i nope now so using a protractor what you want to do is you want to measure that angle in there okay uh can't really show protractors so well in this particular application uh but if you plonk your protractor on and measure it you should find that you get an angle of 84 degrees in there 84 degrees uh and then so to write that down as a bearing 84 degrees has only got two digits okay we haven't got 84 we've got 85 or 95 or 175 or zero five okay so i think that perhaps i was just a little bit out of my measurement i'm going to say it's that one there 085 it says work out the actual straight-line distance from paris to marseille work out the actual straight line distance from paris to marseille so paris is up there in the north look and marseille is down in the south so let's join them up with a straight line now with your ruler you're just then going to measure that carefully then count the number of centimeters is between paris and marseille now again because i'm using an emulated ruler here i'm not going to be able to accurately measure it to show you but if i did get a ruler out i think is around about 8.2 centimeters now if you're working from a practice paper from your school because they print it out for you sometimes these things don't get reproduced correctly at the right size so you might find that yours is slightly bigger or slightly smaller don't worry too much about that it's just one of those things that happens when schools give their students practice papers uh what you just need to make sure you do is you take whatever measurement you've got there and you multiply it by the length to get the and you multiply it but sorry by the scale factor in order to get the correct distance so the scale factor is one centimeter represents 80 kilometers and i've got 8.2 centimeters each one of those centimeters is 80 kilometers gonna have to multiply these two things together okay so again using napier's bones grid and do my eight point two times eighty uh eight eights are four two eights are sixteen eight zeros well zero and zero what's that then adding up along the diagonals that's zero six five six uh so reading around my answer around the edge here i've got the digit six five six zero now remember i had one number uh behind the decimal point there so i'm also going to have one down here that's behind the decimal point so my answer is 6656.0 or 656 kilometers so what is the actual distance it's 656 kilometers again if you're working on a reproduced practice paper that's been photocopied by your school it might be that it got stretched or squished a little bit so your answer might be smaller or uh bigger than that don't worry about too much as long as you multiplied it correctly by 80 i'm sure you've got the diagram shows triangle abc construct a bisector of angle bac then construct the perpendicular bisector of a ac and then shade the region inside triangle abc that is nearer to ac than to a b and nearer to a then to c so we need a straight edge and compass for this so let's get busy with that then okay first up make sure your compass is set up correctly you need to have the pencil tip and the tip of the compass in the same place before you tighten it up it's a good idea to keep yourself a little stumpy pencil like this one uh so that uh you can swirl your compass all day around without bumping into your hands okay um then what we need to do is uh what we've got do we've got a constructor bisect of angle bac so this is b a and c so that angle bac is this angle here we need to cut this in half okay now to do this you stick the pen uh the compass tip at the angle so in a and we're going to draw an arc that cuts both those two arms of the angle okay then we need to move our compass tip into that place where we drew we cut it and then into that places where we cut it into our two extra arcs now those two will make a cross and then we can then just join point a through that cross that we've made okay now it doesn't matter what width you set your compass to just that you don't move it while you're doing the construction so you need to try and keep it the same if you know it's moved you probably want to stop and try again now that seems to be a pretty good job you can always check that with the protractor to be sure you've done a good job right next construct a perpendicular bisector of ac this is ac so we want to cut this line in half now to do it we need to set our compass so that it's slightly more than half the distance between the two points between half and two thirds of the line is a good distance okay and then we're going to draw two arcs uh from c and from a and then hopefully they'll cross i'm going to struggle a bit because it's going off the bottom for my paper here um so bear with me okay so there's one of the arcs now move the point over to a and draw the second arc okay i haven't quite got the arc to cross at the bottom there let me just extend that one a little bit okay there we go so then those two uh arcs make two points of intersection which you can then join up with your ruler okay oh i've got my ruler the wrong way around you can see there's an awesome pokemon ruler today okay so there's my second line so we want to shade the region that is inside triangle abc that is near to ac then to a b so near to ac then to a b is going to be below this uh angle bisector then isn't it and nearer to a than to c is going to be the left hand side of this perpendicular bisector so the area that we want is this triangle in here so it's below the angle bisector and to the left of the perpendicular bisector all done good job here is a right angle triangle uh and we need to show that x is equal to 12. okay so we've got two sides of a right angles triangle and we're wanting to find a third side so clearly we are dealing here with pythagoras theorem pythagoras theorem states that c squared equals a squared plus b squared where c is the long side and a and b are other two sides okay so we're just going to sub into there and find what c is and then say that's what x is so c equals a squared plus b squared in this case c is is x x squared is equal to 7.2 squared plus 9.6 squared so on my calculator 7.2 squared plus 9.6 squared comes out to be 144 so then x is going to be the root of 144 which is 12. in the diagram the area of triangle abd is 56 work out the length of cd okay in the diagram triangle abd what is triangle abd triangle abd is a b d it's this triangle here now we're told that the area of this one is 56 centimeters now we know that the area of a triangle is half its base times its height so area of triangle is half base times height uh so we know that baseline of this triangle is 14 but we don't know the height let's call the height h okay let's call it h so then my area of my triangle which i already know is 56 is going to be a half times my base which is 14 times my value of h which i'm actually trying to find okay uh right what can we do with this then so half of 14 that's seven isn't it so actually what i've got here is 56 equals 7h and then dividing both sides by seven sevens into 56 go eight times so that means that h is equal to eight okay so now we know how tall this triangle is it's eight uh now we want to find the length cd which is this length here okay now looking at the right triangle that has got a length of six and a height of eight so i wanna know the hypotenuse of that right angle triangle now you can do this with pythagoras theorem uh i already know the answer because i know that six and eight goes with ten okay so i know already that it's ten because i know what what are the pythagorean triples now these are very handy triples to know so the the first one is three four five and then the any multiple of that is also a pythagorean triple so if you double each of those you get 6 8 10. there's another one 5 12 13. and knowing those means that if you have got a problem that involves pythagoras theorem on the non-calculated paper you can pretty much always work out what these are without even having to think about it but let's assume you didn't remember the pythagorean triple you'd have to use that c squared was equal to a squared plus b squared wouldn't you so c squared equals a squared plus b squared which in this case would be 6 squared and 8 squared 6 squared is 36 h squared is 64. adding them together you're going to get c squared is 100 making c equal to 10. so my answer is 10 centimeters as i already told you okay use trigonometry to work out the length x i tend to remember ratios using the memoric soccer tour maybe you've got a different way maybe you can share it in the comments if you do uh right so let's label up the sides here's my angle this is the longest size that must be the hypotenuse x is opposite my marked annual angle and the other side is adjacent now which ones am i using i'm using the hypotenuse and the opposite so the ratio i'm going to be using is sine okay so sine of my angle sine theta is equal to opposite over hypotenuse which in this case my sine 72 is going to be my x over eight rearranging for x x equals what we're going to do is move that eight up there so x is equal to 8 sine 72 reaching for the calculator sine 72 make sure your calculator is in degree mode that could be a terrible terrible mistake otherwise sine 72 times 8 is 7.608 i'm going to round off to three significant figures does it tell me to do that no if it doesn't say then three significant figures is usually a good one to go for so i'm going to round that off to 7.61 for abc is a right angle triangle a b is 20 bc is 37 calculate the angle bac b a c that's the angle there isn't it uh if i go from b to a to c so yeah that angle between b a and a c okay so that's the angle we want to find uh i'm going to call it theta which is my want for for angles it's my go-to variable that i label them with so that's theta now this is going to be a straightforward trigonometry question isn't it i've got two sides and i want an angle of a right angled triangle okay so step one write down our mnemonic for remembering the the various trigonometric ratios which is sorca you might have something else uh to remember it uh then we need to work out which one which one of our three ratios we're going to be using now we label up our sides relatively angle given so this one is opposite theta and this one is adjacent to theta so we've got the opposite and adjacent so that tells me that we need um tan tangent is equal to the opposite over the adjacent okay so i'm going to write down the formula for tan tan theta equals the opposite over the adjacent okay now what we need to do is sub the values in from our diagram to work out the answer so the opposite is say 7 the adjacent is 20. okay so then theta is going to be the inverse tangent of that then 37 over 20. okay so reaching for a calculator we're going to use the inverse tangent button which you have to access by pressing the shift key first so shift tan and then 37 over 20. okay so what's that then that is a 61.6 degrees 61.6 degrees hello from mr tompkins edtech memberships are a feature on youtube that allows subscribers to directly support creators and to also gain better access to them as a member of this channel you pay a small subscription fee and in return receive a range of additional channel perks here on mr tompkins edtech i provide channel members with their own loyalty badge which is displayed whenever you comment on any of my videos i always reply to channel member comments first and i try to answer all their questions fully i upload exclusive member-only videos for all you ed tech enthusiasts and i also post early releases of my popular exam paper walkthroughs so if you can't find the paper you're looking for yet on my main channel you'll most certainly find it first in the members only area as a special acknowledgement and thank you higher tier members can even have their name appear in the credits of my videos so if you enjoy my content and you would like to support me make more of it and at the same time enjoy some exclusive extra perks then why not sign up today in the diagram a b c is a right angle triangle p is a point on a b and b c is 40 meters ap is 20 meters and angle abc is 30 degrees okay now i do suggest you read through this information above the diagrams carefully and just double check that everything written up there is on the diagram in this case it is i've got the bc is 40 i've got ap is 20 and i've got the angle and i've got the right angle marked but that isn't always the case and it's always worth just double checking because otherwise you might be staring at a problem that you can't solve because you don't have all the information okay but all the information is on the diagram here so we can proceed show that ac is 20 meters where's ac ac is this side here isn't it i need to find that line ac how can i do it uh well if we look at triangle abc that's just a right angle triangle and i know an angle 30 degrees and i know a side hypotenuse so i can find a second side no problem just using um trigonometry can't i okay so if if 30 is the marked angle then the 40 degrees that's the hypotenuse isn't it and ac is the opposite and a b is the adjacent okay so time to uh think about our memonic soccer tour now we have got a hypotenuse and we want the opposite so which one is about o and h it's the soft part isn't it so that's the sine of an angle is equal to the opposite over the hypotenuse if we're right now nice and long okay so going to write out the correct trigonometric ratio next thing is to sub in what we know we know the angle is 30 so let's put that in so sine 30 is equal to the opposite that's what i want to find it's the side ac and then hypotenuse we know is 40. so rearranging multiplying both sides by 40 we'll cross multiply that 40 from where it is to the other side so i'll get a c is 40 sine 30 okay at this point you'd reach for your calculator and then realize oh i don't have my calculator this is the non-calculated paper so am i stimmid nope because you should remember your exact trig value shouldn't you for angle 0 30 45 60 and 90. if you didn't remember those why not you need to know them for your exam don't you okay there are various ways of remembering these values but you should remember that the sine of 30 is a half okay so i'm going to replace sine 30 from my knowledge bank of sines and cosines i need to remember and sub it in okay so 40 times a half is equal to ac so that makes ac equal to 20 as required a water tank which is a cylinder with radius 40 centimeters in depth 150 centimeters we're told that it's filling the rate of 0.2 liters per second and we're being asked does it take longer than an hour to fill the tank well in order to answer that question we need to know what the capacity of the cylinder is to start with so let's do that first what's the volume of this cylinder well a cylinder is a prism and the area of a prism is equal to its cross-sectional area times its length for its height so we need to work out what this bit is here okay now that area is a circle it's pi r squared times the height so the volume of this is going to be well that's pi r squared doesn't it so the volume of this cylinder is going to be pi r squared h i'll leave the pie in for now do i want to leave the pie in now let's put it in 3.14 times 40 squared times 150 so what's that 3.14 times 40 times 40 times 150 so that's 753600 now that's in cubic centimeters now we're told we should really know this but one liter is a thousand cubic centimeters so if it's got a volume of um that then it can hold let's divide that by a thousand and find how many liters it can hold so that's 753.6 liters okay so if it's filled at a rate of 0.2 liters per second how many seconds does it take to fill up so uh time taken to fill up is going to be the volume divided by that flow rate then isn't it so taking time is equal to 753.6 divided by 0.2 and that equals 3000 seconds 3768 seconds now this question is asking about time in hours so we need to convert that to hours so time in hours is going to be the seconds divided by 60 to get minutes and then another 60 to get hours so 376 divided by 60 divided by 60 is 1.046 reoccurring which i'm going to round off to 1.05 so that's 1.00 hours okay so does it take longer than one hour to fill the tank well yes it does because it takes one hour and five one hour at 0.05 so the answer is yes it takes longer than one power a cube has edge length 0.9 meters work out the total surface area of the cube give your answers in square centimeters okay so notice first that that's in bold and normally things in bold are there because students will otherwise not see them so just take extra care when things are in bold so square centimeters this this measurement is in meters we've been given up here so we need to convert that into centimeters first and if we're going to work this out in centimeters so let's convert that to 90 centimeters so this is 90 centimeters by 90 centimeters by 90 centimeters okay now uh a cube is made up of six equal faces and each face is a square and 90 times 90 is the area of each square so the total surface area the total surface area of this cube is going to be the the area of one face which is 90 times 90 times by the number of faces which is six so we're going to find six faces and each face is 90 by 90 so the total surface area is going to be six times ninety times ninety so tapping that into the calculator six times ninety times ninety gives me the answer of forty eight thousand six hundred centimeters squared 48 600 centimeters squared okay let's just write it down in the bottom oops let's write in the bottom properly triangles a b and c and d e f are similar and we've got them shown here now it says work out the value of x now if two shapes are similar it means their sides are in the same ratio so these two sides match up on my diagram and these two sides match up and if we have a look at the relationship between three and six that's times two isn't it so the six is twice as big as the three so it's gotta be the same here from here to here it's got a b times two as well so whatever x is when i multiply by 2 i'm going to get 9 in other words x is 9 divided by 2 which is 4.5 okay so that's my value of x it's 4.5 uh and then it says write down the size of angle y well as well as uh all the sides being the same ratio in similar shapes all the matching angles are going to be the same so that one and that one actually match up don't they so i know straight away that y must be the same angle it must also be 53 so that needs to be 53 degrees on the grid draw a shape that is a reflection of the of the shape white show your mirror line okay so they haven't actually drawn a mirror line on for us so we've got quite a lot of freedom for this question so let's pull out a ruler and draw on a mirror line for ourselves i'm gonna draw the mirror line here okay and then let's reflect each of the points uh when you're doing a reflection should really take each point in turn and work out where its reflection goes so this point is one unit away from the mirror line so its reflection will be one unit away this point here is three units from the mirror line so its image is going to be one two three units away as well and so is that one at the top and it's going to be three units away okay once we've got all the corners we can then draw the lines in like so and then we have our reflection say as long as you've reflected it in a mirror line and drawn your mirror line on you should get both at the mark for that triangle a and triangle b are drawn on the coordinate grid yes they are part a says draw the image of triangle a after a rotation of 180 degrees about the point zero zero okay so one way of doing this uh if you're not very good at rotations would be to go and ask the invigilator for a sheep of tracing paper lay that over the top of your diagram next what you want to do is to put a little cross where the center of rotation is at this point here is it's at zero zero isn't it so if you put a kind of a cross going through zero zero it will help you keep track of your rotations uh then you wanna put your pen tip or your pencil tip at zero zero and turn the paper around okay so when when you get that little cross line up again for the first time you'd have done a 90 degree 90 degree turn and then let it go around one more time so it lines up one more time and you've got a 180 degree turn now when you've got 180 degree turn they don't bother telling you if it's anti-clockwise or clockwise because you're going to land in the same place in this case i'm going to have the points as indicated here so i'm just going to remove the tracing paper carefully and put the image points in so my three image points are going to be here here and here then aren't they okay so just carefully put the image points on and then once you've got the image points on you can then join them up with a ruler like so so rotation it should change the orientation and position of your shape but not as in not its overall shape because it's still the same shapes just kind of upside down and move to a different quadrant okay so that's my rotation uh the next thing we need to do is to draw the image of a triangle after a transition of uh by the vector two negative seven uh well a vector two negative seven if you were gonna draw that vector it would be two along and seven down so i need to change take each one of my points and move it two units to the right and seven units down so let's take this point here first of all go in two units to the right and one two three four five six seven takes me to that point there okay just going to rub that out again okay similarly this point here is going to be translated to that point there and this point here is going to be translated to that point there then isn't it okay once you've got your corners then you can join them together with your ruler again part b says describe fully the single transformation that match triangle a onto triangle b now this is a three mark question so you expect to give three bits of information in order to get the to the mark let's go back and have a look at the the diagram uh you can see a and b they're sitting next to each other a is bigger than b i remember a is the original image our object though and b is the image so a has been translated onto b um so i think this is going to be an enlargement it's one of those crazy mathematical enlargements where your image ends up being smaller than the object okay but we still call that in maths and enlargement uh weirdly anyway let's let's draw on some ray lines pick two points that go together so this point here matches up with this point here on the image so if i draw a ray line through those points i'm going to use green so like so i'll just draw these faint really uh and i'm going to do the same i'm going to find another two points that go together so that tip of the triangle goes with that tip of the triangle down there doesn't it so i can put another ray line through those two points and you should find that your ray lines converge on a particular point and that particular point is known as the center of enlargement in this case it's this point here isn't it and that is the point negative one two okay so that's my center of enlargement now i need to give three bits of information it's an enlargement it's one thing uh it has a center of enlargement at minus one two that's the second thing and the third thing is going to be the scale factor now notice that this line here is four units and its matching side is two which is half as big so that's telling me that the scale factor is a half okay so there are the three things that i need to mention when i describe this transformation it's an enlargement one mark center -1 2 scale factor one half okay there are my three bits of information for my three marks 50 people took a test before the test they predicted whether they would pass or fail 30 people predicted they would pass 26 of the people who predicted they would pass did pass and 37 people passed all together complete the frequency tree are you predicting that you're going to pass this test all right so 30 people predicted they would pass so prediction 30 people predicted they would pass so 30 goes here so if there's 50 or together then in the fail box it's going to be 50 subtract 30 isn't it give me 20 people in there so 30 people predicted they would pass and 20 predicted they would fail okay so i've dealt with that first line there 26 of the people who predicted they would pass did pass so out of my 30 people who said they would pass 26 did pass so then that means that 30 subtract 26 or four of them thought they would pass but they didn't they failed okay uh and it said 36 so we dealt with that one now haven't we so uh and then finally 37 people passed all together so i know that 26 people passed up here and i know that 37 people passed all together so 37 subtract 26. is the number of people who passed who thought they were going to fail then so 37 subtract 26 is 11. so if 20 people thought they'd fail but actually passed that means nine people who thought they failed did fail 20 subtract 11 is nine isn't it okay the table shows information about the distances walked by 120 students on their way to school in one week and you can see that we've been given a grouped frequency table for those uh distances uh so i've got zero to five there are twenty students who walk between zero and five uh 48 students between what between five and ten miles 30 students walk between 10 15 miles and 22 between 15 and 20 miles now we're asked to work an estimate of the mean for this data now the problem is that this data has been grouped so we don't have the precise values but we're estimating so we're going to use mid x as an estimate for each of these so you find mid x by adding your two n values together and dividing by two so zero plus five is five divided by two is two point five you're basically averaging that two endpoints uh five plus ten is fifteen divided by two is seven point five ten plus fifteen is twenty five divided by two is 12.5 and 15 plus 20 is 35 divided by 2 17.5 okay once you've got the mid x then we can work out f times x so grabbing the calculator 20 times 2.5 that gives me 50 48 times 7.5 is 360. 30 times 12.5 375 and 22 times 17.5 give me 385 oops okay and i just need to add all those up to get a total so 385 plus 3.75 sorry 375 plus 360 plus 50. it gives me a total of 1170 1170 so then my mean then my estimated mean is going to be my sigma fx divided by my sigma f in other words the sum of fx which is this value here 1 1 7 0 divided by my sigma f sum of my frequencies which is this value 120 okay just type that into the calculator again 1 1 7 0 divide by 120 is 9.75 okay so that's my answer 9.75 in one hour the shop sells 180 scoops of ice cream the number of scoops of each flavor is shown in the table okay so 45 scoops of vanilla 75 scoops of strawberry and so on now we need to share this information in a pie chart but before we do that we need to work out what angles and things we're going to use so before we get on to the pie chart we've got some work to do on the table first now we've already told what the total was the total number of scoops was 180 wasn't it okay so 45 plus 75 plus 50 plus 10 that's 180. now what we want to do is we want to take those number of scoops and turn them into the number of degrees that we need to draw uh on the pie chart to represent them now in a full turn you've got 360 degrees haven't you so the the angle that we're going to use to show this needs to add up to 360. now you'll probably notice straight away that 360 is twice as big as 180 so that means we can use two degrees to represent one uh scoop of ice cream that was sold okay so that makes it very easy to work out the various angles that we need to to use we just need to double up then each of these so my 45 scoops of ice cream if i'm using two degrees per scoop that's going to be a 90 degree angle that we're going to use uh strawberry 75 times 2 that's 150 degree angle chocolate is going to be a 100 degree angle and mint is going to be 20 degrees okay okay so moving on to the pie chart now we've been helpfully given one of the lines to start off with so this is going to be like one of the lines for our vanilla sector now vanilla was a 90 degree angle so if we lay up a tractor along that starting line with the center of protractor right at the end of the line in the center of the circle we should be able to count around to 90 draw a mark then remove the protractor and then draw on our 90 degree angle line okay so that was 90 degrees wasn't it okay so this was then vanilla right now the next angle we need to draw is a hundred and fifty so again we're going to take the protractor we're going to lay on top of our newly drawn lines of the second vanilla line and we're going to count around 250 we're going to make a little mark remove the protractor and then join up the line that we made okay so 150 it's going to be there isn't it okay so then that is my strawberry section that was 150 degrees uh the next section was our chocolate degree our thing and it was a hundred degrees so again we're going to lay the protractor along our last drawn line with the center of the protractor at the center of the circle we're gonna count up to a hundred make a little mark remove the protractor and then we can draw our line in okay so that was my chocolate section uh and that was 100 degrees okay and now the mint section it kind of draws itself okay so it's just the leftover part isn't it but you might just want to double check that that angle there is 20 degrees so if you measure it and it should be there or thereabouts if you look at degree either way it's not going to make much difference but just double check that it's there or thereabouts for 20. okay and that's that we're done the scatter diagram shows the results of 10 of the people in these experiments here are the other two results okay so i've got two more people to add one was 47 years of age and took a time of 21 seconds and the other one was 60 years of age and took a time of 34 seconds so i need to add these to the scatter graph uh just make sure you're clear about the scale before you start plotting points you can see from here to here there are 10 divisions aren't there so each one of those little squares is going to be worth one and the same up here as well got 10 divisions uh spanning a gap of 10 so each one is worth one it's a nice easy scale to use so just find 47 on my scale this way and read up and read across from 21 and mark our cross okay there's a cross there and then uh the next one is it's 60 and 34 60 and 34 one two three four there okay right so now i've plotted those points part c says what type of correlation is shown in the dia in the scatter diagram okay if i get my ruler out uh you can probably see that i've got a kind of band going up the data like that from from left to right sloping upwards so got clear positive correlation here so i'm just going to write down the word positive question d estimate the time it would take a person age 35 to complete the puzzle show you're working to justify your answer okay well my working is going to be on the the scatter diagram itself okay so coming back to the diagram if i put a line of best fit up my graph just trying to angle it so you've got the same number of points on either side of your line something like that doesn't have to go through the origin so don't force it through there something like that you do get a bit of leeway uh when you draw these points um probably it's like anywhere that this end is anywhere between there and this one is anywhere between there and you usually get the mark so it doesn't have to be an exact science uh but a reasonable attempt at drawing your line the best fit is required okay so once we've got our line of best fit we can use it to help estimate our value so i'm going to find 35 years of age on my diagram and i'm going to read up to the curve and across okay so there's 35 and reading up to the line when i hit the line i'm going to change direction and read across and i'm reading that as about 11 on my scale okay so estimate the time it would take a person at age 35 to complete the puzzle it would take them 11 seconds parties says lee says that at least 80 of the 12 people completed the puzzle in under 30 seconds easily correct show working to support your answer now coming back to the diagram if i put my ruler at 30 you can we can then count how many people are below it then can't we so how many is that then got one two three four five six seven eight nine people are below 30 seconds aren't they and we've got three above that line there okay i'm just going to draw kind of a dotted line along there show our method okay so we've got nine out of 12 then haven't we okay so what's that as a percentage nine out of 12 as a percentage well i just need to tap that into my calculator don't i 9 divided by 12 times by 100 to turn that into a percentage so 9 divided by 12 times 100 comes out to be 75 so is he correct no it's not at least 80 percent it's 75 so lee is not correct beth uses these four cards to make four digit numbers how many different four digit numbers can she make that are greater than eight thousand okay beth what you're gonna do here uh two four five and eight so we've got four places to fill let's think about what choices we have for each of these now my number needs to be greater than eight thousand so i i don't have a lot of choice for the first digit it's gonna have to be this one here isn't it it's gonna have to start with an eight so it's eight thousand and something if it starts with two two four five it's gonna be less than eight thousand so it's no good okay so it has to start with the eight that was that was only one choice i had for that uh right what's the next digit the next digit could be any of the other three numbers so i've got three choices for that one it could be the two or the four or the five so let's say i pick the two how many choices have we got left for the next choice i've got two choices let's say i pick the four how many choices have i got left for the last one i've only got one okay so then we can work out the total number of ways by then multiplying those choices together so 1 times 3 times 2 times 1 should give me the number of ways of working out all together so 1 times 3 times 2 times 1 is 6. so there are six different numbers greater than 8 000 that she can make another way of doing it would be an exhaustive approach where you just wrote down all the various values until you had all the different ones so 8245 8254 etc uh until you've got through all of them either way you're going to get six values the four possible outcomes of an experiment are a b c and d and we're told that the probability of a is 0.28 i'm told the probability of b is twice the probability of a so that's going to be 2 times 0.28 what's that 2 times 0.28 that is 0.56 and we're told that the probability of c is equal to the probability of d okay so we've got the events a b c and d okay so that is our outcome and this is the probability so we know that a is 0.28 we know that b is 0.56 and we want to work out we know that c is equal to d we don't know what those are let's call them x and x uh okay and we want to work out what x is then for the probability of d now because of probability one of those things must have happened all of those probabilities need to add up to one so if i take uh 0.28 and i add 0.56 and x and x and add it together i should get one for completeness so let's add like terms together what is not 0.28 plus 0.56 0.28 plus 0.56 is 0.84 so 0.84 x plus x is 2x plus 2x is equal to 1. so i've got myself a little linear equation an x that i can now solve so taking my 0.84 moving to the other side of the expression i'm going to get 2x is equal to 1 subtract 0.84 which is uh 1 subtract .84 is 0.16 so i know that 2x is 0.16 so then x is going to be 0.16 divided by 2. so divide that by 2 i'm going to get 0.08 that doesn't look right yes it does 0.08 so that is going to be my value of d and c and our answer wants us to find the probability of d so that's 0.08 one of the x's hannah wants to display all the possible outcomes when rolling two fair six-sided dice give a reason why a tree diagram is not the best method to use well in a tree diagram you're doing a branch aren't you for every possible outcome so when you throw the first dice you're going to have six possible outcomes so if you're going to draw this as a tree diagram you'd have one two three four five six different branches like that and then each of your branches is then gonna have six more branches one two three four five six one two three four five six uh one two three four by six so it's gonna it's gonna be like a forest not a tree isn't it there's so many branches at the end it's just just confusing so uh the reason why i wouldn't use it is that it is too many branches so draw a sample space uh to display all the possible outcomes that's a much more sensible idea isn't it now a sample space you're going to have a table and you're going to have a column for each outcome from one event and a row for each outcome from the other so i can get a one a two a three or four a five or a six on my first dice i can get a one a two a three a four a five or a six on a second now what you put in the middle depends on what the question is now if it was the sum of the two dice then you would put the sum in the middle so one plus one is two uh if you were subtracting one subtract one would be zero and so it really depends now in this case i can see in the next question that we're looking at adding them together so i'm going to put the sums in the middle here so 1 plus 1 is 2 1 plus 2 is 3 4 5 6 7. 2 plus 1 is 3 2 plus 2 is 4. five six seven eight three plus one is four five six seven eight nine five six seven eight nine ten six seven eight 9 10 11 and 7 8 9 10 11 12. okay so that's my sample space for adding the two scores together and then the last part here says show that the probability of the score of the two dice add in to 11 is 1 over 18. okay well there are 36 outcomes all together now probability is equal to successful outcomes over total outcomes isn't it that's the definition of probability in this case there are 36 different outcomes each one of those numbers in the sample space 6 times 6 is 36 so that's the number on the bottom uh and how many 11's can we see well i can see one there and one there so that's two all together so there are two successful outcomes out of 36. uh dividing bot top and bottom by two that's going to give me my one over 18 as uh as requested there's a venn diagram always my favorites uh a school has 86 teachers 42 a male and 44 are female one third of the male teachers have blue eyes and one quarter the female teachers have blue eyes uh so we're going to put that on a venn diagram for a we're told that a universal set is teachers in the school m is the male teachers and b is the teachers with blue eyes we need to complete the diagram okay so where should we start i think if we look at this bit here 42 are male and a third of the male teachers have blue eyes that means we can work out what the intersection is going to be then don't we so we've got 42 male teachers all together and a third of them have blue eyes so 42 times one third or 42 divided by three three goes into 42 is that uh 14 times so that means they're going to be 14 male teachers with blue eyes so that means the intersection between m and b is going to be 14 and now i can start working out the rest so if there were 42 male teachers altogether of which 14 had blue eyes then 42 subtract 14 will be the number of teachers with who are male and don't have blue eyes which is 28. uh one quarter of the female teachers have blue eyes how's that going to help us oh okay i suppose that tells us the bits that are in this the number who are on b and not in m and we know there are 44 female teachers and a quarter of that would be 11. so that is that one there then isn't it so 11 teachers have blue eyes and are not male and now we can work out what's left uh we've got 86 teachers all together so then 86 minus my three other parts 28 and 14 and 11 so that's 42 plus 11 53 86 to subtract 53 which is a 33 female teachers with not blue eyes that's what's gonna go there isn't it part b says one teacher is uh who has blue eyes is chosen at random work out the probability that the teacher is male so we want the probability that the teacher is male given that they have blue eyes in other words if we know that the teacher is in this group what is the probability then that they are here so that's going to be there are 14 teachers with blue eyes out of a total population of 25 teachers with blue eyes so that is going to be 14 over 25. two ordinary fair dice are rolled complete the tree diagram okay so we've got our rose for the first dice on the left and the probability of getting a four is one over six and we know that the probability then of not getting a four is going to be five over six because all the branches of each part of a tree diagram should add up to one because of completion so again up here that should be five over six one over six and five over six okay so then circle the probability that both dice land on four now both dice land on four is this root here so that's the probability of four and four probability of 4 and 4 is going to be those two values multiplied together then it's going to be 1 6 times 1 6 which is 1 over 36 is that one there part c asks us to work out the probability that at least one of the dice does not land on four now the probability at least one not four that's basically one minus the probability of them both being fours which is one minus one over 36 which is 35 over 36. now you could do it the long way and work out that root that root and that root and add them up but because all of the outcomes add up to one if i know the first one i can just take it away from one and it will give me the remainder phew well that was a long session so well done for getting all the way through it to the end please do like and subscribe if you found it helpful and share the link with your friends this really helps the channel out past paper practice is one of the best forms of revision in the run-up to the exams so be sure to check out my past paper walk throughs by checking out the playlist linked on your screen see you on the next video [Music] you