okay welcome to this video where we're going to be having a look through a load of questions which are very very common topics that appear on paper one now when you look through the checklist it's not that you can go through the checklist and just tick things off and say they don't come up on paper one any of these topics can appear on a paper one exam for example something that you might think is a calculator topic could be something like circles but these can be written in terms of Pi so really when you go through this checklist there's nothing which you can cross off and say that's definitely coming up on a calculator paper which does make the paper one exam the nastiest one in my opinion because you do have to revise everything so we're going to go through a load of questions where you can just sit and you can do your revision with me the day before the exam making sure that we practice a lot of common topics that appear on paper one now we've got a lot to get through and not enough time with the exam coming up tomorrow so we are going to get started straight away so good luck stick with me and if you don't make it to the end which I'm hoping you do best of luck in your exam thank you [Music] okay so we're gonna round these numbers to one significant figure so 6.32 is going to round to six six point five one rounds to seven and then 0.503 to one significant figure on the bottom is 0.5 and whenever we've got these we've got to work out what's on the top and what's on the bottom so on the top there 6 times 7 is 42. and it's 42 divided by 0.5 obviously if that was a nice little division there we could do that straight away but dividing by a half does something in particular so dividing by half actually doubles your answer so if you think half fits into one twice sort of fit into 40 to 84 times so there we go dividing by 0.5 double xylem Bros if we had something like three on the bottom or something like that that'd be a little easier but dividing by 0.5 doubles our answer so that's estimating just remember to round to one significant figure and then to complete your Sims that way whenever you see that word estimate so write 180 as a product's prime factors and give your answer in index form so we can do our nice little prime factor tree and again thinking of any two numbers at times to make 180 so I'm going to go for 18 and 10. 10 becomes 2 and 5 locking off those numbers when we see that they're Prime 18 is 2 and 9 and 9 is 3 and 3 and again locking those all off when they're Prime remember and you can write down some of the prime numbers if it helps now writing these in size order we get two times two times three times three times five and then we can simplify these so two two times two becomes two squared three times three becomes times three squared and then times five at the end and that's an index form there using in this all right so a washing machine costs 640 pounds plus twenty percent v80 James Payne's a 68 pound deposit and the rest in 10 equal monthly installments work out the cost of each monthly payment so it's a percentage of an amount questions so we need to work out twenty percent to six hundred and forty ten percent is the most important percentage we ever have to work out so we can work out everything from there so we'll work out ten percent ten percent is 64 pounds twenty percent will be double that so double that is 128 pounds so that's the extra amount he's got to pay for v80 so if we add these together 640 add 128 we get 768 pounds now it says he's going to pay a 68 pound deposit so let's take that 68 pounds off because that's what he pays right at the start and that leaves us with 700 pounds it says in the question he's going to pay that in 10 equal monthly installments so if we finish this off if we divide this by 10 so 700 divided by 10 tells us that he's going to pay 70 pound per month there you go so the main part here was the working out the percentage so always work out 10 and remember from there you can figure out any percentage we could have half 10 we could have got five percent as 32. or we could have actually divided it by 10 we could have got one percent dividing 64 by 10 1 would have been six pound forty and we could have built that up to any any percentage that we wanted from there to six pound forty let's have a look at something else okay moving on to fractions multiplying and dividing fractions and multiplying fractions is the easiest one isn't it so we can times the top numbers three times two is six sometimes the bottom numbers 4 times 3 is 12 and we always want to simplify our answer in a question will normally ask us to write in its simplest form so here we can divide the top and bottom by six and we get one over two as our final answer there so always looking to simplify on to the next one we keep the first fraction of the same we flip the second one over and we multiply them together so times in the top numbers just like before 2 times 5 is 10 1 times 9 is 9 and again we'd normally be asked to write our answer in its simplest form or as a mixed number and here it is an improper fraction 9 fits into ten once with a remainder of 1 9 left over so one and one ninth would be my final answer there and that's dividing fractions when it comes to adding and subtracting the process is the same I'm just going to go for this one addition so when we're doing this we need to obviously look at the fact that there is a mixed number here we're going to make that into an improper fraction first and the same goes for multiplying and dividing so we're going to do one times the four the big number times the bottom and then at the top so one times four add the three is seven quarters and we're going to add to that this two-thirds and again if that was a mixed number we'd also convert that into an improper fraction now we need a common denominator when writing and subtracting so that's going to be 12 on the bottom so we can times this left fraction top and bottom by three we can times this right fraction on the top and bottom by four and that's going to give us two fractions out of 12. so 7 on the left times three is twenty one over twelve and on the right 2 times 4 is 8 over 12. and now we can add these together and again if we were subtracting them we'd just subtract those numerators now but we'll add them together and we get 29 over 12. and again it says to give your answer as a mixed number so we are going to convert this now into a mixed number 12 fits into 29 twice up to 24 and there'll be a remainder of 5 5 12. and again there's my final answer just be very very careful at this point just to have a look does this fraction simplify here the little one it doesn't but let's imagine it was 4 12 there instead of five twelfths that would actually simplify wouldn't it because if it was four twelfths we could divide the top and bottom by four and that would end up being one third so just have a look in that situation does it simplify this one obviously doesn't it's two and five twelfths sound like another one so in a backpack account is two-fifths of blue and it says 18 a blue work out the total number of counters in the bag so it's not asking us to work out two-fifths of a number it's saying that two-fifths are eighteen remember it could have said work out two-fifths of let's say twenty and you divide by the bottom to get one fifth you'd say that one-fifth is four and then Times by the top to get two fifths and we'd get two-fifths would be eight that's a different question so it's nothing to do with that but it says two-fifths of blue and eighteen are blue so two-fifths equals eighteen so if I want to work out one fifth in this scenario I need to divide by the top this time because that'll give me one fifth half of that so one-fifth would be nine and then to work out five fifths the original amount we could times that by five so times it by five and we would get five fifths equaling 45 counters there you go so that's our scenario there so slightly different the other way around obviously to working at a fraction of an amount it says that that fraction is 18. so dividing it by 2 to get one-fifth and then times eight by five to get the original amount okay so writing some numbers in standard form we're going to make this a number between 1 and 10 times 10 to the power of however many jumps we have to do in whatever particular direction so this number here 5600 I'm going to hop the decimal in between the five and the six so one two three so that becomes 5.6 times 10 to the power of it's a big number there it's not of null Point number so 10 to the power of three final answer the next one is a small number so we can have a negative power because we hope the decimal the other way this time so one two three would make it 3.4 and again this time it would be times 10 to the negative 3 because it's a naught Point number that indicates that it's a small number when it comes to the other way around writing them as ordinary numbers I like to rewrite this little bit the start to start with so two and three and I like to just imagine where that decimal is between the two and the three so times ten to the power five makes it a big number so we need to hop it five places one two three four five to the right and then filling in all those zeros underneath and if it is messy just remember to rewrite that but there's my answer two hundred and thirty thousand the next one eight point zero four times ten to the negative three so this is going to be a naught Point number and again I'm just going to write these digits out at the start these bits here and imagine where the decimal is at the top 8.04 so negative three means it's gonna be a small number so we're going to jump it left three one one two three so it's going to go there fill in all the zeros and tidy up at the start so 0.0804 final answer okay so looking at this one four times ten to the five times three times ten to the minus two and there's an example of one more we probably won't want to write this out as eldinary numbers and work them out because it might get a little bit over complicated here let's look at applying that same little trick so let's do the four times the three which gives us 12. and that's going to be times 10 to the power of something that's gonna be a little bit careful here because although we can add these Powers let's just have a look because one of them's a negative so the powers if I write this to the side we're going to do five for the first Power and we're going to add the next power which is negative two and 5 plus negative 2 is 5 take away two so my power there is going to be three so that's fine 12 times 10 to the power of three and again we just need to put this in standard form because this 12 is not between 1 and 10. so instead we'll make that one jump smaller we'll make it 1.2 and to balance that out we'll make the power one jump bigger so three goes up to four and there's our final answer 1.2 times 10 to the four same process again we've got three divided by 6 though so let's do this to the side three divided by 6 let's write that as a fraction three divided by six if we simplify that it goes down to one half so to keep this as a number rather than a fraction I'm going to write that as 0.5 so over here we have 0.5 of 3 divided by 6 and then we've got that times 10 to the power of and again subtracting the powers on these so we've got three take away a negative 2 on the second one and three take away negative 2 turns into a plus so we have 3 plus 2 which gives us a power of 5 for this one so we have 0.5 times 10 to the power of five and again we can balance this out because it needs to be in standard form but this time to make 0.5 between 1 and 10 we make it one place value bigger to make it five and if we make the number bigger the opposite this time for the power we make the power smaller so 5 drops down to a four so five times ten to the four so a similar process but here we have to make the number bigger so we made the power smaller okay so this question here we have got it in reverse so it says right 35 as a decimal so again I'm going to write the percentage I'm going to write it without the percent sign and just write the numbers 35 so to turn it back into a decimal we just divide it by a hundred so we help the decimal back into places which would make it 0.35 obviously when we're writing a decimal though if we have a decimal point at the start there we do want to just tidy that up and put a zero at the start so it's 0.35 and there we go that's writing it as a decimal so very similar to our previous question now we want to write it as a fraction it says write 35 as a fraction now of course this has already given it to us as 35 so we can straight away put that as 35 per 100 or as a fraction 35 out of 100 and again that is all we need to do in terms of writing it as a fraction again this question though does say give your answer in its simplest form so we need to spot what they both divide by now hopefully you can see that because one ends in a five and one that ends in a zero that they definitely both divide by five so we'll divide them by five to start with and then we can always see if it divides any smaller so 35 divided by 5 is 7. in fact let's just swap back to our other color there 35 divided by 5 is 7 and 100 divided by five don't know why I put divided by 2 down there we definitely wanted to divide by five but a hundred divided by 5 is equal to 20. so we've got the answer 7 over 20 again the top knew the numerator there 7 only divides by one or seven and twenty doesn't divide by seven not perfectly anyway and we are looking for a number that fits in perfectly so our answers there was 0.35 for our decimal and 7 over 20 as our simplified fraction okay so it says here write the following numbers in ascending order now ascending order means smallest to biggest so we need to find out which of these is the smallest which is the biggest and put them in order of course at the moment we have a decimal we have a fraction we have a percentage another fraction and another decimal so it's not very nice to order them as they look so the best thing to do on a question like this is to turn them all into the same format now while you might have noticed on the previous questions but with all of them we used percentages so personal preference for me I like to write them all as a percentage so first of all why don't we just try converting them all into a percentage of course you could try something else here but I do think the percentages are the much easier way in order to do this so for the first one 0.30 of and of course don't forget you can put a zero at the end there because then when we multiply that by a hundred it gives us 30 percent so our first one is 30 percent for the second one we want to turn that into a fraction over a hundred of course if you just know this already then that is fine but 1 over 4 would become 25 over 100 so that one there is 25 percent of course if you just know that that's fine as well you don't have to show the working out for that as long as you're confident with it now the third one is already written as a percentage so that's nice and easy 28 percent the next one's a fraction so again let's think about how we would turn that into a percentage 6 over 25 we would want to multiply the top and bottom by four to make it over 106 times 4 is 24. so that becomes 24 percent and then for our final question here or our final part of this question we've got a decimal so if we times that by a hundred and we help the decimal it's going to jump over the nine so that is going to become 29.5 and there's our first part where we've had a decimal inside our percentage but that's fine as well it's just 29 and a half percent so let's get rid of our working out and then let's think about which one of these is the smallest and then we'll put them in order now when it when you're doing this question it says write the following numbers in order it doesn't say to write these percentages in order so we do have to write the original parts of the question in the appropriate order but before we do that I'm going to order them from the smallest so hopefully you can spot 24 is our first smallest number we then have 25 which is our second then we have 28 then we have 30 percent and then actually I've missed help the 29.5 our fourth smallest was definitely the 29.5 percent and then we have our thirty percent so obviously you've just seen always be very careful and check that you've read all the numbers carefully so now we've got that done let's just write the original parts in order so the first one was the 24 percent and that was 6 over 25. the next one was the 25 and that was one over four the next one was 28 and that was already a percentage so that's fine to leave it the next one was the 29.5 which is 0.295 and the final and biggest part was the 0.3 and there we go we've written them all in ascending order using our conversions okay so we're looking at some negative numbers now it says here at 7am the temperature was negative four degrees it says by 3 pm the temperature had gone up by 10 degrees right down the temperature at 3 pm so in order to do this we're going to first Bridge zero so I mean by that is what do we have to add to negative four to get to zero that's going to be four and then we've got an additional six to add on so the calculation we're actually doing here is negative four add ten you might be quite happy just doing that we've got to add four to get to zero then an additional six and we do get to six degrees Celsius so there we go that'll be our answer for that six degrees for the second part of this question it says at 9 00 PM the temperature was negative 2 degrees and by midnight the temperature had gone down by seven so the calculation we're doing here is negative 2 as our starting position we're going to take away 7 as it's going down by seven so negative two take away seven would get us down to negative nine degrees and that would be our final answer so it's obviously just looking at adding and subtracting when we have negative numbers involved not forgetting and this is something that people tend to get wrong just because there are two negatives in this question that does not mean that it becomes a positive that rule only applies when you have two negatives next to each other so if I was to create a completely different question I could have something like three take away negative five now if this is the scenario when they are right next to each other whereby they turn into a plus so that would become three plus five which equals eight so this is the one scenario where it happens and of course when you are multiplying with negatives so something like negative three multiplied by negative five would become positive 15. so these are the scenarios where two negatives do end up or result in a positive answer but it certainly doesn't apply when we have negative number as our starting point and we are simply taking away so obviously don't forget that when it comes to negative numbers obviously side track from the question slightly there but hopefully that's going to help us a little reminder when you're dealing with negatives okay so moving on to algebra we're looking at index laws to start with so we know the index laws we've looked at this when we have powers that are getting multiplied together when they have the same Base number these powers can be added together so two and three would make five we also had the process when they're being divided so x to the power of 7 divided by x squared we take away the powers and we get x to the power of five we also have powers in Brackets so if we have x squared to the power of 5 we multiply the powers so we get x to the power of 10 in that particular example now having a look at this one we've got 30 x squared y to the power 3 on the top and six x y squared on the bottom so we're doing a divide now we've got these numbers on the top and the bottom 30 on the top six on the bottom so numbers always just divide numbers so 30 divided by 6 gives us 5. now individually looking at the letters we've got an x squared and on the on the bottom we've got an X now it's not written in there but that is a little power of one there that's not written so if we subtract the powers 2 take away one gives us x to the power of one I'm not going to write that in I'm just going to leave it as X because that means x to the power of one moving on to the Y's we have y to the power of three y squared on the bottom and three take away two gives us just y on the bottom there so our final answer would be 5xy okay looking at this one then we've got powers in Brackets so we're going to be multiplying Powers here but we've got this number two in the start as well so numbers always get treated like numbers and that whole thing is being cubed so we need to work out what two cubed is 2 cubed is two times two times two which is eight so we're gonna have eight at the start of this one multiplying the powers for the X then x to the power of two that's going to get Times by three so that's going to become x to the power of six and Y there is a power of one which is not written so one times three gives us y to the power of three and there's our final answer there so a equals five b plus two c b equals three and Y equals negative two find the value of a so we're just having to substitute these numbers in so we're going to find out what a is so a is going to equal five lots of B and when a sub a number in I always put it into a bracket so we have five lots of B which is five lots of three and then two lots of C so plus two lots of negative two let me put these in a bracket like that all we have to do is now multiply these so five times three is fifteen two lots of negative two is negative four and we're adding these together so 15 add negative four is eleven remembering that these just become a takeaway So it's 15 take away four so a would equal 11. there you go just something to remember there whenever you're subbing numbers in always put them into our bracket there just so that you can sub them in and then deal with the calculations later okay so on them and blind show this inequality so I'm enjoying these sorts of things on a number line we look at what numbers we've got we're going to put our little circles above those so negative two and four and we're going to join them up and then we just need to have a look at these symbols so the one here which is less than or equal to or pointing towards that less than or equal to we it has more ink on it so it just remember to color that one in make sure the circle has more ink in there as well nice little way of remembering it there we go so that is between minus two and four remembering that is equal to minus two so the numbers that this could be is we could have negative two negative one zero one two and three but not including the four there because it says less than four so you can have an inequality like this as well where it just says X is greater than or equal to one so again putting our Circle above the one the greater than or equal to has the extra ink on it so we'll color that one in it's equal to that as well and greater than one so we'll just point an arrow to the right saying it's got to be greater than one and that's that for that if that type of inequality okay so a nice quick one here solving an inequality now this symbol shouldn't put you off here it's an inequality yes but we're going to treat it exactly the same as if it was an equal sign so before we saw a question like this where it was 2x plus 3 equals 20. we're going to do it in exactly the same way but we're just going to leave that symbol in there so we're going to take away 3 from both sides which leaves us with 2x is less than 17 and then dividing by two we get X is less than 17 over 2 and we could probably work that out 17 over 2 half of 17 is 8.5 so X is less than 8.5 okay so on to factorizing a quadratic this is the opposite of expanding a double bracket so we know what our answer is going to look like already it's going to be in a double bracket both of the next at the start and when we're doing this we've just got to find the pair of numbers that are going to go into our bracket so looking at the 12 we know it's going to be a pair of numbers that multiply to make 12. so I always write these down we can have 1 and 12. two and six or three and four we just need to have a look at these numbers there because we've got a 7x in the middle so they have to add up to make 7x so 1 and 12 there's no way I can make seven two and six I could either make eight by adding them together or I could make four or minus four if I do a minus with one of them so I'm not going to make seven but I can make it with three and four I can have plus three and plus four and that would be seven so plus three and plus four and that's fact that's not a four plus three and plus four and that would be factorized have a look at the one below we've got some negative symbols going on there so we've got a slightly different process but our answer is going to look the same it's still going to be in a double bracket it's still going to be X at the start and we're still going to have two numbers in here with some symbols so 28 the numbers of times to make 28 are 1 and 28 2 and 14. what else could we have four and seven and let's have a look if these can make 28 so what have we got we've got minus three in the middle so we want to make minus three well we're not going to do that with 1 and 28 that's not going to work 2 and 14 we could make 16 or we could make 12 that's not going to work but 4 and 7 we could do plus four take away seven that'd make minus three so plus four and negative seven and that's our double bracket there finished so when it comes to solving a quadratic the way to solve it is we have to factorize it and we can only factorize it and we'll discuss this a little bit more when it is equal to zero so sometimes there won't be equal to zero but particularly in the foundation level style questions they're always going to be equal to zero here and if they're not we'll have a look at how to obviously just rearrange them slightly in a SL in a later question here when it comes to this here all we're going to do is we're going to have a look at that quadratic there so we've got x squared plus 7x plus 12. now when we're factorizing one of these they go into a double bracket and all we do need to do is have a look at this number at the end which is 12. and we have a look at what numbers multiplied to make 12 so we could have 1 and 12 and again I'm not going to talk about this too much because I'm going to link the description for factorizing quadratics but we can have six and two and we can have three and four so we just need to figure out what combination of numbers are going to go into our double bracket so we'll put our double bracket here it's all equal to zero and we've got an X at the start of both brackets so obviously we're looking at that number that number in the middle there the coefficient of x which is positive 7. so the two numbers that multiplied to make 12 and also add to make that positive seven are our three and our four okay so we're going to put three and four in our bracket and that would be plus three and plus four there we go now in terms of obviously our Solutions here now you don't really need to know this for this uh question in particular but if this was a quadratic equation a line equation sorry it would relate to a quadratic curve okay now I'm going to draw what this would look like it looks something like this okay there we go not to scale there but it looks something like that now when we are looking at the solutions here we are looking for essentially where this curve crosses through the x-axis and that's what equal to zero is representing there and it's in a columns very simple level equal to zero kind of means ground level it means where Y is equal to zero and if you think about the Y axes the y coordinate just here we've got y equals zero just down there so we are having a look at on this graph where does the quadratic curve cross through the x-axis when Y is equal to zero now there's a little trick with these brackets but in terms of obviously explaining how we go about getting our Solutions here we just say okay well when does this bracket here X plus 3 equals zero and if we write that out when does X plus 3 equals zero and if we take away three from both sides to solve that we get X has to be equal to negative three and if we do that for the other one as well so if we do that for this one just here we see okay well when does X plus four equal zero and again if we solve that and take away 4 from both sides we get X has got to be equal to negative four so there we go if I label that on the graph here we've got minus three and we've got minus four and that's how I knew what the graph would look like but there we go there's our two solutions X is negative three and X is negative four but if you have a look at the bits in the brackets there obviously we've got a plus three in the bracket and our solution ended up being negative three and on the flip side on the other one we had plus four in the bracket and our solution ended up being negative four so obviously it's important to know how to do this little process here setting them both equal to zero and solving it for some of the harder ones but in terms of just getting through these questions nice and quick once you've factorized it all you have to do is flip the symbol in the bracket there we're gonna have a look at a couple of others where maybe there's a negative in the bracket or two negatives or something like that but that's all you've got to do is just flip the symbol so there we go there's our two solutions X is equal to negative three and X is equal to negative four so you will notice in these questions we'll tend to get two solutions unless obviously the two solutions are maybe the same which can happen as well but we're always going to get two solutions here and in this case it was negative three and negative four slightly different one so draw the graph of y equals x squared plus two x minus three between minus three and two now again I'm going to do a little table for this so we have X and Y and it's between minus three and two so minus three minus two minus one zero one two this is a slightly more complicated one for us to sub in because to find the y coordinate you have to do x squared add 2x take away three so I want to write these down nice and carefully so I'm going to start with two again for the positive one so we have 2 squared Plus two lots of two take away three I'm just going to work that out nice and slowly 2 squared is 4. plus two lots of two which is four take away three so four plus four is eight take away three is five So my answer there underneath the two is going to be five now I need to follow this process for all of them but the one thing I do need to be careful of is when we get to these negative numbers so up here where I put 2 squared in just here we should really make sure we keep that in a bracket because as we get down to those negatives it's going to have a big effect on that and we'll see what happens when we get to them and I'm going to do the other ones a little bit quicker let's have a look so 1 squared is one add two is three take away three is zero zero squared is zero and zero is still zero take away three is negative three and here's where we need to be careful so the first one here is negative one so I need to put if I'm going to do this on a calculator so I'll write my working out down negative 1 squared add two lots of negative one and then take away three now the reason you have to put this in a calculator is because if you put negative 1 squared without the brackets it will give you the answer negative one it will Square the one and put the negative with it rather than doing negative one times negative one squaring a negative number makes it positive so negative 1 squared is one add 2 times negative one is negative two and take away three so one take away two is negative one take away three is negative four so we get negative four for that one there we go let's have a look at the next one negative two squared is four add negative four is going to be zero and then take away three because it's negative three again and then on to the negative three negative three squared is nine add two lots of that's a negative six so nine take away six is is three and then take away three again gives us zero so you can probably see a little bit of a pattern here look if you look at that negative four in the middle either side of that we've got negative three either side of that we have zero and then we have the additional five there but let's just plot these and see what it looks like so negative three to zero is there negative two to negative three is there negative one to negative four is here zero to negative three one to zero and two to five just I'll lay up there there we go so this forms a quadratic curve a parabola and we just need to draw that with a nice smooth curve as best you can going through all the points making a nice smooth curve and there's our graph drawn just as a note we could also have a cubic graph so we could have an X cubed up here and we just treat that in exactly the same way subbing the numbers in very carefully using brackets and it tends to be that a cubic graph has this sort of shape so just be careful there are some others as well we could have a reciprocal graph or we have something like y equals three over X and that has a different look again it has this sort of Curves shape within the positive part of the graph so there's lots of different graphs that we can have a look at but just treat it all in the same way and plot the points very carefully joining them up with nice smooth curves so find an expression in terms of n for this sequence so we're going to do the nth term and it says hence find the 50th term so what I always like to do is draw these little Loops just to see what times table this is related to and from three to seven it goes up by four and then it keeps going up by four so this is related to the four times table now the expression for the four times table is four n if we do it in terms of n now this is obviously not the four times table though is it because the four times table if I write it above this 4 8 12. 16 20 and all of these numbers are one less look from four to three is minus one from eight to seven is minus one so it's the four times table but it's had one taken away and that is our nth term there in terms of n there is also a little trick that you can apply on this you can go backwards for so if we go back four we get the number minus one and that always tells us how much smaller or how much bigger it is than the times table we're looking at so it says hence fine the 50th term well here's our little code 4N minus one to find the 50th term in the four times table you do four times fifty and it's no different for this one but it's the four times table take away one so we're going to sub it in and we get four times fifty or four lots of fifty but then we have to take away one four lots of fifty is 200 and 200 take away one gives us 199 and that is our 50th term just thinking if you didn't have this nth term you'd have to just keep counting up from 19 so you'd go 23 27 and so on and you'd have to keep counting up to the 50th term and it'll take you quite a long time so this is a nice little shortcut for that okay so when it comes to length conversions there's a few things that we need to know so one thing that we need to know is that one centimeter is equal to 10 millimeters okay and a lot of these things you can see on your ruler but obviously not when it comes to centimeters to meters when it comes to centimeters to meters we need to know that 100 centimeters is equal to one meter or we could say that one meter is equal to 100 centimeters it's obviously we need to know these you can also need to know kilometers as well and that for that we need to know that one kilometer is equal to a thousand meters so you could obviously have questions on any of these conversions but if we look at these two that we've got here for starters we've got change 350 centimeters into meters well there are a hundred centimeters in every one meter so if we divide this by a hundred that would just mean hopping the decimal in two places and we would get 3.5 meters so that's one conversion that we could have this one's a little bit harder we've got 8.3 meters into millimeters and you'll also notice that in the first question the actual units were given to us in words and here they're just given to us as M and M and double M there for millimeters so we could have either of those but for this one it's a little bit trickier because we're going from meters which is in this conversion here down to millimeters which is in this conversion here now you can always learn the conversion between the two but I think it's easier here to First convert it into centimeters and then convert it into millimeters so to convert it into centimeters we would want to multiply the 8.3 by a hundred as there are a hundred centimeters in every meter so 8.3 multiplied by a hundred just means hopping the decimal over the three and then over again which would make it 830. so it's going to be 830 centimeters now every centimeter has 10 millimeters in it so we're going to multiply that by 10 to come invert it into millimeters so we'd write 830 multiplied by 10 and obviously just adds a zero on the end there so that's just going to be eight thousand three hundred and that is millimeters and there we go that would be our final answer I'd obviously just take the same approach if we had kilometers involved so we would divide or multiply by a thousand to turn the kilometers into meters or the meters into kilometers and there we go there are some of our length conversions okay so working out 45 of 64. it's a little trickier here because it doesn't end in a zero but just straight away let's think about our process so we're going to work out 10 straight away and we'll work that out in just a second we're also going to need five percent so we'll have to work that out as well which again will work out in just a second and then we're going to need to get to 45 percent so in terms of what I'm going to actually need here I'm gonna need four lots of the ten percent and one of the five percents four lots of ten would be forty percent one of the fives is five percent so combine there that's going to give us our 45 percent so let's work out our ten percent now we've got 64. so to get 10 divide it by 10 so hop the decimal in and we get 6.4 so our 10 here is going to be 6.4 and again we're going to want to halve that for five percent and again if you are struggling to halve these numbers at any point don't forget you can always just divide it by two using bus stop just to the side if you need to 2 goes into six three times keep your decimal point in place and two goes into four twice so 3.2 and there we go five percent is going to equal 3.2 so we want four of those 6.4s and we want one of the 3.2 so now again you could multiply 6.4 by 4 or you could just add them all together using column Edition so we want four of those again if we go beyond forty percent you've probably got quite a few there and you probably want to multiply them instead but not too difficult for us to just add those together so what do we get 4 8 12 16 plus the two is eighteen so carry the one and then 6 12 18 24 27 plus the one is 28 so we get 28.8 and there we go there is our final answer and that is 45 of 64. again just being very careful when you do that obviously I've added them together quite quickly so take your time with it but just as I said before as well you could actually have just done 6.4 and multiplied it by four instead if you prefer to use a multiplication method but if you are going to do that don't forget to take the decimal out before you multiply so we would have instead done 64 times 4 and then hop the decimal in at the end and then added the extra 3.2 so arguably that might be a little bit longer than just adding it together I think if you just add them together it's pretty quick and everybody's normally pretty happy to do column Edition so if we want to get this without a calculator all we actually have to do is follow our process for working out a percentage and that is to obviously start by working out 10 so work out 10 percent which is 12 and for 20 we can double that so 20 would be 24. and then we're obviously increasing by a percentage so we would increase by this 24 so we just add that on to the 120. so 120 add 24 and that comes out as 144. so we know the answer is 144 so for this type of question why would we bother using a multiplier well to be fair for this type of question we probably wouldn't it probably could be quicker for us just to work it out without a calculator but not all questions are going to be this simple and not all percentages are going to be as simple as 20 percent so we need to have a method for if it's going to be a little bit trickier surely there must be a faster way for us to do this with a calculator and there is so we're going to look at that 20 and we're going to figure out what we would actually multiply by to increase by 20 and therefore you know if it was 17 or 5 what would we actually be able to multiply by and it all comes down to thinking about that as a percentage more than the original so the original amount that we have we could think about as being a hundred percent and I'm going to draw it as a little bar so we've got a hundred percent and we are going to add on an extra 20 so that's almost like we're extending the bar we're making it 20 bigger now if we think about that what is that as an overall percentage well we've got a hundred percent and we've got 20 there and overall that is a hundred and twenty percent and if we want to increase by 20 to get this 120 we need to think about what 120 is written as a decimal while a hundred percent isn't actually written as a decimal that would just be the number one so the hundred percent would be written as the number one but what would go after the one so it'd be one point something well thankfully it just reads as it does just above it's just it's 1.20 so that point 20 there is the extra 20 on top of the one which is our 100 so it's the extra 20 on top of the hundred percent and that right there is the decimal that we would need to multiply by we just multiply by 1.20 obviously you don't need to put the zero at the end so you would just multiply by 1.2 and 1.2 is the decimal multiplier to increase by twenty percent and if you get your calculator now because we're obviously using a calculator method and you type in 120 and you multiply it by 1.2 then you will get the 144. so let's type that in 120 multiply it by 1.2 and in this instance we actually do get 144. so you can see it actually calculates that percentage increase for you so obviously when we are using a calculator method we need to know if we need a few things here we need to know obviously how to turn the percentage into the decimal and obviously thinking about the fact that it has to be one point and then that increase in that decimal there so another example that we could think about is let's say we were increasing by and let's go with 30. so what would it increase the 30 be well instead of being 1.2 or 1.20 instead it would be 1.30 and it just follows on this little pattern so you don't have to write down the full 120 and draw the bars all you have to know is that if you are increasing by a percentage it's one point and then whatever that percentage is afterwards so in the case of 20 1.20 in the case of 30 1.30 and that's what we'll multiply by okay so this question says Alice buys a pack of 10 drinks for four pounds she sells all 10 drinks for 50p each workout Alice's percentage profit well her original cost then is four pounds so we've already spent the four pounds on our 10 drinks she then she's then going to sell them all for 50p so we need to do 10 times 50p so 10 times 50p and again you can work without whatever you choose but that comes out as 500 Pence there we go and we will probably want to convert that into pounds because the other one's already in pounds although we could change the four pounds into 400p I'm just going to turn this into five pounds there we go so now we can actually work out how much profit that she's made because she's bought them for four pounds sold them for five pounds so she is going to make one pound profit and there we go and now we've got our one pound profit we can just apply that to this question just like we have on all the others so the profit is one and the original was four so we can do one pound out of four which straight away comes out as one quarter so that's a very nice one for us because that is straight away going to be 25 of course if you want to multiply that to make sure we get the right number on the bottom you multiply by 25 to make it a hundred let's not drawn that 25 very nicely let's write that carefully there we go times it by 25 and you'll get a hundred on the bottom and you get 25 on the top which is 25 percent and there we go there's our final answer again you could work out that percentage of the original ten percent of four pounds obviously you get into decimals here but ten percent of four pounds would be 40 Pence we're trying to get to one pound profit so we'd probably also want to get the five percent so five percent would be twenty p and then we can build that together we could add another 10 percent in which is another 40p and in total that now makes one pound and the total percentage there was 25 when you add those three together so there we go that is obviously approaching one of these problems we had a little bit of working out to start with let's have a look at one that's slightly trickier than this then before you have a go at your final questions okay so share 240 pounds in the ratio five to seven so if we're splitting this in a ratio let's have a look we've got it in total here five plus seven equals 12 parts okay so if we split this into 12 parts we could do 240. divided by 12 and 240 divided by 12. is 20. so in each part there's 20 pounds now there was five parts and seven parts so those five parts are going to be five lots of 20 which would be a hundred pounds and the other part there was seven parts and that times twenty would give us 140 pounds so splitting in the ratio five to seven gives us 100 to 140 and the last little check you can do here is just to add these two together just to make sure you do get that 240 pound that you started with so we're happy with that as a final answer so Craig and Ian share some money in the ratio eight to three Craig receives 32 pounds how much does Ian receive C to I created Ian was in the ratio eight to three and it said that Craig receives 32 pounds so that is this first number eight here that is 32. so we've got to think about is how do we get from 8 to 32 so 32 divided by 8 which we can do to the side 32 divided by 8 is 4. so we must have timed by four to get there so if we do the same to the other side Times by four three times four is twelve so there we go how much does Ian receive Ian receives this 12 so I'll just say 12 pounds final answer so Emily and James share some money in the ratio four to seven James receives 21 pounds more than Emily how much do they share between them so Emily to James is in the ratio four to seven now it says that James receives 21 pounds more and if we look at these two numbers here from four to seven is an extra three parts there goes an extra three so that extra 21 pounds is three parts so if three parts of this ratio equals 21 pounds then one part must be to divide that by three must be seven pounds and from there we can work out how much they both receive if it was four to seven let's get rid of this little bit down here and each part is now worth seven pounds or four times seven gives us twenty eight pounds for Emily and seven times seven 49 pounds for James so Emily Emily receives 28 and James receives 49 it says how much does station did they share between them so if we can add these together it gives us the total amount so 49 plus 28 gives us seven and seven so 77 pounds there we go to this they share 77 pounds between them three tins of beans and four jars of jamway 2080 grams and the total weight of five tins of beans is one thousand eight hundred we'll cut the weight of one tin of beans and one jar of jam well straight away we can work out the weight of wanting a beans from this because it says five tins of beans is one thousand eight hundred so if we do 1800 and divide it by five we get the answer 360. there we go we might have to do a bit of a bus stop for that but we can do that five's into one thousand eight hundred five into eighteen goes three remainder three fives into Thirty goes six and then zero so 360 grams and that is for the tin of beans so tin of beans there we go it then says three tins of beans and four jars of jamway 2080 back in the first line so if we work out the weight of three tins of beans now or three of those would be 360 and 360. if we add those all up we get nine ten one thousand and eighty so that's the weight of those uh tins of beans so if we subtract that away from this original amount so 2080 take away 1080 that leaves us with a thousand grams so those four jars of jam so four jars equals a thousand grams and we want to know the cost of one so we can divide that by four and it gives us one jar equaling 250 grams so there we go one jar is 250 and one Tina beans was 360. so a car has to travel 30 miles along a Motorway in 24 minutes will the car have to travel faster than 70 miles an hour to get there in 24 minutes I'm going to do this as a non-calculator one then we'll have a look at one that's maybe a little bit more calculator based but if it's to travel it's 70 miles an hour or more now 70 miles an hour means that we go 70 miles in 60 minutes so 70 miles equals 60 minutes now we can do is we can actually start to break this down so if I have a look what can I divide both sides by so I can divide both sides by 10 that would mean I would go 7 miles in six minutes okay dividing that by 10 as well so you can have a look to see if you can ever break these down this way so 60 divided by 10 gives us 6 minutes there we go now it says we've got to go 30 miles and we've got to go there in 24 minutes now just having a look at this speed here seven miles equals six minutes so traveling at this speed if I wanted to get to 24 minutes have a look 24 minutes we would times that by four so if I do the same to the other side and times that by four that gives us 28 miles so traveling at 70 miles an hour in 24 minutes we would go 28 miles and the question said a car has to travel 30 miles so it's not actually going to be able to make that 30 miles in 24 minutes it's only going to be able to go 28 miles it would have to go over 70 miles an hour so I would make obviously that statement no okay it's only going to go 28 miles or it's not going to be able to do it in that sort of speed okay so as you can see this particular graph is incomplete so let's have a read and see about how we're going to complete this it says here is part of Cole's Journey from his house and back here is the distance time graph for part of his journey and it says part of his journey because it's not been completed so for part A it says work out curl speed for the first 30 minutes of his journey so we need to be careful again because that's not necessarily the entire line there but in this case I do think that is going to be the full line as 30 minutes takes us up to this point so that is for his full Journey so we're looking at the speed from the start of the journey which is down here up to the top of the journey where the other Mark is so obviously in order to work out speed we're going to want to use our speed distance time formula which again we'll write down in a formula triangle and that is going to be speed is equal to distance divided by time so we need a distance and we need a time and again we're going to write this in kilometers per hour although it doesn't actually say to so we could actually write this in kilometers per minute but we're going to assume for all of these speeds it's going to want it in that format so in terms of the distance that is a distance of 20 kilometers which we can obviously read by looking just to the left here so we have 20 kilometers as our distance and we're going to divide that by and let's just write it down here so we don't forget that number it's 20. divided by the amount of time now again this amount of time as we've been Untold in the question was 30 minutes so if we're going to write that in terms of kilometers per hour then we're going to divide that by 0.5 again conversing those minutes by doing 30 out of 60 and spotting that that's a half or typing it into your calculator so 20 divided by 0.5 and again we can type this in on the calculator it comes out as 40. a half fits into one twice and therefore it doubles your answer so it fits into 20 40 times so just a quick way of dividing by a half if you don't have your calculator on you so that's our answer for Port part A and again that was in kilometers per hour so we could write that as kilometers per hour onto Part B here it says Carl spends 15 minutes at the shops and we can see that on part of the journey so if we get rid of this we'll highlight that point and as you can see he spends 15 minutes you can see that each of those squares is five minutes down the bottom so we're happy with that that is completed as being the 15 minutes it says he then travels back to his house at 60 kilometers per hour and to complete the travel graph so actually what we do we need to find here well we need to find the time it takes him so we know exactly what time it's going to finish for example down here we're at 45 minutes now if it takes him 10 minutes to get home he would finish here at 55 and we could connect our line using a pencil and a ruler and complete our distance time graph but we actually want to figure out the exact time there so at the moment we're at 45 minutes but how are we going to figure out the time well it again comes back to our speed distance time formula and you can see if you look just at the top here that if we cover up time then time is going to be working out by distance over speed so we want to get the distance and divided by the speed and that's quite good because we've been given the speed in the question and the distance we already know because we've got it over here it's 20 kilometers from his house so we need to do the distance which is 20 and this is for Part B and we're going to divide that by the top the speed which is 60. so 20 divided by 60. and if we do 20 divided by 60 what answer do we get well 20 divided by 60 is going to come out as a third so how are we going to interpret that in terms of a time because that is in hours so 20 divided by 60 comes out as one-third now this is the difficult part for these questions because you have to remember that that is in hours and we want it in minutes so what is a third of an hour well a third of an hour we can work out we could do that and we can do the working out to the side because we've got 60 that we want a third of and a third of 60 you can work out in two different ways either you could just straight away times that by 60 on your calculator so just type in times 60 once you've got your third value and it will come out as 20. or you could actually work it out using a fraction of an amount and you could do 60 divided by three again that comes out as 20. so the working out here we want to get a distance there that ends after 20 minutes so at the moment we're at 45 minutes and 20 minutes on top of that is going to take us to 65 minutes and again you could write that down I'm just going to write down that that it was equal to 20 minutes so we have the working out down because you always want to make sure you're sure you're working out for these questions so again you want to get your pencil and a ruler and you just want to connect that up using a nice straight line and that would complete this distance time graph now you don't have to get rid of all of your working but just make sure that that doesn't get confused with any of the other lines you don't want lots of lines drawn on there as it might get confused with the actual line that completes the graph so there we go that would be how we would complete our graph now obviously the difficult part on this question was that completion part so let's just run over that again so we spend 15 minutes at the shops and then travel back at 60 kilometers per hour so we took the distance which we got from our formula triangle and we divided it by speed 20 was our distance from home 60 was our speed given to us in the question and that came out as a third that was in hours so we multiplied it by 60 because there are 60 minutes in an hour and it gave us our 20 minutes because down the bottom it was in minutes that allowed us to then draw 20 minutes on top of our 45 minutes which allowed us to complete the graph so hopefully you're okay with that obviously this is the most difficult part of it so it was worth talking about that again okay so this is quite an interesting question because this is talking about density and obviously we're doing this non-calculator and normally density is a calculator based question so hopefully we can have some nice numbers now as soon as you see that word density in a question you're going to want to write down your formula whether you use a formula or whether you use a formula triangle so I'll write down the formula triangle because that works for everybody then so that's going to be density is equal to mass divided by volume now this question says it wants us to work out the mass so straight away if I cover up the mass that means I'm going to do density and multiply it by my volume so this question gives us a cube and it says the rubber Cube has side lengths of four centimeters so in regards to this image here we've got four centimeters across there four centimeters here and a four centimeter height so to work out the volume of a cube you would just multiply all those numbers together or in other words four times four times four obviously take that in stages 4 times 4 is 16 times that by 4 again and that is going to be 64. so 64 and the units are centimeters so it's 64 centimeters cubed now it gives us the density since the density is 1.5 so we know to do this we do the density 1.5 and multiply that by the volume and the volume is 64. so we just need to do 1.5 times 64 which isn't too bad when you are multiplying by 1.5 that just adds on an extra half obviously you can hop the decimal out you could do the working out to the side so if we do that we'll get 64 times 15 5 times 4 is 20. 5 times 6 is 30. add the 2 is 32. put our placeholder in 1 times 4 and 1 times 6 is nice and easy and if we add that together 0 6 and 9. so you get 960 and then you just have to hop the decimal back in one jump as we took it out at the start so when you jump that in you get 96.0 so that'll be 96 and in the units it said grams per centimeter cubed so that would be 96 grams as our final answer so there we go we'll get 96 grams so I you can see we can have density questions that are non-calculator but you're going to tend to get nicer decimals involved they're just going to be a little bit nicer for your calculations okay so reflect the shape in the line x equals one now this is your x axis and this is your Y axis so x equals one is right there and the only line that you can do from that point other than going across the x-axis and it doesn't see the X say the x axis is to go up and down so x equals one is this line here if you think about any coordinates on that line the x coordinates are always one so that's our reflection line there and we're going to reflect it in that line which is quite nice and easy to do pick a point and we just go one two to the line so one two away and just follow that process for all the points so picking this point that is four away so another four away gets me to there and the same one at the top that's four away and four away gets meter there and then just joining it all in obviously using a pencil for this one and that'll be my triangle haven't drawn that in particularly neatly there we go okay not forgetting as well though you could have a line it could say the line I don't know y equals three which would be across at y equals three which would be across here there's another couple of lines that we could have as well we could have this one y equals x the line Y equals X is the diagonal line here where all the X and Y coordinates are equal so we could have that one as well where we have a diagonal there's one more that we could have as well let's pick a different color for this we could have y equals negative X which is very similar to the one above but it's pointing in the opposite direction if I do that in blue as well that's going down this way so we could also have to reflect in a diagonal line but again you just follow the same process counting diagonally right okay so something slightly different again we've got the front elevation uh and plan of a solid are shown on the grid and on the grid draw the side elevation from the direction of the arrow okay so we're looking at it from the front there and again thinking about the front elevation we're looking at it from this angle here now again thinking about the Logics of the last question it doesn't matter how wide it is for the moment but the height of that from the front is going to be about is it going to be exactly the same height as it is over here okay the height isn't going to change so we can draw that in from base to the base to the top there as being the same height now looking at the plan we can see from the plan that it's only two squares wide so from the front we're only going to see two squares across there we go and that's the shape we're going to see a rectangle that looks very similar to the last one but there are a few little kinks here where the sort of depth of the shape changes uh primarily and let's just get rid of some of these we've got that one down the bottom okay so we want to make sure that matches up as well so we've got that one there it's going to be in there and we've got this one here it's going to go in there and there we go that's what it's going to look like okay it's almost like a little set of stairs it'd be quite difficult to actually draw this but imagine in what it kind of looks like I'll do my best as I think it will go something like that it's like a little set of staircases I probably shouldn't have started drawing this because it's not going to look very good but here we go try my best it's gonna look something like that there you go like a little set of steps okay and as you can see this little step here is slightly higher up than the two below it okay as you can see from the diagram now it's two squares up for that one rather than just one square up so that's the shape that we're actually drawing and obviously if I sort of color code this look we've got this face here that we're seeing as this one down the bottom we've got the next one up face here that we're seeing here and then we've got our final face okay in fact let's do a different color I've got the final face there that we're seeing as this part here okay so it does obviously help if you can kind of visualize it as a 3D shape although it's not the easiest to do um but there you go that's how you go about drawing this one okay so there we go that's that one drawing our elevation we're drawing we're drawing a side elevation there we go so that's how we draw our side elevation using our other elevations and it can help to be able to draw that 3D shape before you actually go about drawing the elevation okay but if not you can always think about the logic I just use making it so make sure it was the same height make sure I had the same width as the plan and obviously just matching in in any of those uh little changes of depth there with those lines and indicating those but there we go oh dear got rid of that and right in the last step there we go put that back in there we are so let's have a look at our next question right okay so in this question we've actually gonna we're actually gonna have to draw the 3D Shape okay so we need to have a think about what this is actually going to look like so it says the diagram shows the plan the front and the side elevation of a solid shape draw on a centimeter grid it says draw a sketch of the solid shape and give the dimensions of your solid on your shape dimensions of the solid on your shape okay so just be careful with that because obviously giving the dimensions means we've got to put the lengths on so we've got about how many centimeters wide or high or whatever it is now looking at this look from above it's a circle and then from the front and from the side it's a rectangle and there is only one shape and this is just about knowing your shapes really there's only one shape that's a circle from the top and a rectangle from the side and that is a cylinder okay obviously when we draw a sketch of that we don't see it as a circle as a sketch it kind of looks a bit like an oval sort of shape but then we have there's a bit coming down I'm not drawing that very well let's draw that line in again there we are and a slightly bit of a curved base there and you could even put the sort of a dotted base at the back that we can't see but there we are that's a cylinder and if you can imagine looking from above you would see that Circle but from the side there if we sort of dropped our heads down and we were standing in front of it we wouldn't see that circle on the top we just see the big rectangle there from the front the side the back the left whatever side you stand on you would see the uh that that rectangle so finishing this off obviously we've drawn a little sketch of it we've just got to add in the dimensions there's two Dimensions we want to add one is the width all the way across here and if we have a look that goes across four centimeters it's on a centimeter grid so that's going to be four centimeters there and then we also want to draw the height in let's have a look at this rectangle over here was one two three four five so yeah it goes five centimeters up so we'll also add that in as well and that is five centimeters right there we go uh that wow that's how you draw one of these obviously from a sketch it does it does actually mean that you've got to know your shapes quite well you can imagine there's probably a couple of other shapes that you could have thinking about like a cone obviously the front and the side of a cone wouldn't be a rectangle but we'd have a sort of there we go this sort of shape from the front and the side if it was a cone okay you still have a circle from the top you might actually have a point in the middle if it was a cone because you'd see that little tip of the cone as well but there's just another kind of shape that you can imagine looking above being a circle could also have a little cone as well but there we are that's how we go about drawing one of those obviously just making sure you read the question carefully obviously when it says give the dimensions you need to make sure that you actually add those on from the grid as well there we go let's have a look at our last question looking out the size of the angle between this regular octagon and regular hexagon so there's different ways that you can approach this but we need to know what the angles inside a hexagon all add up to and what the angles inside and octagon add up to different ways of doing this you can do the one 180 listerol so you can keep adding 180 so three sides is 180 four sides is 360. five sides is 540. I can keep doing it this way or you can apply a bit of a formula so I'm going to go for the formula even though I've started writing this down but that is an option there that you can use so for six sides and a hexagon we take away two so six take away two which is four and then four times 180 tells us the total amount of oh that should be a time sign let's change that four times 180 gives us the total amount of angles in a hexagon which is 720. so all the angles inside a hexagon equals 720 and there are six equal angles in there so if we divide that by six 720 divided by 6 gives us 120 degrees so all of these angles in the hexagon 120. obviously we're concerned about this little area here in this question because we're trying to find the angle between them but we've got the angle and the hexagon and that's really important when you see these words regular hexagon and regular octagon always just work out the interior angle and label it all over the diagram so moving on to the Octagon same process eight sides takeaway two is six and then times that by 180. they're going 6 times 180 which is okay if you have a calculator if not you just have to work that out but six times 180 is 1080. and again it's got eight equal angles so if we divide that by 8 1080 divided it by eight gives us 135 degrees and again we can just label that all over the diagram so 135 135 and lots of these but I'm concerned again with this little area here so angles around a point equal 360 and at the moment we've got two of these angles kind of like a jigsaw so if we add these two together 120 plus 135 at the moment we've got 255 degrees and we need it to add up to 360. so to finish that off if we do 360 and take away that 255 that we've just worked out there we get a missing angle of 105 degrees and that's our final answer 105 degrees which we could always label on the diagram as well 105. and there we go so whenever you see these questions where it says regular octagon regular hexagon okay just spot these like these bits of language here just work out their interior angles and label it all over the diagram working out the volume of a keyboard is quite nice and simple I do like to do it in a particular way though just so it's the same for all of these shapes where we have a constant cross section so this cross section I'm going to have a look at this front face just here so I'm first going to just work out the area of that which is 8 times 12. so 8 times 12 gives me 96 and that is the area of the cross section 96 centimeter squared and then to get the volume which is times it by however far back the shape it goes which is the same as that 14 right there so times up by 14 96 times 14 will give us our volume to 96 times 14 is 1344 and it's a volume so we put centimeters cubed and that's how you work out the volume there that works for any shape we've got a constant cross section if you imagine sometimes we have these triangular prisms and sometimes we could even have a trapezium face prism there okay anything with that constant cross section you always work out the area of this front face to start with the cross section and times it by the distance it goes back that same logic can be applied to a cylinder so our cross section this times a circle so all we've got to do is work out the area of the circle remembering area equals pi r squared so working out the area of that Circle it's pi times 4 squared which again is 16 pi we can write that as a decimal now what I'm going to do is I'm just going to leave it as 16 Pi for the moment so I don't know any rounding errors here but you can write it as a decimal at this point we're not going to leave it as 16 pi and we've got the area of the cross section you just times it by how far it goes through the shape which in this case is this 15. so all I've got to do is times up by 15. so 16 pi times 15. which comes out as 240 Pi which is also a decimal so it just depends on how the question wants the answer here so it's 240 Pi now there are a few non-calculator questions and again there's more of these on the higher but a two here that we really need to know and that is sine 30 and cos 60. now the value of sine 30 is one half as is the value of cos 60. so they're both equal a half and they're really good ones to make sure that you know for these non-calculator papers okay so do remember those ones sine 30 and cos 60 and they both equal one-half let's see how we can use that so this question says given that sine 30 is a half which hopefully we know anyway work out the length of a b and a b is this side here so let's label this triangle up we've got the opposite we've got the hypotenuse which is asking us to look for and this is our adjacent again now again we're not going to use the adjacent here we're just going to use o and H and if we remember our little formula look s o h has the O in the H in and we're looking for the hypotenuse there we go so if we're looking for the hypotenuse we cover up the H and we need to do o divided by S so s is my sine 30. so if I was to do this on a calculator I'd have to do 4 and I'd have to divide it by sine 30 okay but this question's actually told us that sine 30 equals a half so actually what I'm doing is 4 divided by a half I'll write that as a decimal if it makes it a little bit nicer to see but 4 divided by 0.5 so actually it's just asking me to do a number some now which is 4 divided by a half and if you divide by a half it doubles the answer there doubles the number so 4 divided by a half is eight So my answer would be eight centimeters there we go now of course that could go the other way around as well I could have to do a times so had it not given me this 4 here maybe it had already told me that this was a then instead I would do eight times a half doing s times H so eight times a half and that would give me the four here that's been given us the opposite so we could have it either way okay so when we're looking at a pictogram we really need to look at the key so the key here on the right hand side which I'll just highlight says that a circle represents ten tins and it says here that some tins of paint were sold on Monday Tuesday and on Wednesday it then tells us that on Thursday 60 tins were sold and on Friday 35 tins were sold and it wants us to complete the pictogram so for 60 tins that's going to be six of those circles as they represent 10 each so we just draw six circles into our pictogram nice and neat and there we go there's our six for 35 tins we're gonna have to use part of a circle so we'll get the three circles in that represents 30 and then we just need to draw half a circle as best as you can which is going to represent those five so obviously drawing that nice and neat and you can probably do it a little bit better than me when you've actually got the paper in front of you now for Part B it says more tins were sold on Tuesday then on Monday how many more tins so we just need to count from the pictogram so on Monday there are three circles that would be 30 tins and on Tuesday there are four and a half circles so that'll be 45 tins so in total that's an additional 15 you can show you're working out that's 45 take away 30 which is going to equal 15 extra tins that are sold on Tuesday as opposed to Monday and there we go there's pictograms okay so this question says Harry and Sheamus recorded the number of minutes they spent revising last week so the table shows some information and it wants us to draw a suitable diagram to show this information there's a couple of options we could do obviously a uh we're going to be drawing a bar charts there are a couple of different bar charts that we can draw now regardless of which option we're going to go for we're going to have to scale the axes and when it comes to one of these I'd probably draw a dual bar chart where we have the bars next to each other so if we have a look at this we just going to have a look at the information and identify which one of those is our largest number then that is 40. so we're going to need to go up to at least 40 on the left hand side so if we just go for 0 10 20 30 and 40 and then just make sure that you label those 0 10 20 30 and 40 and then we're going to want to label the axes there and that is going to be minutes so we'll just put here minutes to the side and then we're going to go down the bottom so to make sure that we can draw this correctly we want to label the days so we're going to go for Monday Tuesday Wednesday Thursday and just label them in so for here we're going to go for Monday and let's just imagine we're going to have our bars kind of drawn side by side so it's going to go about that much so that would be Monday and then we're going to leave a gap between those so our next ones are going to go sort of here we can just sort of label this in every five squares I'm counting just to make sure that it's nice and neat and hopefully this all fits so that could be Monday there we'll leave a gap and then we'll have Tuesday here leave a gap Wednesday lever Gap and then we'll go for Thursday just over there we'll separate their Tuesday and the Thursday just like that okay so that also needs labeling so we'll put days underneath and we'll just write that that's going to be the days down there now we just need to draw the bars in so if we go for Harry first and we'll do 35 for Harry so if we draw a bar coming up obviously just between 30 and 40 and that'd be 35 and then 10 for Sheamus and we'll draw that in now obviously we can't tell at the moment which one is Harry and which one is Sheamus so we're going to have to draw a key so what I'm going to do is I'm just going to slightly shade in Harry's bars and then I'm going to draw a little key and we'll draw that at the end so we know we've got some space so for Tuesday we're going to have 30 for Harry and again I've left a gap so now I've got my bar here for Harry and then I'm going to do Sheamus as well which is going to be 20. and again just going to shade in Harry's and don't need to shade it in massively just a little bit of shading next one is 25 so 25 for Harry and there we go let's draw that across there we go and we're going to have 25 as well for Sheamus there we go and I've not drawn that perfectly there you could probably do this better when you have a ruler and a pencil I'll try and make it as neat as I can and then again shading in Harry's and then we've got our last one here which is 20 for Harry and we've got 40 there for Sheamus and that was the largest out of all of them so up to 40. drawing that in nice and neat not forgetting to shade Harry in and then somewhere on your graph or to the side of the graph you're going to want to draw a key I'm going to do it over the top of the graph just because I'm limited on my space but I normally draw it to the side in fact I could probably fit it into the sides I'm just going to draw a little box shaded in and I'll just put Harry next to that and then I'll just do my blank box that's Sheamus there we go and I'll just put the key above that just there right there we go so that is how we would go about drawing our bar charts now you could draw a different type of bar chart as well where the bars are stacked on top of each other that's fine to do I would always go against drawing that if I had the option because obviously if we do that we're going to have to scale the numbers up a lot more because we have to look at their totals instead so for example we would have to scale Thursday up to the number 60 because we'd only go up to 20 for Harry then an additional 40 on top for Sheamus but you could go about doing that as well but obviously given the choice I would draw one of these bar charts so there we go that's how we go about drawing one of these bar charts okay so reading from a stem and leaf it says the speed of 27 cars work out the median and work out the range so to work out the median I want to know which one's in the middle we can take quite a nice approach for this we can just cross numbers off either side until we get to that middle number so if there's 27 I'm going to cross off 10 from each side at least so I can start to get from the middle so so one two three four five six seven eight nine ten and then going from the back one two three four five six seven eight nine ten now I'm just gonna do one from each side until I get one in the middle so one and one two and two three and three and I'm left with this one here don't forget the key though it says three dash eight or three line eight means thirty eight so that's next to the five so it's five and six which is 56 so the median there would be 56. working out the range we want to look at which one's the smallest and which one's the biggest let's have a look if I get rid of some of my lines here the smallest number there is 38 the biggest is 70 so to work out the range which I'll do up here so the range will just be 70 the biggest takeaway 38 the smallest which is 32 that's my range there again there's median and range from a stem and leaf okay the table shows the probability of picking different colors so we have red blue white and black and it says the probability of white and black is the same complete the table so at the moment we've got these two which add up to 0.7 and probabilities have to add up to one now what we've got left over there is 0.3 I like to think of it as 0.30 because if they're both the same we're gonna have to halve that and half of 30 is 15 so it's 0.15 for both of those and now they add up to one now for Part B it says that there are 60 counters in the bag how many are white now if we look at the probability of white which is over here 0.15 is 15 so really all this is asking us to do is to work out 15 of 60 and we can do that nice and easy 10 is six five percent is half of that which is three and add them both together and that's nine now part C here you won't be given this at the same time as question B because we already know that there are 60 counters in the bag but it says that there are 12 red counters and how many counters are now in the bag so if we imagine this is a completely different question now but it's saying that red counters here which is 20 percent is now 12 counters so all that's actually saying is twenty percent equals 12 counters now that's okay we can turn that back into 100 quite nice and easy I could just times that by five but let's imagine it's not a nice percentage I'd have to try and break it down to something which turns into a hundred so we could divide it by two which would give us ten percent or just half a 12 which is six and then times that by 10 to get us back to 100 okay again you could actually just times twenty percent by five I think that's quite nice just to break it down and build it back up so 60 counters there would be in the bag okay so here we've got 60 people each took a driving test one day 21 were women 18 of the 60 people failed 27 of the men passed their test one of the men is chosen at random what's the probability this man failed his test now this is a little bit nicer to draw frequency three because we only have two options here we have men and women and then whether they pass or fail so the trees are actually a lot nicer when it's just the two let's have a look we've got men and women and then off each of those branches they either pass or fail I'm just going to put P for pass and F for fail there we go and there's the tree and there we go so pass fail so if we go about filling in the information we've got then we've got 60 people took a driving test so 60 at the start 21 more women so 21 down here 18 of the 60 people failed their test so let's have a look 18 of the 60 people failed their test so we've got these two here in failed and that has to add up to 18 there's 18 of them are going to fail we don't know whether it's male men or women yet but 18 failed then it says 27 of the men passed their test so up the men 27 past there we go that's that one uh right okay so that's all the information that we've got now we know that there are 21 women so we can work out how many men there are they have to add up to 60 so it's going to be 39 there we go and now we can work out this one just here because it's got to be 39 we've got 27 in there already so that is an extra 12 okay just double check that up to 39 yes they do now we can deal with this bit of information look these two have to add up to 18. we've got the one of them there is 12 so this has to be six and then these two here have to add up to 21 so we've got six in that one at the moment so that's an extra 15 there to make sure that's up to 21. now I can take the same approaches before it says one of the men is chosen work out the probability that this man failed his test so one of the men so it's at least 39 and 12 of them failed so there we go we've got 12 that failed out of the 39 men there we go there's our probability if you're only going to watch one video this year this is the one with GCSE maths right around the corner I'm going to show you three easy steps to maximize your grade using minimal effort step one head to the tgmt hub go to the bottom of the page once you've logged in and access the revision quiz from here you can answer a question on every topic within the GCSE you can go through working through each question and making sure that you practice absolutely everything once you've done that you are going to get a full report that shows you exactly what you got right and wrong on all of these questions so work your way through the quiz and then access your reports from here you can see an individual units what you need to work on and what you've already secured so identify what you need to work on then we're on to step two head back to the hub you can go through the units you can go to the exact one that you know that you need to work on and you can access all the individual topics within that you units once you've done that you can pick the topics that you want to work on watch the video and you can complete a topic quiz for that particular topic so work your way through the quiz answer those questions and then you're going to get another report that's going to show you exactly what you got right and wrong so that you know if you need to do any more work step three head back to the hub and go to the bottom you can now go on to the upgrade quiz on here you're going to be able to answer questions again different to the First on every single topic within the GCSE once you've done that you're going to get another quiz breakdown under the report that shows you exactly what you got right and wrong so you can put any of those finishing touches in before that exam comes along with the exams right around the corner we need to make sure that we make revision simple join upgrade okay so when looking at order of operations we just need to think about those rules and the orders in which we do things so for part A here it says work out two times five plus seven that's a really nice one because multiplication comes before addition so in this particular question we would do multiplication first which is two times five so we'd have 10 for the answer for that and then we'd need to add seven and when we do ten plus seven we get the answer 17 so that's hopefully quite a nice one now for this one here it says right brackets in this statement to make it correct so at the moment we have seven times two add three and that equals 35. now if we to follow our normal rules here 7 times 2 would equal fourteen then we would add three and obviously doesn't equal 35 so it doesn't work as it is and that means there's only one place for us to put the bracket and that would have to be around the two plus three of course no matter which one you're given you do need to check this out and just make sure that it is correct so we do brackets first two plus 3 is equal to five so actually we're going to be doing 7 times 5 and 7 times 5 is 35 so we're happy with that we know it's correct and we've checked our answer so that's how we go about doing order operations okay so when looking at some negative numbers we can have any form of calculation or any kind of question to do with negatives but here's an example with something to think about so it says write one of these numbers in each space to make the correct calculation so you just need to remember whichever number we put first here is going to be our starting number so if we want negative seven it's probably a good idea to try something down in the negatives so if we were to put negative 9 here what would we have to add to negative 9 to get negative seven well if we're going to add something to negative 9 we're getting closer to zero to get to negative seven so we don't want to be adding a negative we want to add a positive so if we we could add 2 and that would get us to negative seven so that's one of our Solutions there but is there anything else we could have tried well if we had have started with negative 2 we'd have to add negative 5 to get down to negative seven so we couldn't have started with negative two what about if we started with two is there any way we could have got down to negative seven well we can add negative 9 to 2 because if we do two add negative 9 that would be the same as 2 take away 9 and 2 take away 9 would equal negative seven as well so we could have had negative nine add two or we could have had two take away 9 which would have been equal to negative seven and in each of these questions whenever you're dealing with negative numbers you just got to remember your starting number and then whether you are going up or down so if you're adding a positive you'll go up closer to zero or obviously above zero and if you're adding a negative that's actually going to be doing a takeaway so just be careful there with your starting number and which direction you're going when you are adding or subtracting negatives okay so in this question we've got some estimation it says here Paul organized an event for a charity each ticket for the event costs 19.95 and he sold 395 tickets Paul paid costs of six thousand pounds and he gave all money left to the charity work out an estimate for the amount of money Paul gave to the charity okay so let's have a think now he's got the tickets for 19.95 and we are going to do an estimate here which means we're going to round all our numbers to one significant figure so for the amount of money or each ticket we're going to round that to 20 pounds so we'll write 20 pounds for that any Penny sold 395 tickets so let's round that I'll just label it here to 400 tickets now just as a side note here if you don't round them you can still get the answer correct but bearing in mind we're doing a non-calculator method here we don't want to waste any time doing 19 pound 95 multiplied by 395 tickets when we could just do this nice easy method of using our one significant figure rounded numbers but again you can do it the other way using doing the full calculation but you're going to spend a lot more time doing that you're not going to gain any extra marks so for this one here if we've got 400 tickets each one was 20 pounds well we're going to do 400 tickets multiplied by the 20 pounds which is easy for us to do because we just do four times two which is eight and then add on those three zeros so that's 8 000 pounds so we sold 8 000 pounds worth of tickets and he paid costs of six thousand pounds so that means he's made two thousand pounds profit because if we take the 8 000 pounds that he made selling the tickets take away the six thousand pounds costs that he had to pay that's left in with a profit of two thousand pounds and that two thousand pounds is what he's going to give to the charity that says he gave all money left to the charity so that two thousand pounds which we would have made he's given it all to the charity now it says here is your answer to a an underestimate or an overestimate give a reason for your answer now in our calculation we've done a multiplication so we've multiplied two numbers together so what have we actually done to our numbers well we rounded the cost of the ticket up from 19 pound 95 to 20 pounds we've said they're more expensive and we've rounded the amount of tickets up from 395 to 400. so we've said more tickets and a higher price so that is going to be an actual larger number there that we got for the 8 000 so actually he would have made slightly less than 8 000 pounds and then when we took away the six thousand we'd have had a number less than the two thousand pounds so our answer is an overestimate and we would have to give a reason for that so our reason there would be that because we rounded both the numbers up it's given us a bigger number when we worked out this 8 000 so 8 000 was more than it should have been so therefore it's an overestimate and there we go there's a bit of estimating with the first one we've got workout three and four fifths add three sevenths and give your answer as a mixed number in its simplest form now when we're looking at this sort of question here uh one thing we need to remember no matter which type of calculation we're doing adding subtracting dividing or multiplying if we have mixed numbers involved we need to make them top heavy first now when it comes to adding and subtracting there is a little bit of a different approach that you can take that some of you may use but I'm going to use the same method for all of these types of questions and that is making any mixed numbers top heavy fractions first okay or improper fractions first so the process for doing that is obviously taking our fraction here three and four fifths we want to turn that into an amount of fifths so to do that we want to figure out how many fifths are in three so we can do three times five which is fifteen add the extra four-fifths there and that makes nineteen fifths so three and four-fifths three times five is fifteen add the four that we've got there as well is nineteen fifths okay so it's the big number times the bottom to make 15 at the top number on the fraction there nineteen so we've got 19 fifths and we're going to add that to the three sevenths and when it comes to adding and subtracting we need to have a common denominator so we need that number on the bottom to be exactly the same now that means that we can multiply both fractions here as long as we should do whatever do we do to the bottom we also do to the top we can make equivalent fractions with a common denominator so thinking about five and seven the lowest common multiple of those is 35 so in order to get them to be 35 the left fraction there I can times the top and bottom by seven and the right fraction we can times the top and bottom by five and that would give us 35 on the bottom of both so we've still got to work out 19 times 7 there and obviously take your time doing so but 19 times 7 is 133 and at the moment that is going to be over now 35 okay so you get some quite big numbers here but obviously don't be afraid just to decide to do some multiplication there additives 7 times 10 and then seven times nine and added them both together okay but you can show your working out to the side on the right fraction there we're going to times them both by five so three becomes 15 and 7 becomes 35 and now we're in a position where we can add these both together so we can just add together the numerators 133 plus 15 which gives us 148. and that is over 35 remembering that you don't add together the denominators there we go we've got 148 30 fifths now obviously it says here to give your answer as a mixed number in its simplest form so it's up to you whether you're trying to simplify this or whether you uh obviously turn it into a mixed number first but we need to turn it back into a mixed number so to do that I need to know how many times does 35 go into 148 that's not the nicest so I'm just going to write down a few of the 35 times table so 35 add under the 35 is 70. add another 35 is 105. add another 35 is 140 and then it's not going to go beyond 140 there because I get obviously our number there is only 148. so let's go from here um we know that that 35 goes in four times so that's going to be a big four and what's left over from 140 to 148 is an additional 8 35 there it goes so four and eight thirty-fifths left over okay so looking at that number there you just got to decide as well does that little fraction at the end there simplify does 8 over 35 simplify now in the case of this fraction here it actually doesn't there's nothing that goes into both 8 and 35 other than one and obviously that's not going to simplify it for us so that actually is fully simplified there so even though it says give it in its simplest form in this circumstance here that little fraction on the end there doesn't simplify okay but it's going to say that anyway because we didn't have to use 35 as our denominator we could have potentially used a bigger number like 70 or even larger if we wanted but because we use the lowest common multiple it's just ended up that doesn't actually simplify at the end there but do look out for that because that fraction quite often does need simplifying okay but there's our final answer 4 and 8 over 35. let's have a look at some subtracting right okay so this question says 2 and 1 7 take away one and a quarter okay work that out and give your answer in its simplest form now it doesn't mention anything about it being a mixed number so that obviously gives us a bit of a hint that we're not going to have a mixed number here we're just going to go about this process in exactly the same way as when we were adding okay adding and subtracting the process is the same except obviously we're not going to add our fractions together but then we're going to take them away from each other so we're going to make them top heavy to start with turn them into improper fractions so 2 and 1 7 2 times 7 is 14 add the one is 15 over 7. and the one on the right I'm obviously taking these away one times the four is four add the extra one on top There is five quarters and there we go we just need to make these have a common denominator and again we'll look for the lowest common multiple here so 7 times 4 is 28 that does actually give us the lowest common multiple there so we're going to make the denominators out of 28 so we're going to times the right fraction by seven okay that's top and bottom and we're going to times the left fraction by four and again that's going to be top and bottom so our two fractions are going to be out at 28 so I'm just going to write two blank fractions here I know what the denominators are going to be I need to figure out those numerators so 4 times 15 on the top of the left one gives us 60. and 5 times 7 on top of the right one gives us 35 there we go so we've got 60 over 28 take away 35 over 28. so again the denominator isn't going to change we're just going to write how many 28s we have left and put 60 take away 35 there leaves us with 25. there we go so that is our final answer there we get 25 over 30 over 28 sorry okay so again just have a quick look does that simplify are there any numbers that go into 25 and 28 um no okay 25 dividing divided by 5 1 and 25 and and none of those apart from one obviously going to 28 so it doesn't simplify so again it says give your answer in its simplest form but again we got the lowest common multiple there for the denominator and made sure that they are the ones we use so a little hint for you there's always a look for that lowest common multiple okay in the case of these two they were quite nice because it was just the two denominators times together but you always want to have a look for that lowest common multiple just to avoid having to do that simplifying at the end there for this one we are going to work out three fifths of 65. so again just like before to start with we are going to work out one-fifth so to get one-fifth we're going to divide by five so for one-fifth we're going to do 65 divided by the denominator there which is five and again if you're not sure on that just do some division to the side you might know how to do that mentally if you want but 5 goes into six once remainder one and into fifteen three times so there we go the answer is 13. so 65 divided by 5 is 13 and that's one fifth and then again flipping to the numerator because we want three of those so we're going to times that by three so for three-fifths we are going to take the 13 multiply it by three and that gives us an answer 39 and again you can do your working out to the side if you need to for any of those but there we go that's how we're going to work out a fraction of an amount so divide by the denominator to get one of them whatever that denominator may be and then multiply by the numerator and that's going to give us our answer for working out whatever fraction we're looking for whether it be 2 7 4 9 3 11 whatever it would be so divide by the denominator and then multiply by the numerator okay so this question says three-fifths of a number is 48 work out the number so this really tends to confuse people when it does come up because obviously when you see three-fifths of another number A lot of people are going to think okay straight away divide by the bottom Times by the top but this is actually in Reverse this is saying that three-fifths of a number is already 48 and if we think about this in terms of our bar again let's just think about if we had to split this up into five portions this time because it is in terms of fifths there we go so if I split this up into five parts which I may not get perfect there we go good enough and it's what it's saying is that three of them has already been calculated as 48 and we want to know what the entire original number is so looking at this then okay well if only three of them were 48 well we wouldn't divide by 5 anymore but instead we would divide by three because we want to know how much is going to go into each of those three boxes just above so this time and it's obviously the complete reversal before because this time we're going to divide by that numerator so instead we're going to divide it by three and that's going to tell us what number goes into each of the bars Above So 48 divided by 3 is equal to 16 and again you can put that obviously the working out just to the side so if I put now 16 into each of these bars I would also put it into those additional two because we are looking at now all five of them so that number there if we add up all of those 16s and again you can obviously add them all up or you can do 16 multiplied by five so that's working out I would do to the side 16 times 5 because there are five boxes and that adds up to 80. there we go and that is my answer and again you could put the 80 next to your diagram if you prefer to use a diagram method but that is how we're going to go about approaching these questions so if you think about what we've actually done it's just the complete opposite of what we did before and that's why it's in Reverse this time instead we are dividing by the numerator and multiplying by the denominator but you need to understand why and I think that this visual idea really does help you to understand why we would divide this down by the numerator and why previously we were dividing by the denominator okay so it's a bit of a special one it says write down the value of 3 squared times 3 cubed or over 3 to the power of 5. let's just have a look what we get here so if we tidy up the top adding the powers we get 3 to the power of 5 on the top and that's also being divided then by 3 to the power 5 on the bottom now it says write down the value if I think about what this is as a power of 3 though if it's a power of 3 to the power 5 divided by three to five power five we'd subtract the powers so that's five take away five and five take away five zero so three to the power of five take away three to the power 5 gives me 3 to the power of 0. now 3 to the power of 0 here is a special one when we've got something to the power of zero because if we think about what we've actually got here we've got a number on the top and it's being divided by the same number on the bottom and what happens when you divide any number by itself let's just think of an easy number just three on its own three divided by three gives us the answer one anything divided by itself equals one so therefore three to the power of zero must equal 1. okay and this rule applies for everything here when we've got something to the power of zero anything to the power of zero is going to equal one it doesn't matter what I pick whether I pick five to the power of zero that's going to equal one I could pick anything 49 to the power of zero that also equals one okay just thinking about this logic here of anything divided by itself has to equal one and that's how we get a power of zero so I'm not gonna give you any questions on this but it's another little rule to have a bit of a think about if you are asked to find something to the power of zero the answer is always one and just be careful not to mistake that power of zero is a degree symbol it's not it's a power of zero there okay so write down the value of 4 to the power of negative two now the negative part's still going to do the same thing so we're still going to do the reciprocal so 4 again is not a fraction so we'll write it as four over one and the reciprocal of that is one over four now this actually has a negative two in there so that 2 is just a normal Power okay so the two is just a normal Power of Two And when we do a power of two we Square what we've got so I'm going squaring a fraction we are just going to square it like we would a normal number is that we're going to square the top and we're going to square the bottom so the square of 1 1 squared one times one is still one and the square of four four times four is 16. so there's our Final Answer 1 over 16. so a negative power flips it over the number is just a normal Power right so write down the value of 125 to the power of a third so again 125 to the power of a third we have this number on the bottom which is the root and a 3 would represent a cube root again it's just a one on the top though which is a normal Power of one and a normal Power of One does not change our number at all so a normal power but a cube root let's write that in so that would represent the cube root of 125 so it just helped an a square and Cube numbers here 125 is a cube number and the cube root of five is five there we go remember we can always think about that backwards we can think okay well 5 times 5 is 25 times 5 again is 125 so the cube root of 125 is five so when you're doing these fractional Powers just look at what numbers on the bottom that could be a square root a cube root we could have a 4 on the bottom and that'd be a fourth root we could even have five or numbers Beyond so a bigger number here 504 000 in standard form so there isn't a decimal in this number at the moment but the decimal would be here at the end if we had one so if we think about where that decimal would be if we were to write it to write this in standard form we'd have to do quite a few hops here so one two three four five okay so that's five hops for this number and that would make the number 5.04 okay not forgetting that that four exists we can't just uh not include that it's still there so 5.04 still times 10 and that was five hops it's a big number so five okay not a negative power for this one because it's not an all point number it's a large number so 5.04 times 10 to the power of 5. so just remember if it's a large number you can have a positive power with however many hops you've done to make it between 1 and 10 and if it's a no point number you can have a negative power again for however many hops you've done to make that number between one and ten okay 8.03 times 10 to the minus five so it's a negative 5 there in the power so it's going to be a naught Point number so I'm gonna have to hop it the other way this time so let's write out what we've got so far we've got 8.03 so the decimal is already in drawn in for us it's just in there so we're going to hop it from there to the left to make sure it's at null Point number because it's a negative 5 is the power and we're gonna do five hops so one two three four five and then let's fill in these zeros and put that decimal in there where it's going to go now now we've got this number here obviously we just need to tidy this up because we would normally write this with a zero at the front so tighter that was a zero and let's rewrite actually what we've got here so we've got 0. and then eight zero three and that is my final answer there for writing that as an ordinary number so just remember it's a positive power we're going to hop it to the right to make sure it's a big number and if it's a negative power we're going to hop it to the left to make sure it's a naught Point number a little number there okay so looking at this one 4 times 10 to the 5 times 3 times 10 to the minus 2 and there's an example of one more we probably won't want to write this out as old knowing numbers and work them out because it might get a little bit over complicated here let's look at applying that same little trick so let's do the four times the three which gives us 12. and that's going to be times 10 to the power of something I'm just going to be a little bit careful here because although we can add these Powers let's just have a look because one of them is a negative so the powers if I write this to the side we're going to do 5 for the first Power and we're going to add to the next power which is negative 2. and 5 plus negative 2 is 5 take away two so my power there is going to be three so that's fine 12 times 10 to the power of three and again we just need to put this in standard form because this 12 is not between 1 and 10. so instead we'll make that one jump smaller we'll make it 1.2 and to balance that out we'll make the power one jump bigger so three goes up to four and there's our final answer 1.2 times 10 to the 4. one more question before we have a go so 4.3 times 10 to the minus 5 times 3 times 10 squared again giving your answer in standard form so we can apply the same little trick again we've got 4.3 times 3. if you've not got a calculator here we could we could just add together to three four point threes we're timesing it by three so four point three four point three and another 4.3 369 and then four eight twelve so we get 12.9 so 4.3 times 3 is 12.9 times 10 and let's figure out what that power is going to be so that first power is negative 5 and we're going to add 2 to that so negative 5 add 2 is negative 3. so my power there is going to be negative 3. again it's not in standard form notice I'm going to balance it out so I'm going to make the number one jump smaller down to 1.29 I'm going to make the power one jump bigger so bigger than one bigger than negative three is negative two getting closer to zero there so there's our final answer just following the same process building it up in difficulty we did four point three times three I used a little bit of working out to the side which is absolutely fine to do then we balance that out so we made the number at the start one place value smaller balanced it with making the power one jump bigger just watch out there when it is a negative power there making it one bigger would not go down to negative four that would be going smaller so we are going up to negative two to make it bigger okay so in this question we just need to put some symbols in the correct boxes so it says here here are the symbols and we have the less than symbol the greater than symbol and the equal two symbol write one of these symbols in each box to make the force true statements so 14 is less than 21 so we can put the less than symbol in there that one's okay now this one here we're gonna have to work out so four plus eleven uh sorry four plus seven is equal to 11. and 103 take away 92 well that is also equal to 11. so that statement there they are equal to one another for the next one we've got 2 squared 2 times 2 is 4 and here 2 times 2 is 4 as well so again that's equal to one another and then for our final one negative three and negative five or negative three is greater than negative five it's closer to zero so it's a bigger number and there we go there is filling in symbols where we're using inequalities okay so when we're looking at some systematic listing it says here Jennifer is going to have a meal she can choose one starter and one main course and then we've got our three options for the starter Pate melon or ham and for the main course let's just highlight a different color we've got beef salmon or lasagna so it says write down all the possible combinations Jennifer can choose and that word systematic listing means we're going to make a list in a system or a systematic way that means it'll quite be basically a logical way we're going to use a system to create it so we're going to start off by picking the first starter and I'm just going to use the first letters so let's just check to make sure they are all different letters yes they are so for the Pate as the starter we would then have a main course of beef or we could have the Pate starter with the main course of salmon or we could have the Pate starter with the main course of lasagna so the system that I'm using is I'm starting with the first starter and matching it up with all the main courses now that I've matched it up with all the main courses I would go on to the next data which is the melon and again I'm just going to match that to all the same main courses Above So melon with beef melon with salmon and melon with lasagna and then I'll go into the final one which was ham so ham and beef ham and salmon and ham and lasagna and that is all of our possible combinations there and that is completed so we've used a nice system to create those to create this list okay so that's how we go about using systematic listing okay so when looking at some length conversions we need to remember for this first one where it says to change 450 centimeters into meters how many centimeters are in one meter so there are four 450 here that we need to convert now in each meter there are a hundred centimeters so we've definitely got more than one meter here but we could actually just divide it by a hundred if you can't just spot this answer divide it by 100 that would hop the decimal to here and that'd be 4.5 so for this one here we would say 4.5 meters and that would be our first answer for the next one we're going from meters to millimeters that's quite a big conversion there so to start with rather than jumping straight from meters to millimeters let's go back from meters to centimeters so 9.4 meters if we want to turn that into an amount of centimeters again we'll just do the opposite of what we've just done which is multiplying by a hundred so if we hop the decimal over twice that would be 940 centimeters now this next conversion relies on you knowing how many millimeters are in a centimeter of course if you have a ruler that's quite nice and easy to remember because you can look at your ruler but there are 10 millimeters in every one centimeter so we would times that by 10 and that would turn it into millimeters so that would be 9400 millimeters now of course you might have spotted there if we are timesing by a hundred and then times in by 10 then we could have actually just multiplied by a thousand but that's going to give you a lot of conversions that you need to remember so I would prefer to just remember the conversion between meters and centimeters and then also centimeters to millimeters but there we go that's how we go about some length conversions now when it comes to all of these percentages my first step is always the same and that is to work out 10 so to start with here we're going to write down what 10 percent is so 10 is nice and easy for us to find you take the number which in this case is 80 and you just divide it by 10 and that just means essentially you can do your little trick which is to hop the decimal in so if you hop the decimal in over the zero you get that 10 is equal to eight which is always very nice when it does end in a zero because obviously you can apply the trick which is just to get rid of the zero there but ten percent is eight and now we need to figure out how we're going to get to 35 percent now because it's a five percent in there as well we're also going to have to find a five percent and that's easy enough now that we have ten percent because we can just halve eight and that would give us that five percent is equal to four we now have everything we need in order to in order to build up to 35 percent we know that 10 is eight so if we want to get to thirty percent we would want to have three of those and the additional five percent there which would get that up to 35 in total so you can apply different methods here in order to get to 35 you could either just add together the pieces that you need or you could do some multiplication as well and what's a nice easy method that you can do is just say okay we want three of the eight uh the ten percent so eight another eight another eight and a five percent which is a four and just add them all together okay you might argue actually it's a bit easier just to times eight by three but we get 8 16 24 plus the 4 is 28. and that would be our final answer 28. so there we go nice and easy okay so this question here then we've got a lot of information it says on Saturday some adults and some children were in a cinema it says the ratio of the number of adults to the number of children was five to two and each person had a seat in standard seating or had a seat in the premium seats three quarters of the children had a seat in the premium seats and 117 children had a seat in the standard seats and there's exactly 2 600 seats in the cinema let's just highlight that in a different color and it says on the Saturday were their people on more than 60 of the seats so there's a couple of different ways that you could approach this what we could do is we could straight away work out that 60 which seems a little bit counter-intuitive as it's the last line in the question but if we work out that 60 straight away we'll know what kind of number we're aiming for here so we could work out that sixty percent and again depending on whether you are using a calculator or not will depend on how you approach that but if we do that without a calculator because that's relatively simple for us to do we can just work out this sixty percent to start with so 10 would equal 260. and then we times that by six to get our sixty percent and if we just do that using column multiplication just to get our answer nice and quick six times zero is zero six times six is thirty six and six times two is twelve so that's 1 560. now on a question like this you never know in an exam that may have just got you a mark so well worth working that out if that's the nice easy step to go for this question so let's get rid of that and obviously always leave you working out again but we'll get rid of that and we'll just have a look at going back to the start so it says adults to Children was five to two and then it gives us some information about where they sat we know that three quarters of them had seats in the premium seats and 117 children were in the standard seats now both of those bits of information are just about the children so if three quarters of them were in the premium seats the other quarter must have been in the standard seats and it tells us that that was 117 children so we'll know if this one is three quarters and this one was one quarter because it's the rest of the children and it tells us that that one quarter is equal to 117 we just want to know what the three quarters is equal to and we know if one quarter is 117 in order to get three quarters we just have to multiply that by three so if we multiply that by three that is going to tell us the amount of children that were in the premium seats so 117 times three and again you've just got to do some working out again you can either multiply it by three or you could just add that together three times whichever you prefer I'm going to use a combination of methods here so that adds up to 21. that adds up to five and that adds up to three so that is 351. so now that there are 351 children that are in the premium seats let's get rid of that working out because we now have quite a lot of information we know that the first thing we worked out we're trying to see if they filled 1 560 seats and now in this second bit of information we've actually worked out the total number of children we have 351 that are in the premium seats and 117 that are in the standard seats so if we add those both together we have eight six and four so 468 children now we obviously just want to work out how many adults there are so we know there are 100 468 children but how are we going to get to the adults so it tells us the ratio of adults to children at the start just over here and it tells us that the ratio of adults to Children is five to two and we've already worked out that the children in that ratio is 468. so we're using a different part of our ratio here we're not just sharing in a ratio we're actually working backwards because we've already been given one of the numbers so in order to get back to one part we could divide that by two and that would allow us to then figure out the other way around what we would have to have multiplied by and essentially to get our number down here to figure out the amount of adults so if I want to work that out to the side I can do 468 divided by 2. which tells me what one part would be worth and that answer comes out as 234 so we must have multiplied by 234 per part in order to share that ratio out so times 234 so that means in order to get the total amount of adults we're going to have to Times by 234. again if this is non-calculator you're gonna have to do some working out there or you could just apply a little bit of a trick for timesing by 5 which is the Times by 10 and half of it although you might argue it's probably just quicker to do 234 times 5 to the side and just get that correct so 5 times 4 is 20. 3 times 5 is 15 add the 2 is 17 and then 2 times 5 is 10 plus the 1 is 11. so in terms of working this out we're going to need a bit more space there let's get rid of that Arrow we've got 1170 adults and to get there we Times by 234 Okay so we've got quite a bit of information now we've got a 1170 adults and we've got 468 children and it said is 60 of the seats filled and we already worked out that in order to fill that we needed to fill 1560 seats so now we have all this information and let's get all of this out of the way we can actually finish this question off because we can actually just add these two together and that's going to give us the total amount of people at the cinema so to finish this off we do 1170 the total amount of adults add 468 the total amount of children that adds up to five six one one thousand six hundred and thirty eight which is greater than 1560. so where it says it was what were there more were there people are more than 60 of the seats we would say yes and you could either then write a sentence to say 1638 is more than 1 560 or you could just say as an inequality 1638 is greater than one thousand five hundred and sixty and there we go and don't forget to write your conclusion here as it does say that and don't forget the question does say You must show how you get your answer so all of that working out is essential so this question on the screen to here it says only blue Vans and white Vans are made in a factory so the ratio of the number of blue Vans to the number of white Vans is four to three and it says for the blue Vans the number of small Vans to the number of large Vans is three to five work out the fraction of the number of Vans made in the factory that are blue and large okay so we're looking at these ones that are blue and large now as as these sort of uh question develops and this sort of gets a little bit harder there's a specific approach that I'm going to take when I'm actually looking at these and that is I'm going to kind of structure this in the same way that you would structure a probability tree and looking at this as more more on the lines of work out the probability of picking a van that's blue and large out of all of these uh sort of vans in the factory I saw a way that's what I'm going to draw this then is I'm going to draw a bit of a probability tree so I'm going to start by saying uh Blue to White Okay so we've got the Vans they're blue or they're white now over here we've got the ratio of blue and white we've got four to three now that's out of seven in total so four out of the seven there are blue and three out of the seven there are white so for the blue here we've got four out of seven and for the white here we've got three out of seven there we go then it gives us the next piece of information now it only mentions anything to do with the blue Vans here and I see sort of questions develop we're going to have a look at ones where it mentions something about both branches here but in this particular question look it just says for the blue Vans we've got small and large so I'm only going to do another Branch coming off the blue Vans here and that's going to be our small ones and our large ones and again we've got a ratio here it's three to five so it's out of eight in total so for the small ones there that's the three and that's three out of eight and for the large ones there it's five and again that's five out of eight now when it comes to actually working out the fraction here that are blue and large we treat it in the same way that we do a probability tree we have a look okay we go up the blue branch and we get four out of seven for the blue and then we go up the large root or down the large route depending on how you've drawn it here and that's five out of eight okay and just double check that you've put those in the right place that's large five out of eight that's right so that's blue and large so all we do normally when we go on a longer tree like this is we just multiply those fractions together and we're not going to do anything different in this question so what we're going to do is we're going to take that first fraction there for blue which is four sevenths I'm going to multiply it by the large fraction there which is 5 8. there we go our multiplying fractions is nice and easy we just times the top and times the bottom so four times five is twenty and 7 times 8 on the bottom is 56. there we go and that's our final answer 20 out of 56 is our fraction of the vans in the factory here that are blue and large okay so hopefully that makes it quite nice and simple to understand but essentially we just pair together the two fractions that we're looking at and multiply them together just like we would on a probability tree okay so looking at a money problem it says here Carla is planning a holiday for four people for seven days here are the costs of the holiday for each person so the travel is 150 the hotel is 50 pound for each day and the spending money 250 pounds work out the total cost of the holiday for four people for seven days so let's work out each person to start with once we've got the cost for one person we can just times that by four as there are four people so we've got the 150 pounds is on the travel so we know that that's going to be 150 pounds the next one we've got 50 pounds spending money or sorry for 50 pound for the hotel for each day so if we're going for seven days fifty times seven well five times seven is 35 so 50 times 7 would be 350. of course you can work that out to the side using column multiplication or whatever method you prefer and then we've also got the spending money doesn't say that's for each day so we'll assume that that's just the spending money for the whole trip and then if we add all of those together that's going to be the total cost per person so we have zero 15 to carry the one and then three four five six seven so 750 pounds per person now of course we've got four people so we've do four lots of 750 you could always just add together four of them or just use some multiplication four times zero four times five is twenty and four times seven is twenty eight plus the 2 is 30. so that's three thousand pounds so the total cost for the holiday for four people for seven days is three thousand pounds and there's our final answer okay let's have a look at our next question okay so looking at a bit of density without using a calculator now when we've got density in a question straight away we want to write down our formula or you could write down a formula triangle so I'll write down a formula triangle density is equal to mass over volume so if we put that in a formula triangle we can obviously work out any of them just by covering one of them up so it says a lead pipe has a volume of 40 and the density of lead is 11.3 grams per centimeter cubed as soon as you see that particularly this is a non-calculator question it's going to put a lot of people off because we've got decimals involved we know we're going to have to do a multiplication or potentially a division but it says work out the mass of the lead pipe so if we cover up the mass on the top there we know that to work out mass is density times volume so to work this out we need to do 11.3 and we need to multiply it by 40. now that obviously doesn't look very nice but on its own that's not a particularly difficult question all we'd need to do and there are other methods of going about this you could do things like multiplying by four and adding the zero one if you're happy to do that but no matter what the number here if we just hop the decimal out if obviously you had a decimal here we'd have 113 multiplied by 40. we'll work that out pop the decimal back in we've got our answer so zero times all of those is going to be zero so we'll just put zero for all of that then we've got our placeholder because we're moving on to the tens four times three is twelve carry the one four times one is four plus one is five and then four times one is four again so we have four thousand five hundred and twenty obviously we just need to hop that decimal back in So if we hop it back in we get 452.0 it's going to show there that I'll put that back in so our answer would be 450 2. now of course we're looking at a mass so we need to give a unit of mass and we can always find that unit of mass within the question this particular question gave us in the density unit set said grams per centimeter cubed so that would be 452 grams and that'll be our final answer for that question so don't be put off by these sorts of questions where they normally appear on a calculator paper just follow that same process and remember your written methods okay so solving 4X plus 2 all over 5 equals 6. now same process again it's all locked in by this denominator which is dividing by five so the first thing I'm going to do is times both sides by 5 to remove that divide now again it's not going to change the top there we're just removing a divide so 4X plus 2 equals 6 times 5 which is 30. and now as you can see it looks very similar to the question that we had before we're just going to follow the same approach now to solve it so it's a plus two so we're going to minus two so we take away 2 from both sides and we get 4 x equals 28. and then just like before dividing it by four and that gives us x equals seven there we go and there's our final answer for this one we've got nine x minus five equals 3x plus thirteen again here 3x is the smaller value of x so I'm going to subtract 3x on both sides of the equation again by 3x would leave us with 6X over here minus 5 and that equals positive 13. so I'll just write 13. again then moving the numbers to the other side it's a minus five so we're gonna have to add five and we get six x equals 18. and this is quite a nice one here because 6 does go into 18 so when we divide by 6 we get x equals three we have to worry about any any fractions there and simplifying those okay so when looking at recognizing graphs this isn't really one of those ones where you can just teach it on one question to do this and to know this properly you need to have looked at straight line graphs quadratic graphs reciprocal graphs cubic graphs as well and you need to just be be able to recognize what they look like when you were learning them when you were doing them but for this question here we'll have a look anyway so it says our first one here we're going to match up each of these graphs in the table and put the letter now we have two linear graphs these two here we've got one with a positive gradient and one with a negative gradient so we'll start with those two now in our equations here the first one has an x squared so that is a quadratic graph the next one is a linear graph so we'll have a look at that one the next one's a linear graph as well and then we have a reciprocal graph down here where there is a division with x so we're looking at these middle two now let's look at the gradients for those now the first one we've got y equals three minus 2x so the gradient of that is negative two that number that coefficient that's stuck on the X so that one's our negative gradient so that's going to be graph d as that's going to be sloping downwards and our next one has a positive gradient you can see that's 2x plus 3 so positive 2 a positive gradient and that'll be graph a and this is exactly how I'd approach this question anyway I would pick the one that you're most familiar with you might disagree and say actually I knew straight away which one the quadratic was and that's fine you could have done that one first but always work from the one year note to the one you're not so sure on so for this one here then the first one which we've already highlighted the x squared minus 7 is a quadratic graph so that is graph B the quadratic graphs have this U shape or if there is a negative quadratic a negative x squared it has an N Shape but you're probably only going to be looking at these u-shaped graphs anyway um but of course you need to remember that in case there is a negative x squared as well now again you can see that that is correct for this graph because the y-intercept that number at the end is negative seven so that matches up just down there with our negative seven so that would be fine as well so we're happy with that that is graph B and for our final one process of elimination we have our reciprocal graph and that is graph C and that is obviously this one just here that reciprocal graph has a very unique shape and obviously you just need to be aware of that when you are dividing by X you get these reciprocal graphs here which never touch the axes as when you're dividing by X you can never get you can never divide by zero so it never ends up touching the axes okay so that is recognizing graphs so this first one says on the grid draw the graph of y equals two X plus one now there's lots of ways of doing this but essentially it's just about understanding what this actual line equation means here so we know hopefully that all line equations need to be in this form y equals MX plus C okay so y equals MX plus C again in this line equation M represents the gradient which in our line equation here is the number two so we have a gradient of two and C at the end there represents the y-intercept which for us here is positive one so we've got a y intercept of positive one now in terms of what that means the graph is going to look like we know it's going to cross through positive one on the y-intercept which is right here okay I'll try and Mark that on there we go so we know the line's going to go through this point and all we have to do is figure out where all the rest of the coordinates are now in terms of y equals 2x plus 1 remember X and Y relate to the X and Y coordinates so all that this equation actually means is to find the y coordinate you do 2 times the x coordinate and add one or double the x coordinate and add one so all we need to do is actually pick some x coordinates now if you have a look on the graph you can see there's lots of x coordinates here we've got minus 1 0 1 2 3 and 4 at the end there okay so we have to actually do is pick them double them and add one and that'll tell us where the corresponding y coordinate is and we can do that using a little table and you don't have to use a table but it just just tidy it up a little bit if we just make a little table here sometimes you'll be given the table but it's quite unlikely that you'll be given it so we've got some x coordinates and we've got some Y coordinates we need to find now for X we've got starting at -1 and then it goes zero and then we've got up to four so one two three four so if we start with negative one this one here we need to double that and add one I'm going to just sub that into my little equation up there so y equals two X plus one so it's two lots of minus one two times minus one and then add one and if we double negative one we get negative two add one to that gives us negative one so our first y coordinate there is negative one as well so minus one minus one and we can plot that if we want straight away there it is minus one minus one moving on to the next one um what do we have for this next one we have uh the Y cut the x coordinate of zero so if we just without doing all the working out for all of these let's double zero is zero and add one is one so two times zero is zero add one is one if we carry on doing this for the next one so when X is one double one is um two and then add one is three so double one is two and one is three and you can start to see there's a little bit of a pattern emerging okay between all of these points if you have a look from minus one to one that is adding two from one to three that's also adding two so actually all we need to actually do is keep adding two so add two to three is five add two again is seven and add two again is nine and then we can go ahead and plot as many of these as we can obviously we can't do them all here um because the graph's not big enough but for the next one let's have a look one two three is there and two and five is there okay so they're all the points we can plot we could actually only plot it up to this point here um but once we've done that all we need to do is grab a pencil grab a ruler and join it up nice and straight all the way through the graph unless it states otherwise so with your ruler nice straight line going all the way through and extending it up to the edge of the graph there there you go and that is your line and it's always good practice just to label it you don't have to okay but you could just label that as the line Y equals two X plus one it just helps if there are multiple graphs or multiple lines on the graph but you don't have to label it in a shot too okay but there you go y equals two X plus one there are other ways of drawing this and it just comes to having an understanding of the equation there so we already talked about one of them the fact that it starts on the y-intercept of plus one and then the fact that it has a gradient of two just means that for every step across from a point on the line so if we look at this as I know it's two squares there you've got to be careful of the um careful of the scale but that's one across going from naught to one and if you have a look it goes up from one to three there you go so it has a rise there of going up two for everyone and you can always draw this in its way of understanding it as well so it's two there and one here every one across it goes up two and that's what a gradient of two actually means it means for every one step across it goes up two steps even though the scale is slightly different here it's one in terms of the numbers across and two in terms of the numbers going up and that is what a gradient of two actually represents but it's best bet is just to draw the little table out uh and then get all your numbers nice and neatly drawn in okay so I'm looking at intersecting graphs we can have a look at this particular one this is always one that looks way more complicated than it should so it says here use the graph to solve these simultaneous equations when you're solving simultaneous equations normally you're finding that X and Y value and that X and Y value you relate to the coordinate where they cross over and you can see on this graph here we can actually see that coordinate just there so normally when we're looking at simultaneous equations like this one here we try and make the coefficients of X or Y the same same sign subtract different times add or whatever process you use to do that now this particular one has already shown us the graph so we don't have to draw them we don't have to do it algebraically all we literally have to do is read that coordinate so we'll go down we'll read our x coordinate first so x equals negative two and then we'll read the y coordinate so go along and we get y equals four and there we go and that is using the graph to solve it of course you might be asked to write down the coordinates of the solutions in which case you would write it as negative of 2 4 just make sure you read the question but if it says find the solutions we would write x equals y equals okay of course you could also have to draw linear graphs and actually do these yourself but this is just intercepting graphs and understanding where the lines cross over is that solution okay so in this question we are given a line already drawn on a graph and it says write down the equation of the line now we know already that what the equation of line is going to look like it's in this format y equals MX plus C again that c being the y-intercept now we can see the Y intercept on the graph it's over here five so for starters I can actually just draw that straight in we can say C equals five and we can just put that in our line equation so y equals MX Plus 5. all we need to figure out is what the gradient is now if you have a look at this question we've got to find some coordinates that we can actually see so here's another coordinate that's on the line one three and if I want to find the gradient between these two points all I have to do is draw a little right angle triangle in so I can go across or down from this red point but if I go across and then down I'm just going to say how big the movement is there so that's a movement of one there and then that's going down too so I'm going to say that that's negative 2. now the equation to get the gradient is two different ways that you can look at it you can have a look at it and just use this little triangle it means the change in so it's the change in the y coordinate over the change in the x coordinate and it's sometimes referred to as rise over run okay so you might know one of these two but change in y over change in X or rise over room now in this case look the change in the y coordinate is this movement here going down it's minus two so that'd be on the top of my little fraction here minus two and the run or the change in X is on the bottom there is the one going across so it's minus two over one and minus two divided by one is negative two okay so my gradient here is negative two so it's obviously sloping downwards which is why it's got a negative gradient but that's um what our gradient actually is for this question here so all I've got to do is put negative 2 in place of M and we get y equals minus 2X plus five obviously thinking about that last question that we looked at you could actually write that as 5 minus 2x but you can write it in either way it's absolutely fine to do so just a little side note you can actually do these triangles as big as you want if I picked some different coordinates let's have a look that one there and this one here for example and if I was to go down instead and make a big triangle like this let's have a look my run there is four and my rise goes from Seven down to negative one which is a movement of -8 and if I did rise over run for this one or the change in y over change in X it would be minus eight over four a minus eight divided by four is minus two so you can do this over your life okay because it's a straight line it doesn't actually matter where you do the gradient from it's just you just need to draw it and read those numbers very very carefully so you can get the correct gradient there okay so using some formulas so this one here says make you the subject of and then we have a formula to rearrange so we're making cue the subject we need to get rid of that constant to start with so we're going to get rid of this plus seven we're going to do that by minusing seven to the other side so our first step would say p minus seven is equal to 6 Q I always write out your steps now we want Q to be the subject not six cubes we're going to have to divide by six to get that six unstuck there from the queue so if we divide by six we'd have P minus seven all divided by six and always write it as a fraction we don't want any multiplication or division symbols in a formula and that is going to be equal to q and if you want you can write that the other way around you can write Q is equal to P minus seven over six but it's fine to leave that equals Q on the right hand side now we've got Part B it says Part B using your answer to part A find the value of Q when p is equal to negative 11. so we've already got Q is equal to and then P minus seven over six so all we need to do is substitute p is equal to negative 11 into our rearranged formula so if we put that in we've got negative 11 take away 7 and all divided by 6 will be equal to q and simplify the top negative 7 and negative 11. I'll say that the right we run negative 11 take away 7 would be equal to negative 18. I'm going to divide that by 6 18 divided by 6 is 3 so negative 18 divided by 6 is going to be equal to negative three so Q will equal negative three and that'd be our final answer there substituting that in so it says find the 20th term in a sequence and it starts to give a stud the start of a sequence and obviously what we could do is just see what is going up by and keep writing it but if we're trying to find the 50th or the 100th term it's not very efficient way of us doing that so we're going to actually have a look at how we can use the nth term to find the answer to this so first of all we can actually find the nth term of this sequence so to start with we note that it goes up in threes so the start of our nth term is going to be 3n which means it's related to the three times table and obviously using our little trick we can go backwards so take away three and that gives us positive one here so it's three n plus one now thinking about what that means and I've mentioned it in the previous video and that means the three times table but with one added on now if we want to find the 20th term in the sequence all we actually really have to do is sub 20 into this expression here so n is going to become 20 or 3 times 20 it's obviously the 20th term and the three times tables 3 times 20. so if we do 3 times 20 so I'm going to sub that in three lots of 20 but then we just need to add 1 to that so 3 lots of 20 is 60 and add 1 to that gives us 61 and that's how we can find numbers in a sequence quite quickly there okay so sub find the end of the term sub the number in and then obviously just write down what that number in the sequence is as a sequence has an length term of 2N squared plus five find the tenth term in the sequence it's obviously we've got a different nth term here this is a quadratic sequence but we're going to approach it in exactly the same way now it does have a little N squared there but it's already given us the nth term so we've not had to find the nth term of this sequence it's just been given to us and it does look a little bit different to the previous two but we're just going to apply it in the same way so if we want to find the tenth term we just need to sub 10 in so if we sub 10 in we've got two lots of 10 squared N squared 10 squared and then add 5. okay so obviously remembering remembering your order of operations here you've got to work out the powers first which is why I've put the 10 squared in the bracket it's not to forget to wear that out first so 10 squared is 100 I'm just going to write that there 10 squared 100 so 2 times 100 working out this bit would be 200 and then add five would give us 205 there you go so a little bit of rules on substitution there obviously don't forget to check out the video on substitution if that's thrown you at all but just being careful to sub that number in so we've sub 10 in place of the N that's N squared so 10 squared was 100 2 lots of that is 200 and then add 5 at the end for 205. okay so this question says is the number 51 in this sequence now again we could keep writing and see if 51 appears but that we can actually just use the nth term again to actually figure out a question like this is is the number 51 in the sequence so if we find the nth term okay it goes up in fours so it's 4N and then back four again gives us minus one so four n minus 1. now there's our nth term what we could do is we could just sub numbers in see if we get 51 you know sub 10 in that would be 40 take away 1 is 39 and just keep trialling and erroring it but actually we can take a quick approach again now if we want to know if 51 is ever in the sequence we want to know is there a number that I can sub into there that ever equals 51 and I can make that statement just by writing an equation so I can just say well does 4N minus 1 does that ever equal 51 okay so it does 4 n minus 1 equal 51. and if we go about trying to solve this let's see what we get so if we add 1 to both sides we get 4 n equals 52. and then solving this just like a normal equation divide by 4 by so divide by 4 and we get n equals and 52 divided by 4 is 13 there we go so we get n equals 13. now what we found out there is that if we sub the number 13 and in place of n that would give us 52 and you can go back and check it 4 times 13 uh gives us sorry it gives us 52 and then when you take away 1 you get the 51 so there we go when we sub 13 and we do get 51. so yes we would say 51 is in sequence and we could even say it's the 13th term in the sequence there okay so looking at some parts of circles so here on the diagram below draw a chord of the circle now a chord can be drawn in in lots of different places but we've got lots of different ways of drawing a chord as well or two specifically now you could draw a diameter okay a diameter is a special name for a cord that passes through the center so you won't be wrong to draw the diameter in and of course by doing that I've already described you what the diameter looks like but a chord can be drawn anywhere as long as it touches two points on the circumference and it's just a straight line so we could draw it anywhere it doesn't need to pass through the center and that right there would be a chord you could also potentially be asked to draw something like a segment and maybe it would ask you to shade that in if you ask to draw a segment that would just be this bit just here you could shade that in as well obviously if you were asked to draw a segment as well as that you could be asked to draw something like a sector so you could do use the radius and another radius to form a sector so that's that's a piece that you could be asked to do as well but of course you are only asked to record and our next one here asks you to draw a tangent so a tangent is a straight line that touches the outside of the circle on the circumference at one point and then keeps on going so you can draw this nice and carefully Abbott there we go that would be a tangent so lots of different things that you could be asked to draw we talked about the diameter the radius a segment a sector the chord the diameter obviously could be a chord as well and now the tangent so lots of different things that you could draw on a circle but there we go there are our two questions now for this question here we've broken it up into two parts I'm going to do them both on this one grid and as you can see on the grid we've got a triangle there drawn um at this position here obviously over in the positive section now when we translate a shape what's going to happen is this triangle is going to move somewhere else now it doesn't necessarily have to go into any of the other sections but it's going to move somewhere within this grid so if we have a look at this first one to start with it says translate the triangle by the vector and then we have minus four on the top and three on the bottom now if you don't have a very good understanding of vectors there is a full video that I have done on column vectors which I'll link in the description Link in the description for you to have a go at but in regards to this one we're going to go through it relatively quickly so that number on the top there represents going left or right so positive number moves it right and if we have a look on the grid look the positive numbers go to the right and the negative numbers go to the left so a positive number on the top indicates moving right and a negative number indicates moving left and that goes for the same on the bottom except the bottom number there is representing moving up and down so again positive numbers move it up and negative numbers move it down now the way to go about doing these is just to pick one of the corners or one of the vertices on the shape I'm going to pick that top one there I'm going to move that point four to the left as we have negative 4 on the top and I'm going to move it by three up so let's move that so we go one two three four to the left and then one two three up and that's going to finish off just there now I would encourage you to draw these lines in although you certainly don't have to obviously if you do draw them in just do them nice and lightly in pencil now I'm going to get rid of that and we're going to draw it in from that point now just be careful obviously we're going to redraw it in from the point that we actually move so that top position of the triangle so from there we need to go down by two again just keep looking at the triangle it goes across by one so across one and then join it up and obviously if the question says to label it with a letter or to shade it in or something like that and obviously just make sure you do that as well now let's look at the next one the next one's slightly different because it's going to move it slightly differently because this time our Vector says to move it two on the top and minus four on the bottom so two on the top would indicate a movement of two to the right and negative four on the bottom is four down I'm going to pick a different Corner this time let's go for this bottom right one so 2 to the right so that would be 1 2 and then four down so one two three four there we go and let's just put a little cross there and obviously this time then we're going to be drawing this particular triangle in from that bottom right corner so we need to draw the base in to start with up two for that triangle and again joining it up and again doing that nice and neatly with a pencil and a ruler ideally Okay so we've got a question here it says describe fully the single transformation that Maps shape P onto shape Q so we're going to go from the shape on the left of the shape on the right now the process is pretty much the same here we just need to figure out how far it's actually moved so let's match up some similar Corners so we've got this one on the top right and we'll match it up to that one there let's have a look how we get there so from P to Q so we'd have to go right one two three so that's three to the right and then one down there we go so three to the right one down and we could even put a minus one there as we know that one down is going to be a minus so in order to draw or write out our description here we are going to have to State what kind of transformation it is so we know that that is a translation so we can say it's a translational we can we can also say it's been translated so I'm going to write a translation there we go so a translation and then in order to describe the movement we give the vector so I'm going to say a translation I'm going to say by the vector or you could say using the vector or anything along those lines and then we just need to write that Vector in in our bracket so that's 3 to the right so that's positive 3 on the top and then one down so that's negative one on the bottom and there we go that'd be four marks for us on that question alternative is with this question obviously you could be asked how to get from Q to P that would have been slightly different because actually to get from Q to P we would have gone three to the left and we'd have gone one up so it would have been pretty much the same Vector but the symbols in front of those numbers would be different we'd have minus three and we'd have positive one if we were going in the other direction so there we go that's how we're going to describe a translation it's very important you include all of these words you do need to say the word translation the word moved is not going to get either marks for this because that's not that's not the transformation the transformation is called a translation and we also do need to give that vector and ideally stating that by the vector and actually then giving it okay so working out the perimeter of a circle sector now it says a is the shape of a quarter circle of radius 15 and we can see that work out an estimate for the perimeter of shape a now quite a nasty one so if we're going to start working this out for starters we've got the word estimate so we're going to use Pi is equal to 3. so Pi is equal to 3 for this particular question now when we're working out the circumference that's going to be part of the perimeter so this little Arc here which is part of the circumference of we just said is one of the numbers that we need so we need to think about our circumference formula so the circumference is equal to Pi times diameter so for this particular Circle the radius is 15 so the diameter will equal 30. so to work out the full circumference we would do the circumference is equal to Pi is 3 times 30. so the circumference is equal to 90 centimeters now this is a quarter Circle so for just the length of that Arc there we're going to want to divide that by 4. so 90 divided by 4. now you could write that as a fraction 90 over 4 or you could start to simplify it something like that but if we were to write 90 divided by 4 that would come out as well we could halve it we could get 45 divided by 2 and that is 22. 0.5 centimeters okay so not very nice there to get that particular length but that there is 22.5 centimeters now to get the perimeter we just need to add them all up so we have the 15. we've got another 15 here for the radius then we just need to add them all together of course it did say to estimate so we could have estimated 90 divided by four you could have rounded that up maybe or I mean you'd probably leave that as 90 but you could have maybe rounded 22.5 to 20 but I think I think that wasn't too bad there dividing 90 by 4 so we probably have a range of answers for this if it was an actual estimate question but we'll add these all together you've got 15 plus 15. Plus 22.5 if we add those together that's 30 plus 22.5 which is 52.5 centimeters and that'd be our final answer just there it's obviously when it comes to Circle sectors or looking at circles in general and a non-calculator paper two different types of questions that you could have you could have one like this where we're doing an estimate or you could be asked to write things in terms of Pi so if we just think about another type of question that you could have here for example if we were asked to work out the area in terms of Pi what would we do I'll leave that last line of working out there that we did for this question but if we were to work out the area in terms of Pi what would we do now we would still use our area formula so area is equal to pi r squared and for this one here obviously I'm making this up so we'll have to think about the numbers as we go along and then we'll have to divide our answer by 4. so we'll work out the areas if it was a full circle so we would do pi times 15 squared now pi times 15 squared 15 squared you'd have to work out to the side that's 225 so we'd write the answer as 225 lots of Pi that's what doing in terms of pi means we just don't press times pi so we've got 225 which would be the area of the full circle but of course this is a quarter Circle so we'd have to divide that by four and because we're leaving it in terms of Pi and because 225 doesn't divide perfectly by 4 we'll just write as a fraction over four of course that might divide by four you know if the number squared for example if it was 10 squared and that was divided by 4 100 divided by 4 quite nicely so you could have 100 divided by 4 which would give you 25 Pi but this one particular didn't divide by four so it'd be fine to leave it as a fraction of course that's not what this particular question asked but just a thought there if you had to write something in terms of pi as well so it says that A to B to C is a straight line so that tells us that these two angles next to each other have to add up to 180. so that's something we might use at some point but it says work out the size of angle and it gives us something different here it says angle BDC so to find angle BDC you go from point B to point B and then down to point C and you can see that by drawing that it has created this angle just here so that is the angle BDC and that is the one we're actually going to be looking for so again you can obviously trace your finger over the shape to find that at any point so let's think about that we've got a triangle on the right there this one just here which is a scalene triangle there are no lines on that particular one there is a line just here obviously because this triangle on the left is an isosceles but that line there doesn't necessarily relate to the triangle on the right so the triangle on the right is a scalene triangle all the angles will be different or could be different and on the left here we've got an isosceles so because we have that top angle in the isosceles we can definitely work out these two base angles again just finding the ones underneath those two little lines so at the moment there's a top angle of 70. so if we do 180 take away 70 it's going to tell us what's left to be split between the two base angles and that is adding up to 110. so if we divide that by two that will split it into our base angles and 110 divided by 2 is 55 degrees so both those base angles are 55. now that's going to allow us to move into the other shape as we mentioned at the start these two angles just here are going to add up to 180. so to find this angle just next to the 55 we would do 180 and take away 55. and that gives us an angle of and again you can do your working out for this but that gives us an angle of 125 degrees so this one here is 125 and now you can see we have enough information on this right hand triangle to find our missing angle because we now have two of the other angles so if we add those two together we've got 125 and 29 and if we add them up that adds up to 14 5 and 1 so 154 degrees we want to take that away from 180. so to finish this off our final bit of working out we'll do 180 take away 154 and again if you can do these mentally that's absolutely fine but we'll just do our written method for the purpose of this video 10 take away four is six seven take away five is two and one takeaway one is zero so that final angle there is 26 degrees and that's the final answer for our question because there we have found angle b d c okay so here is our next question it says here is a scale drawing of a garden it says we need to plant a tree five meters from point C nearer to a b than a d and less than three meters from DC shade the region where we can plant the tree so this time we've got three things to be aware of and also we are going to shade a region rather than a point now if you do want to draw this out and you want to replicate it in the same size I could just measure this rectangle for you and as you can see it's eight and a half centimeters down and if we want to go across the rectangle is going to be let's have a look around about 13 centimeters across so again you don't have to draw this but if you did want to follow along by drawing it then you can absolutely do so so we'll start with the first bit of information here where it says we need to plant a tree five meters from point C so if we're going to go from point C then again we're going to draw a circle around Point C but it does say it wants to be five meters so we need to have a look at our scale and as you can see down the bottom it says the scale is one centimeter is equal to one meter so for five meters that is going to be five centimeters so we just want to get our ruler and measure out our Compass so again putting the compass point down at the bottom of the ruler and measuring out where five centimeters is so five centimeters is only that far this time so we'll put our compass point on to C and we'll draw a nice circular Arc that shows where it has to go so there we go that is going to be five meters from point C so there we go that's our first part it now tells us that it has to be nearer to a b then to a d so A to B is the line that goes across the top of the rectangle from A to B and a to d is the line that goes downwards from a obviously from a to d so in order to do that we need to do an angle bisector because we're going to want to find the points that are closer to a b than to a d so we need to bisect the angle in half so that we can see all those points on that half now obviously this is a rectangle and not a square so you wouldn't want to just join the point a to c because that's not going to be a perfect angle bisector so obviously watching the video on angle bisectors would be very helpful for this but if you haven't watched a video on angle bisectors I'll just explain it so we go up to point a and we're going to put our compass on point a and there are two ways to draw an angle bisector I do it in one particular way so I'm going to follow the process that I did on my video so all we do is we take our compass and we're going to draw a nice Arc that joins up the two lines and we're going to draw a nice little art like that and then we just extend our Compass slightly further and we're going to draw another Arc and there we go and that's all we need our compass for so we can put our Compass out of the way now as that's enough for us to draw our angle bisector so if we move this out of the way and we'll actually now join these up so what we do at this point so we do like what's called a while about what I call it crisscross so I take the lower point of the first Arc and join it to the higher point of the second Arc and then again the other lower point on the first Arc and join it to the other higher points on the second Arc and where those lines cross over that is going to be where our angle bisector needs to go from so we go back to point a and we're going to bisect the angle and I'm going to extend that all the way to the end of the shape and there we go that has now split my rectangle into two halves in terms of closer to a b and um and close to a d obviously we need to be closer to a b so we're going to be above that line obviously closer to the B point on the top right of the rectangle now we've got one more bit of information that we need to do and that says that it needs to be less than three meters from d c so in order to get less than three centimeter or meters from DC we actually want to measure three centimeters above the line DC I've said three centimeters because three meters would be three centimeters according to our scale so if we put the ruler down the bottom of this side to start with you can see where three centimeters is and we just need to mark that up I'm just going to put a little line here where three centimeters is and then we're going to do it on the other side so going over to the other side and again just marking that three centimeters so that we know where three centimeters Up is on both sides now you would want to take your ruler and join those up nice and carefully and we have a nice straight line and again mine's not going to be absolutely perfect I probably could have done that a little bit better so it go again oh dear that's definitely not better there we go that's looking perfect so there we go there is our line and above that is going to be greater than three meters from DC and below that is going to be less than three meters from DC so now we actually want to go about shading this position on the map here or the scale drawing so we need to plant it five meters from point C and we need it to be nearer to a b than a d and it has to be less than three meters from DC so it has to be below that pink line and it has to be closer to a b than a d so if we think about where this is going to be we know that it can't be above the lines we can't be here it can't be over here it can't be over here but it has to be below that now we know that it also can't be closer to a d so any of these regions here below the Blue Line it can't be uh it can't be this little one down here either that only leaves us with two so we need to plant a tree five meters from point C so half if it has to be five meters from point C for whatever reason that it can't be close to point C it will have to be this little region here just in the middle otherwise it would be closer than 5 meters from point C and it has to be at least five meters from point C so that's the only region that it can be based on that bit of information and that would be the final answer to that question but as you can see a question like this is pretty difficult because you've got all these construction lines all over the place and you've got all these different bits of information so you just need to go through them slowly and Crossing off some of the regions can be quite helpful so obviously we've picked the easiest one to start with was which was that it had to be less than three meters from DC or below that pink line which allowed us to cross off a lot of those regions and then obviously the fact that it had to be closer to a b than a d allowed us to cross off the ones down the bottom below the blue line and only left us with two then so we just have to think about that first piece of information which said that we need to plant it five meters from point C which just meant that it had to be above or greater than that first Arc that we drew when we drew a circle around C so this question is slightly different look as we don't have an actual shape we've got three points on a map so we have the points a b and c and they kind of form this triangular shape now if you want to draw out again a sketch of this again just try and draw this out to the same sort of size that it would be on a computer screen but otherwise don't worry too much about actually getting this the exact same size it says Point T is 250 meters from point A and then it says Point T is equidistant from points B and point C on the map show one of the possible positions for this point so we've got two bits of information it's got to be 250 meters from point A so if we read the scale there it says that one centimeter is equal to 100 meters so this is going to be between two and three centimeters it's going to be 2.5 again you could divide 250 meters by a hundred and that would give you the 2.5 centimeters that it has to be from point A so if we start by drawing that we're going to get our Compass again put it at the bottom of our ruler measure out two and a half centimeters and then we're going to draw a nice circle around a and that will show us all of the points that are within that 250 meters so if we put it around that point draw a nice circle around a now we know all of the positions there that are 250 meters from point A so it has to be somewhere within this circle or on the outskirt of the circle on the circumference and that all of those points around that Circle are 250 meters from a so for the second piece of information it says that point T is equidistant from points B and C so to get it to be equal distances from points B and C we're going to have to draw a perpendicular bisector of B and C so we're just going to have to join them up to form the line which we could call BC and now we're going to want to draw a perpendicular bisector so again we'll get our Compass we'll put it on point C we'll extend it past halfway and we're going to draw a nice Arc all the way around and then we're going to move that on to point B without changing the size of the compass and again drawing a nice Arc there so that we can draw our perpendicular bisector so we can move all of this out of the way and now we're going to want to draw our line so again if I start on the right hand side so that I can extend it over past the circle then you can see that there are two points where it crosses the circumference of the circle so we want to obviously only label one of those points as it says show one of the possible positions for T and those two positions where our perpendicular bisector intersects the circle of both of our points so obviously we would only mark one of these but if we mark them both we could choose either this position here or we could choose this position here of course being very careful because it does say to mark one of the possible positions so we would only mark one of those and that would be the answer to this question okay so looking at a probability problem says that there are nine green beads and 11 blue beads in a box Jim adds 10 more beads to this box he's going to take at random a bead from the now 30 beads in the box and the probability that you will take a green bead is two-fifths how many green beads did Jim add to the Box well if you add more beads into the box and now it's out of 30. well what would this fraction be out of 30. so let's figure that out well if we want 30 on the bottom we'd have to multiply the five by six and we'd do the same to the top so 2 times 6 would equal 12. so out of the 30 beads there must be 12 green beads and pretty much that has already solved the problem just by turning that denominator into the new amount of beads in this box so if we look at this we've currently got nine green beads we know that it has to go to 12. so we must have added three additional green beads into the box okay so we'll just double check our thought process there so it's out with 30. our probability was at a five so to make that out of 30 we've multiplied by six so the numerator goes to 12. there were nine green beads that's gone up to 12. so that's an extra three beads so three green beads okay and there we go there are obviously other ways of working that out as well like you could add an amount basically trial and error you could add one bead in one green bead in see if that same simplifies down to two-fifths if the rest were blue if that doesn't work add two green beads in add the rest of blue see if it simplifies I think this was definitely the fastest way here just looking at that denominator making sure it matched how many beads were going to be in the Box in total so there we go there's a little probability problem now you obviously need to be aware of how to use Venn diagrams as well when we are looking at inputting numbers from a given set of lists and I'll link that in the description for you to check out as well but in this one we're going to have a look at these more problem solving style based questions so the one you can see on the screen we're going to have a little read through it says 50 students were asked if they have a cat or a dog 22 have a cat 18 have a dog eight have both the cat and a dog use this information to complete the Venn diagram now sometimes you'll be given a Venn diagram sometimes you'll have to draw it for yourself for the purpose of these videos I've put them all in but just be aware that you could have to draw them so obviously if we have a Venn diagram we need to label it up to start with now this question mentions cat first so I'm going to put the right as the other one and the left one is the one that's mentioned first so I'm going to put cats here and I'm going to put dog over there again it doesn't matter what order you put it in just make sure that it's clearly labeled so looking at this question it first says to us that 22 have a cat now the problem with a Venn diagram is we can't just put 22 in this cat section because there are some people that also have a dog so some of them are going to have to go in there so don't just put the first number in straight away instead let's read through the entire question and see what information we have been given that could be put straight in so it then says that 18 have a dog again we can't put that straight in if it did want us to put it straight in it would say only before the word dogs that would say 18 people have only a dog there we go we can't put that in so moving on to the next bit it says eight have both a cat and a dog so that bit of information can go in because we know that eight that have a cat and a dog would have to go into the center so straight away in the middle we can put that eight have both it doesn't tell us how many have neither so that is the unknown piece of information we're going to have to work out now we can move back to that first line as it says that 22 have a cat now you can already see that part of those 22 over here we've got eight that will have a cat and a dog but those eight people are included in the 22 people that have a cat so to work out how many people have only a cat we would have to take away the eight in the middle from the 22. so do your working out to the side I would write 22 take away eight and that is going to give us a total of 14 that have a cat and now you can check because these two numbers should add up to 22. so 14 and 8 do they add up to 22 the answer is yes so we've got that correct we can then move on to the 18 that have a dog now of course we've still got the eight in the middle so we would take away the eight in the middle that have both away from the 18 and that leaves us with a total of ten so 10 people have only a dog and now we've got quite a lot of information there but we need to figure out what goes on the outside now if we add all of those up so 14 plus the 8 plus the 10 that adds up to 32. so we have 32 people yeah to set up to 32 yep I hope we've got 32 people so we need to figure out what goes on the outside so at the moment we've got 32 to make that up to 50 we would have an additional 18 people that would go on the outside that have neither a cat nor a dog okay so that's one type of question that you can have where we've been told what goes in the middle in the crossover section in the intersection so let's have a look at one that's slightly different and and to be honest before we start we need to have just have a look at this little symbol here this just means the universal set it just means all the numbers that are going to go inside this Venn diagram somewhere in there and in in that it says all the odd numbers less than 30. so it might just give you the list of numbers if it doesn't it's probably best just to write them all out and there's quite a lot of odd numbers between um 1 and 30 less than 30. so I'm not going to write them all out but you could just go ahead and write them all out one three in fact let's just write them out one three five seven nine eleven thirteen fifteen 17 19 21 and the rest now these are all the numbers that are going to go in the Venn diagram and they're only going to go in there once okay so there's not going to be any repeats of these numbers but they are all going to go in the Venn diagram now it says set a has got these numbers and set B's got these numbers now if it doesn't label the circles in the Venn diagram just label them yourself so call this a and this one b so having a look now the first thing we want to identify is what can go in the crossover the intersection there so if we have a look in a and b there is a five in both so let's cross off the five and let's put that into the Venn diagram and I'm also going to cross that off my main list up here as well cross that off so we're not going to put it into ice let's have a look there's also 15 in both so let's cross that off and we'll put that in the middle as well and then we'll cross that off the main list so these are all the numbers we've got left to put in a Venn diagram there's there's none that are in amb anymore so all those numbers are in set a I'm just going to put them in the a circle so 9 17 21 and 29 we'll cross those all off and cross them off the main list so nine 17 21 and 29 there we go they're gone and then 25 in b as well so we'll put that one in there again Crossing off both lists so in terms of the numbers we have left we've got um the one we've got the three and I'm not going to circle them all I'm just going to start putting them in because the others are crossed off we've got 7 11 13 19 23 and 27 there we go and that is our completed Venn diagram now I could be asking any sort of question on probability here but I've put all up a little selection in down the bottom it says a number is chosen at random find the probability that is in set and then we've got three questions to have a look at so we've got two symbols that we need to know here we've got the U which is the Union there we go the union we'll talk about that in a sec and we've got the N which is the intersection there we go and these little symbols can be replaced with some words if it helps so the U for Union I replace with the word or and the end for the intersection are replaced with the word and I quite like replacing it with the word and because the middle lettering and is n so I always tend to remember the intersection there the N is and so let's have a look the first one is U so we've got a or b so the numbers are in a or b and it is a probability so we'll talk about writing it as a probability in a sec but if we have a look at the numbers they're in a or b well we've got the 9 the 17 the 21 the 29 which is in a we've got the 25 which is in B and then we've also got the 5 and the 15 which is in a and b so just because that's in both doesn't mean it's not an A or B so in fact finding the numbers that are in a or b is all of these numbers within the Venn diagram okay so they are all the numbers that are within the the Venn diagram then if we count them up there's one two three four five six seven numbers there I'm gonna get rid of that highlighter there but we've got seven numbers let's write that down I've got seven numbers that are in a or b out of a total and again you can count them all from the Venn diagram or from that list that I did at the top but all the odd numbers between 1 and 30. there are one two three four five six seven eight nine ten eleven twelve thirteen forty fifteen numbers there you go so it's 7 out of 15. so that word Union if you think about that as like the United Kingdom the United Kingdom the union is the collection of all the countries there and it's the same for the Venn diagram it's the collection of everything within that sort of uh that land there if you kind of imagine it like a country you've got the collection of all the numbers so you've got 7 over 15 and that is the union of A and B now if we have a look at the next one so B we've got a n b or the intersection of A and B which again can be replaced with the word and so the numbers are in a and b well there are only those two numbers that are in both A and B the 5 and the fifteen so that there is the intersection the numbers are in a and b and if we write down a probability of that that's 2 numbers out of again 15 numbers there you go so there's our two options there for the union or the intersection I've also included this extra one here as well for C which is a dash now if there's a dash with a letter it means not in a so all that little Dash means is any of the numbers that are not in a now if you have a look at the numbers that are not in a well we've got the 25 which is in B that's not in a we've got all these numbers around the outside that are not in a but none of the other numbers so we've got the numbers which are just in a the 9 the 17 the 21 the 29 and we've also got the numbers are in a and b the 5 and the 15 but they are also in a so we can't have those ones so we're just looking at the numbers that are in B and numbers on the outside so we've got one number in b and one two three four five six seven eight numbers there in green on the outside eight plus one is nine so there are nine numbers that are not in a and again that's out of 15 writing as a probability so it says some people went on an activity course and each student had to choose one activity from art drama or music and then it starts to give us all this information that I'm gonna read through in a little bit it says one of the students has chosen at random write down the probability that it is a boy that chose music and we'll discuss that a little bit more at the end now when it comes to this sort of question here and I said already we're gonna have a look at two-way tables we need to get this information organized okay there's so much writing there there's so much going on and a two-way table in this circumstance here is going to make this a lot more well add a lot more clarity to this question so if we have a look okay it says that they uh choose from art drama or music and then later on it goes to talk about whether they are boys or girls and what they chose so when it comes to this type of question all we need to do is draw a little table I'm going to draw one as best as I can here there we go so I'm going to have a few rows there so that I can have boys which I'm just going to label as B girls which I can label as G and I'm going to put a total down here as well there we go and then obviously they chose from three different activities so we've got art which I'll put first we've got drama there we go and we've got music there we are so there are my columns okay this up and this information in and again I'm going to put a total column at the end there right there we go just extend that a little bit okay so we can start putting all these pieces in the right boxes okay and obviously your table is just a way of organizing it so you don't have to worry about whether you put the boys and girls down the left or whether you put art drama or music down the left there okay you can organize it however you like but we've got 41 students in total it says I'm just going to go through this line by line and see what I can put into there are 41 students so I'll stick that in that's 41 in total of the boys and girls in the bottom right there it says 15 of the students chose music that's the total that's not necessarily boys or girls it's just 15 of the students that chose music so 15 down here then we have 30 students or girls Okay so we've got that's 30 in total that were girls there we go 30. the next one we've got eight of the girls chose art so in the art column eight of the real girls chose art so let's have a look that's eight girls there then we've got no boys chose art as our next line there so no boys choose art so I'm not going to leave that blank I'm going to put zero in there and then it says equal number of boys and girls choose drama and we don't know how many chose drama just yet so we'll have to come back to that one in a little bit right okay so if we start adding in some numbers then so down that art column there and essentially we need to find any columns or rows that have one Gap in them so that art one there stands out to me the zero boys and eight girls and that's eight in total so that's eight there we've also got the total amount of boys we know there's 41 people in total and 30 girls that's going to be 11 boys there and then we've got the bottom row as well we've got a missing Gap to the amount that the drama says 8 and 15 here at the moment in total that is 23 and it's got to go all the way up to 41 to get the total at the end there so that is an extra seven to get to 30 plus the 11 is an extra 18. right there we go so now we can actually go back to that last piece of information now it says equal number of boys and girls chose drum because now we know that there are 18 people that chose drama so equal numbers half that Nine and Nine and then we can go about filling in those last to as well on the top row for the boys we've got nine in there so far we need to get a total of 11 so that's an extra two and then moving downwards okay if I look down for the last box here it's got a total 15 so that's going to be have to be 13. now as that's the last box that I did one thing that I do want to check is that these still add up to 30 on the end there so always double check the other way on the last box that you put in so eight and nine makes 17. add the 13 is 13. there we go and that's just a little check for you just saying that you know you've done it all correctly but there we go that is my two-way table drawn and I'll just need to have a look at this question so the question down the bottom says one of the students is chosen so one of the students is chosen okay there we go write down the probability that it is a boy that chose music okay so it's saying one of the students is chosen so obviously there are 41 students so it's out of 41. and it says what's the probability it's a boy that chose music well if we have a look in our table boys that chose music there are two of them so that means that the probability of picking a boy that shows music out of the 41 is 2 out of 41 okay we're going to have a look at another question and say well the language is slightly different but in this question the language said one of the students is chosen it didn't dictate that we were just picking from the boys or just picking from the girls it just said one of the students so as there were 41 in total that's out of 41 and a boy that chose music there are two of them so two out of 41. so it says here there are 10 boys and 20 girls in a class and a class has a test says the Mean Mark for all the classes is 60 and the Mean Mark for the girls is 54. work out the mean for the boys so when it comes to a question like this obviously we know well hopefully we know that when we're normally working at a mean we get the total we divide by how many people there are or how many things there are and we get our mean okay so we do the total and we divide by the amounts okay for whatever it is there we are on its most basic sense there with the mean using the total divided by the amount well in this question here obviously it doesn't actually give us the total it tells us what the mean is it tells us the amount it tells us that there are 10 boys and 20 girls but it doesn't actually tell us the total so what we're going to do is we're going to go backwards to work out some of these totals okay so we're going to think about this in reverse all right and that is how you get the mean there the total divided by the amount okay so if we link some of these things up then it says that there are 10 boys it doesn't give us the mean for the boys so that's no good so far so that first piece of information that we are actually given in terms of the mean it says the Mean Mark for the whole class is 60. well if we were going to work out the mean for the whole class we'd get their total let's call that X we divide it by the amount and in this case the whole class well there's 10 boys and 20 girls so we would divide it by 30 and it would give us this mean here which it says is 60. so in other words we're looking for a number that divides by 30 that gives us the answer 60 and we can just do that in Reverse we can just do 60 times 30. so actually all we're going to do in these questions is we're going to link the mean that we're given with the amount that is appropriate for that mean so I'm going to get rid of this little bit of algebra that I've drawn here but obviously you could think of it in terms of an equation but this is the sort of concept that we're going to run throughout this question and hopefully once you actually get your head around this and why we multiply instead of divide obviously we are doing a reverse this question obviously becomes a lot easier then okay so obviously think about this Topic's actual name it's called a reverse mean normally when you work out the mean you do a divide and obviously then the opposite or the reverse of dividing is multiplying and in these questions we're going to be multiplying rather than dividing to go backwards let me get rid of that so let's have a look so obviously you've been given that the mean for the whole class is 60 and we've just established that there's 30 students or 30 people in the whole class so we're going to do 60. Times by those 30 students and that's going to tell us their total so 6 times 3 is 18 add on the two notes that's 1 800 and that is the total for the whole class and it's usually worth just labeling these particularly if you haven't quite got your head around it let's just label that total for class there we go right so the next bit of information then we've also been told that the Mean Mark for the girls is 54. now in terms of the question look there are 20 girls so if there's 20 girls and their mean is 54 we'll take their mean 54 times it back by that 20 to find out what their total was so 54 times 20. there we go and that gives us 1080. and there we go there is the total for the girls and let's just label that again total for girls right so just thinking logically about the information that we've got here we've got a class of boys and girls it says the total for the classes of 1800 the total for the girls is 1080. so logically we can work out now the total for the boys we can subtract those away from each other so 1800 take away 1080 and let's write that down we go take away 1080 leaves us with 720. there we go and that's the total for the boys let's label that up total for boys and now we can go about answering the question because now we know the total for the boys we can actually think about working out their mean and that's what the question wanted it said work out the mean for the boys so we've got the total for the boys of 720 and in total up here we were given in the question let's highlighting the same color there are 10 boys so to work out the mean just like normal we would just finish this off by saying the total for the boys is 720. I'm going to divide that by how many boys there are which is 10 and that gives us a mean of 72 there we go and there's our final answer I'm just think about what we did there just working backwards through the question multiplying the means by the amount get the totals and then use a little bit of logic there to find out therefore what the total for the boys must have been and then just working out the mean as normal okay so looking at outlier now an outlier can kind of appear anywhere but we tend to look at it when it comes upon scatter graphs so it says here Amy records number of children visiting a local paddling pool each day the information for one of these days is an outlier on the scatter graph now hopefully when you look at it you can see which one the outlier is it's one that lies outside of the usual trend of points so if we look at this one you can see that this point just here is outside of the usual all the usual points which would normally follow this kind of line the best fit that we could draw in just there obviously I've just draw a little sketching so that is the point which is the outlier so we could be asked a couple of things here it could be asked to write the coordinate highlight it anything like that if we're going to give the coordinate we've got to be very careful and just read that very carefully so that is 24 so if I write that over here be 24 on the X axes and if we go across it doesn't land on a very nice number by the looks of things 12.5 it's in between 10 and 15. so again you've got to read that very carefully so using a ruler and a pencil just to trace that across but I'm happy with that 24 and 12.5 so the other thing that we could be asked is to give a possible reason for this outlier and when we're asked about this we're trying to think about something logical we're trying to think about something in real life so we're not trying to come up with anything silly for this so it says here what's a possible reason well let's have a look at the information so why on a day where it was 24 degrees Celsius where there are a lot more children you can see up here there's another Point around 24 degrees and on that day it's like 42.5 so an extra 30 children so why are there 30 less children on this particular day even though it's hot so let's think about a real life scenario when other days where children are probably going to be going to the paddling pool probably at the weekend so why might there be 30 less just here or maybe it was a weekday okay so that is a pretty logical explanation there's lots that you can come up with but just try and think about one that just sounds ridiculously logical okay so give a possible reason okay it was a weekday not a weekend something like that it was a weekday or you could say something like maybe it was raining okay but yeah it's hot but it could be raining as well people are probably aren't going to want to go out while it's raining okay so lots of different things that you could think about there the idea is as long as it sounds reasonable it's probably okay but you're not trying to come up with anything silly or out of the box thinking here just a real logical reason as to why that might happen okay if you're still with me congratulations you've made it to the end of this revision session there was a lot that we've just gone through there a lot of topics but hopefully a nice slow pace and hopefully as you were going through that you've picked up some topics that maybe you might not have got tomorrow had we have not gone through it so hopefully that was useful and helpful and you have picked up some decent questions there that hopefully come up in this exam don't forget to leave me a comment and let me know how you get on and genuinely I really do wish you the best of luck you've got this go and smash that exam but otherwise I will see you on the next one [Music] over it [Music]