Transcript for:
GCSE Math Crossover Content Overview

okay welcome to this video where we're going to be having a look through a question on every single topic that appears on the GCSE now in this specific video we are going to be focusing on looking at the crossover content now the crossover content is the content which appears on both the foundation and the higher paper now the video that I'm about to go through you can download all of the questions in a practice booklet that you can work through I'm just going to show you where you can find it so if you head to the website and the link for this will be in the description you can head into the menu bar and it's in the free sources section now the one we're going to be having a look at is the revision guide but there are loads of rese sources that you can download here for free on the website so you've got the practice papers there are practice booklets there's the ultimate revision guide and the revision checklist now we're going to be looking through the ultimate revision guide as I said we are going to be looking at the crossover content so this one if you are going to download it is labeled as the foundation guide It's the foundation guide because it's just the content that can appear on Foundation but also on the higher tier there is also additional content that you will have to cover so in this particular video we are going to be having a look at all of the content that you need to get to a grade five so on the website all you need to do is click on get the free guide you can then click onto the foundation option and that will download the Rion guide for this particular video if I show you what the revision guide looks like we'll just have a quick look through that so once you have downloaded the guide you will be able to scroll through you can obviously print this off you can have it on paper or you can have it digitally however you prefer and you can see that we're going going to be going through all the units that are broken down here in the guide so you can actually click on these if you're using the digital version we can click through let's go through to angles polygons and parallel lines in unit six so we can click into that and it's going to take us straight down to this particular unit so you can go through you've got the questions you have got some answers there upside down and you've also got a QR code there that you can scan and it will take you to a full lesson on that particular video so you can work through the questions if you want in the booklet or if you prefer you can work through them with me on the video so at any point you can obviously check back to the video to go through any of these if you're not exactly sure on how to answer any of them but there are tons of questions for you to answer with one on every single topic that can appear so for this one obviously bear in mind that as we go through the questions these are just one variation of each particular topic so for example if we look at this question here which is volume of a prism well there are different prisms that you can look at there are different prisms that you can work out the volume of you could have a triangular prism and as I go through the video I do discuss all of these points with each particular question question but you obviously need to be aware that these are only one variation another good example here we have drawing linear graphs the equation there is y is equal to 3x - 2 but of course you could be given a graph which is y is equal to 2 - 3x and that would be treated slightly differently as well so if you have the guide printed off of course if you want to check any other variations of the questions you can scan the QR code and it will take you straight through if you have the digital version you can just click on that QR code and it'll say straight to the video as well so any point you can click this button here it'll take you back to the start and you can just go through the different units as you see fit to help with your revision now this particular video is going to be quite a long one because I go into the questions in quite a lot of depth but if you want a quicker version or for whatever reason you need a quicker video to go through you can always head to the video at the top of my channel there which is everything you need to pass your GCSE maths exam that one there is a lot quicker I do go through the questions a lot faster than I'm going to in this video because I'm going to go into quite a lot of depth for each topic but if you do want to access a faster video that is the video for you but otherwise we are going to be going through all of the questions all of the topics and we will go into quite a bit of depth on each particular topic now before we get started there are lots of resources that you can access on the website if you head into the menu you can obviously have a look around there is now the grade check feature as well so you can go into here you can answer 20 Questions dependent on your exam tier and it will give you a predicted grade in terms of a percentage that you get correct across this selection of questions so that is definitely something worth having a go at it won't take you too long and that is really useful just for giving you a general idea of where you're at and some topics that you might need to work on if you would like to join one of my courses as well you can sign up to either upgrade or tgmt live so if you head to the website you can have a read for either one tgmt upgrade is the OnDemand course where you can access everything that you need that matches everything in the revision guide you can access full lessons you can do bespoke quizzes for each of those lessons for each of the units you can get reports generated that tell you what you need to answer it's an all-in-one package there just to have a look at everything you could possibly need to you can also join tgmt live where you can join me once a week going through lessons together where you can interact with me as well and we go through a bespoke uh schema work where we'll be looking at all the topics that you need to make sure that you are happy going into that maths exam so there's a couple of things to have a look there both of them you can have a look and have a free trial so it's worth going in having a go seeing if you like them picking which ones for you and if neither of those options are for you you've got all of the free sources you've got the grade Checker there's so much for you to have have a look and explore on the website so do make sure you click the link in the description have a look and see if any of those are going to be useful for you moving forward so otherwise we're going to get started with the video so let's get [Music] started okay so looking at our first question we have multiplying decimals so if we have to multiply decimals the first thing that you want to do is to remove those decimals out of those numbers so we've got one jump out of the first number and another jump out of that second number so a total of two jumps we can then do 546 multiplied by 43 so essentially just taking those decimals out and treating them as whole numbers from from there you can use whatever method of multiplication that you are comfortable with I'm just going to use column multiplication so 3 * 6 is 18 3 * 4 is 12 + the 1 is 13 and then 3 * 5 is 15 + the 1 is 16 put my placeholder in 4 * 6 is 24 4 * 4 is 16 plus the 2 is 18 and then 4 * 5 is 20 plus the 1 is 21 from there we'll add these together as normal so 8 7 14 carry the one 3 and 2 so we get an answer 23,4 78 but of course we know that these were decimals we need to put those two decimal places back in so one two and we would get a final answer of 23478 and of course a good method just to make sure you put the those decimals in the correct place is to imagine what the answer would be if you were to estimate this so if we were to do an estimation we would round 54.6 to 50 and we would round 4.3 to 4 and 50 * 4 gives you an estimate of 200 so we know our answer is going to be relatively close to 200 that's going to help us with that decimal placement as well okay so looking at product of prime factors so for this we are going to write the prime number numbers that multiply to make 56 and I do that using a tree so for 56 we just need to think of any two numbers that multiply to make 56 one option that you could go for is 8 * 7 that's quite a nice one it does giv you give you a prime number straight away if the number's Prime we'll just circle that number so we know we're finished on that part of the tree and we'll continue on the eight so 8 is 2 multiplied by four again two is a prime number so we will Circle that and because we've ended up with four that is just 2 * 2 and again they are both primes so we have got an answer here we just need to put a multiplication symbol between all of those numbers so to get to 56 you would do 2 * 2 * 2 and then as well times 7 now you can write your answer in index form so you can use do write this using powers so 2 * 2 * 2 you can simplify that and write it as 2 cubed and then obviously multiply by the seven so either of those answers is absolutely fine you wouldn't have to write one or the other unless it's specified in the question that you have to write your answer in index form or it's specifically asked for you to write it with a power of two so there we go two possible answers that you could write for that one for this question we're going to be have a highest common factor and it says find the highest common factor or in Brackets there hcf of 84 and 180 now you can have some pretty horrible numbers here in order to find the factors of so we need to think of a method that might help us to find the factors and I would always write it in a table so 84 and 180 now completely up to you but one of the numbers is normally slightly easier to find the factors of now in this particular example I don't think necessarily either of these are easier 84 isn't the nicest of numbers to be finding factors of but you just need to look at the two and have a think about which one you prefer now if we go for the 84 I'd always write my factors in factor pairs so 1 * 84 2 * 42 and then I need to try and figure out the other numbers that maybe don't quite as obviously fit into 84 so without a calculator here because this could be either type of question it could be with or without a calculator if we do have a calculator later we just do 84 divided by some numbers and see if they fit in but we want to follow a nice logical strategy so the next number I would want to test is if it divides by three so I'm going to do that to the side three goes into eight twice remainder two and it goes into 24 eight times so three does fit in it fits in 28 times so 3 * 28 I'm pretty confident that four fits in if you don't know again how many times we can do this to the side four fits into eight twice tce and into four once so it goes in 21 times I know that five doesn't fit in I'm not overly sure about six so we should probably test that as well so does six go into 84 well it goes into eight once remainder two and it goes into 24 14 times so yes 6 * 14 seven does seven fit in let's do it to the side so 7 into 84 seven goes in once remainder one and it goes into 14 twice so seven goes in 12 times I know that eight doesn't because eight would go to 80 and then to 88 I know that nine doesn't because nine goes up to 81 in the nine times table I know that 10 doesn't I know that 11 doesn't and then I'm back to the 12 on the other side so as you can see that is possibly one of the worst type of numbers that we can have just because the sheer amount of factors that are in that number but now we have all the factors for one number I just want to test to see if any of them fit into 180 see and I can kind of do a process of elimination here I know that 84 doesn't I'm pretty confident that 42 doesn't although you might want to test that if you're not 100% sure 28 it it possibly could so the only option it could be is if it was 28 multiplied by five potentially that might give you a zero at the end because obviously that number on the right there has a zero on the end 180 so if you're not 100% sure you could obviously test it we could do 28 multiplied by 5 to the side 5 * 8 is 40 carry the four 5 * 2 is 10 plus the 4 is 40 14 there well four but obviously adding the one adding the four on 10 adding the four on so it makes 140 so 28 doesn't fit into 180 so we can cross that off 21 definitely won't if you multiply it by five or 10 so we can cross that off 14 isn't going to either 12 potentially might though let's see of 12 fits in now a couple of different ways that you could test this it's going to have to be a multiple of five that we multiply it by if it's going to end in a zero or we could just do bus stop division so I could do 180 divided by 12 12 fits into 18 once remainder six and then it fits into 60 five times so yes it does 8 180 is 12 multiplied by 15 so the highest common factor is 12 and there we go okay so for lowest common multiple I take a very similar approach to how I uh present it when I'm doing highest common factor except obviously with multiples we are looking at the times tables of these numbers so for the lowest common multiple we want to write out the times tables for 40 and 56 and we will find the lowest number that appears in both lists now because the lists can get quite long I tend to do these going across the page just to give myself some more space to write out as many multiples as needed and of course we want to look at the numbers and think about which one is obviously going to be Spott for when uh they are a common multiple so what I mean by that is 40 it's quite nice to spot if another number um can divide by 40 because it ends in a zero so if I go with the 56 if I add another 56 we get 112 if I add another 56 we get 160 eight and obviously then take your time moving between for the next multiple so if I add six we'd get 174 and add the 50 would get us to 224 now as you can see none of the numbers have ended in a zero yet but if we add another 56 now we do get to 280 and this is what I mean by easily Spott we can sort of see that 280 has a good possibility that it will divide by 40 and if you know um sort of going up in 40s in a certain amount you could sort of say to yourself well we know it's going to get to 200 and then we know that's going to go 240 and then 280 and there we go we get our link there we get our lowest common multiple and our final answer would be 280 okay so looking at some laws of indices now there are a few rules that we need to remember as long as the base numbers are the same we can add powers when we are multiplying and we can subtract Powers when we are dividing so when it is presented to you like this where there is a fraction involved we want to tidy up either the top or the bottom before thinking about that division indicated by the fraction line so on the top they are being multiplied which means that we can add together those Powers so the powers that we have are 7 and -2 and we're going to add those together not forgetting that when you add a negative that is the same as just doing a takeaway so that is going to equal five as our power on the top so if we tidy this up we have 3 to the^ of 5 and that is all over or divided by 3 to the^ of 3 when you are dividing you can subtract the powers so here we've got five take away three and that would give us three to the power of two now a question like this could ask you to write your answer as a a power of three and if it did do that our answer would be 3 to ^ of two but this question here actually says to work out the value of now because it says to work out the value we would give our answer as a number not forgetting that 3 to the^ of two means three multiplied by three or multiplied by itself and that answer is nine so our final answer for that one would be nine so there are a few other types of laws of indices that you can have as well here is one example where you have a fractional power now when you have a fractional power the number on the denominator which in this case is a two represents a root so in the case of a two that represents a square root so we would have a square root for a power of a 1 12 if we had a power of a third that would represent a cube root but this one here just represents a square root so the square root of 36 well the square root of 3 6 definitely the most common one that we would give would be six of course when you square root a number you can have the negative version as well so you could say -6 but here we'll go for the most obvious answer the square < TK of 36 is 6 for Part B here it says to write down the value of 23 to the power of0 now the power of zero that occurs when something has been divided by itself so for example if we had something like 23 to the power of 4 / 23 also to the power of four you would subtract those powers and that would equal 23 to the power of0 so that's a scenario where a power of zero can occur and we know that when you divide anything by itself you get the answer one so anything to the power of zero is just the answer one and that is our final answer so when we are simplifying expressions there are a few different types of Expressions that we can have to simplify or collect like terms or multiply and there's a good variety here to look at so not forgetting when you are given a letter in an expression if there is no number in front of that letter you can put a one in there if it helps you just to collect them up so here we have 5f takeway 1 F now just doing that part to start with five takeway one would give you 4f we then have to add an additional 2f and together that would give us 6f and there we go that's our final answer not forgetting when we simplify this we are looking to fully simplify it so writing 4 F + 2f and leaving our answer just like that would not be fully simplified we are looking to get rid of as many symbols as we possibly can and in this type of question they can all disappear just leaving us with 6f now for the next question we are asked to multiply some pieces together and this is a good one because we have a number here and a number at the end so when you are multiplying here like this we can multiply them in any order we can do 2 * 8 to start with which would give us 16 we then have two different letters so we have an m and we have an N now of course if those letters were the same so if it was an M times another M well that together would give us M squar but this is an M multiplied by an n and M multiplied by n we can't make any expression with it we just write those letters m n next to each other of course when you don't write any symbols between them the symbol which is actually in there is a multiplication sign so this does mean 16 * m * n which was exactly how the question was given to us except we have simplified the two and the eight and written that as 16 so our final answer is 16 MN for part C here we are asked to add together t^2 and another T s now a lot of the time people will give the answer T to the^ of 4 but that would only happen if it was t² multiplied by t^2 as we know that you add the powers when multiplying now of course here we are not multiplying we are adding them together so this is 1 t^2 plus another 1 t^2 and in total that is two lots of t² so our final answer for that one is 2 t^2 Okay so we've got two different different types of brackets to expand here one with a letter on the outside and one without one so when you're expanding brackets it just means to multiply whatever is inside the bracket by whatever is outside the bracket so in the first question here we have five lots of and then 2 m minus 3 so we multiply them separately five lots of 2 m would give us 10 m five lots of three would give us 15 and we'll just keep the subtraction symbol there so here you can think of it as 5 * -3 which is -15 then of course you don't have to worry about putting the symbol back in because you get the -15 anyway when we have a letter on the outside you need to be a little bit more careful because it tends to be that the same letter is inside the bracket like in this case so we'll start the same 2x * 3 or 3 * 2x however you prefer to read it and that is 6X again the symbol is going to stay the same so I can put that in to start with and then I have to do 2X x * X now in this case don't forget it is 2x * 1X even though we don't write the One 2 * 1 will give us two and the X multiplying by another X as we talked about on the last question will increase the power and we'll get X to the^ of two so there we go not forgetting when you have similar letters multiplying by each other you will increase the power as the power of the X's there are both one and those Powers get added together together to make a power of two okay so we can have some slightly more complex brackets as well where we have two sets of single brackets some we're being asked to add or take them away from one another now it doesn't matter on the scenario we're still going to want to expand these brackets as normal so I'm going to start by expanding the first bracket 5 * P would give us 5 p and then we also have 5 * 3 which is equal to 15 and that is A+ three so + 15 for the next bracket you have two different ways of doing this you can either expand the bracket or multiply the bracket by two and then take them away from one another or you can actually save yourself a bit of time and just use this number as NE -2 so to start with we would do -2 * 1 which is -2 and then we would have -2 * -2 p and you can write that to the side if you if you sort of prefer to look at it like this so -2 * -2 P well we know a negative multipli by another negative is going to make a positive number so we can put the Plus in and 2 * 2 p will equal 4 P now of course if you didn't take that approach multiplying by -2 at the start you would have got -4 p and then you would have to remember that you are taking away -4 p and that would then change the sign to an add so be very careful with these ones because particular part in the middle there is what can make a lot of mistakes uh happen so just be very very careful when particularly when these are a takeaway in the middle so now we can collect like terms as it does say to uh to simplify as well so if we look at the like terms doesn't matter which one we start with I'm going to go with the 5 p and the 4 p as the PS are written first and that would give us nine p in total and then I'm going to look at this + 15 and a take over away two and 15 takeway two would give us positive3 so we go our final answer would be 9 p + 13 when we are factorizing expressions this is the reverse essentially of expanding a bracket so in these particular questions when factorizing we're looking to put them back into a bracket by taking out a factor now if the question just says to factorize we only have to spot any Factor but typically if it's says factorize it's because there's only one factor where as you can see in Part B it says to factorize fully which normally indicates there's more things that can go on the outside of the bracket so in the first question five and 10 well the only number that goes into five and 10 other than one is five so five would go outside the bracket and then we just have to think to ourselves what do we times five by to get this five well that would be one and keeping the symbol the same so take away and what do we times 5 by to get to 10 m we're going to have to multiply by two to get to 10 and in order to make sure that that m is there the two will have to have an N with it there we go we can close the bracket and that is factorized and we have taken out a factor of five for the question below we have a little bit more to think about because we have two and six which we're going to find a factor or a numerical factor and then we also have the A's and the B's I'm going to think about whether they can go outside the bracket as well now you could factorize this by just taking out a factor of two which is a number that goes into two and six and if you did do that you would get a s b plus and then to get to 6 a^ 2 you would get 3 a^ 2 and that is factorized but it's not factorized fully because you can also take out A's and B's so let's think about how much we can take out Al together we can definitely take the two out there's an a squ here and just 1 a there so we can definitely take out 1 a and then we have a b here and a b s here so we can definitely take out 1 B so if we have 2 a on the outside of the bracket what would go inside to make these well that says 2 a 2 B on the first one we already have the two we've got the B we just don't have a squ so we' have to put another a in there to make sure the a turns into an a squ we'll keep the symbol the same Plus and now we'll look at the 6 ab^ 2 to get to six we are going to need a three otherwise that two won't turn into six we already have an a outside the bracket and it's an A in that particular piece there with the 6ab 2qu and we don't have a b^ squ we just have a B on the outside of the bracket so we'll have to put another B in there to make sure that when we multiply that out it turns into a b^ squ and there we go that's our final answer we obviously don't want this one on here although normally you do get more marks for these factorized fully questions so taking out any factor is going to get you a mark but obviously we want to secure all the marks and take out all the possible factors that we can okay so for some substitution so essentially we are just going to replace the letters with the numbers now obviously thinking carefully about what that expression or that formula up there actually means it means 7 * R plus 3 * Q or seven lots of r R plus three lots of Q and that's what p is equal to so to find the value of P when r equal 5 and Q is equal to -4 we're going to want to do seven lots of R which is five so seven lots of five add to that we are going to want three lots of Q which is -4 so 3 * 4 we can now look at each of these separately so 7 * 5 and 3 * -4 well 7 * 5 is 35 3 * -4 is -12 so we have 35 add -12 and again we've got the add a negative symbol next to each other so that is just a takeaway so we have 35 takeway 12 and that would be 23 so our final answer the value of P would be 23 so here we've got some algebraic laws of indices quite similar to ones that we've looked at before we have the same base in part A and those being an m and here we have a power of three and a power of four so if we are multiplying we can add together the powers so that would be m to the power of s for Part B we are actually cubing everything inside the bracket now when you have that going on you could write this out in the longest way you possibly could which is 5 NP cubed multiply by itself S three times so NP cubed time another 5 NP cubed now we know you could do 5 * 5 * 5 which is essentially 5 cubed you could do n * n * n which again is n cubed and then P Cub * P Cub * P cubed which again you can add together all the powers so that is a long way of writing it out particularly if the power outside the bracket was a four or a five you'd start to have quite a lot written out so all you need to actually do is Cube the pieces individually so we have five which we're going to cube we have n which we are going to Cube and then we have P cubed which we are going to cube so the quick way of doing this obviously thinking about the fact that if in the long way we would literally just add all of these numbers together that is essentially just multiplying those Powers together so P cubed if we do the powers it' just be 3 * 3 which would make p to the power of n now you can think about all of these as already having a power of one in those earlier numbers and again we have just multiplied those by three so 5 to the^ of 1 * 3 n the^ of 1 * 3 and P to the^ of 3 * 3 we obviously need to simplify this a little bit because we do have a number here which can be written as an actual uh value rather than 5 to the^ of 3 so we'd want to work that out 5 * 5 is 25 * 5 again is 100 125 so we get 125 n the^ of 3 p to the power of 9 and that there is our final answer now the one on the bottom I think this one's possibly a little bit easier because we have a division so numbers will always just divide as normal and the powers we will subtract when doing a division the bit we need to watch out for for is that r on the bottom that doesn't have a power written with it so we can put a power of one in there just to remind ourselves to make sure we actually do that subtraction so 32 / 4 to start with well that is equal to 8 we then have the q's we have a power of n and a power of three and we can subtract those so Q to the power of six for the final Powers we have R to the^ of 4 and R to the^ of 1 and subtracting those gives us R to the power of three so there we go three answers where we were adding Powers we had some multiplication of powers and we've got some subtraction of powers in the last one okay so making a subject of a formula it says here to make t the subject of the formula and currently W is the subject of the formula so what we want to do is have a look at what is next to the T and how we would do the inverse of that to move everything over to the other side so at the moment T is being multiplied by three and then we are adding 11 so first first of all we can get rid of the add 11 we can take away 11 from both sides if we do that on the right hand side the 11 will disappear getting us closer to just saying equals T but on the left we can't actually subtract 11 from W to write any sort of number so we just write w takeway 11 we're almost there we only have one thing left to do is to get rid of this three which is almost stuck onto that letter T and not forgetting 3 t means 3 * t or you could say t * 3 even though the three is written first and to reverse divide or timesing by three we would want to divide by three now we don't want to write this as W - 11 / by three because we don't ever really see it written like that using a formula The Divide symbol just doesn't ever appear so when we are writing a formula and we want to do a divide we will just write it as a fraction so w - 11 all ided 3 is equal to T so there we go we don't need that division symbol we can always write it as a fraction just remembering it's all being divided by three there now you could rewrite this formula as well you could write it as T equals and just switch them over so w- 11 all over three but it is fine to leave it in either form just depends on what the question asks you to do obviously if the question on the dotted line says T equals like this then you are obviously forced to write that in you don't need to write the T equals again you would just write w -1 over three as your final answer there we go so that is some subject of a formula now you can obviously have much harder subjects of formulas as well where you have squares and square roots involved so this is one particular example which we can have a look at now it says here make S the subject of the formula thankfully we don't actually have to undo the squaring here but but we can talk about what we would have to do if we did so here first of all we can get rid of this u^2 okay but the the sort of the difficult part when looking at this is there's no symbol in front of U2 if there's no symbol that means it's a positive u^2 and to reverse or get rid of a plus u^2 we can subtract u^2 from either side if we take away u^2 on the left we have v^ 2 take away the u^2 and that's going to be equal to 2 a s now this is very similar to the last question where we had a three in front of the the letter we were trying to make the subject in this case we just have a 2 a in front of the S we can treat it in exactly the same way though we can just divide both sides by 2 a and we can do that in one jump so all over 2 a and that is equal to S and there we we go s is the subject now it could say in a question like this to have made U the subject and if we wanted U as the subject then instead we would have first subtracted the 2 a s so we'd have V ^2 minus 2 a s and that would be equal to u^2 to make you the subject in this scenario to get rid of squaring the opposite to that is square rooting so if we square rooted both sides we could Square root the left and square root the right the square root on the right hand side would remove the squared leaving us with u and the square root on the left we would have to leave as a square root so we would leave it as V ^2 minus 2 a s and there we go just an idea if we had to also remove a squared okay so with these types of questions where we are given a lot of information and normally it splits it into things like males and females or the choices that that these particular people make it can tend to ask us either to find a certain amount of people that sort of you know in this case go to a certain place or it might ask us the probability now there's a lot of information to decipher in these questions so you can actually figure these out by using one of two methods we can use either a two-way table or potentially a frequency tree although I do tend to avoid using a frequency tree if I have the option so in this question we're going to use a two-way table now reading the first line it says 60 people asked if they prefer to go on holiday in Britain or in Spain or in Italy so straight away I can see there's a lot of information that I need to figure out so I'm going to start drawing a table I'm going to have a column for Britain a column for Spain a column for Italy and then a final column there for the total so you can label that you can put all the words in you want I'm just going to put Britain Spain and Italy and I'm going to have a total at the end it then starts talking about whether they were male or female so I need male row a row for female and again a total so male female and total so I want to go through line by line filling in all the boxes that I can do within my table it does say 60 people arest in the first line so if I tick that off and I put 60 in My overall total 38 of the people were male so that's the total males so the first row and and 38 in the total 11 of the 32 people who said Britain were female so in the Britain column we've got 11 that were female and it says that's 11 out of 32 so 32 in total we can take that off eight males said Italy so Italy column for males we have eight and then 12 people said Spain so that's a total of 12 for the Spain column one of the females is chosen at random what's the probability this female said Spain so we want to know how many females went to Spain which is this box here so we can go about filling in anything until we get there we don't have to complete the table but we just need to fill in anything we can until we get to that particular Square so if we go through you can see in the final column we have 38 males and a total of 60 people in total so that has to be 22 in order for that to add up to 60 we can also have a look across the bottom um row there as well they've got to add up to 60 at the moment 32 and 12 add up to 44 so that's going to have to be 16 more to get over to 60 we can now go down the Italy column so eight males and 16 in total so we must have eight females and now we can actually go across finding that uh row there for the females so 11 and eight that we have just here and here linked together those add up to 19 we need to get to 22 so that would be an additional three people now based on the way that I've done it I don't necessarily need to fill in any of the other boxes but you could you could obviously go to that Britain column they have to add up to 32 so that would be 21 and of course as well next to that now we could go along to the 38 we have 29 in total in that row so that' be another nine people or you could go down as well nine and three make 12 the important part is that that final number that you put in Works in both the row and the column that it was in so you can see if I go down 9 add 3 is 12 that works and 21 9 and 8 adds up to 38 so it works there as well so I know that I've done my two-way table correctly in terms of a probability then we have three people that are females that were went to Spain so we have three out of and this is the important part here now it says here one of the females is chosen so that's really important it doesn't say one of the people are chosen or one of the 60 people are chosen it says one of the females are chosen now if you have a look the total females is 22 so if we are only selecting from the females and there are three of them that said Spain then that is three out of 22 now that'll be our final answer that fraction doesn't simplify but even if it did simplify in this particular type of question we don't need to it doesn't say give your answer in its simplest form so even if it did simplify we could leave it unsimplified and that there is our final answer for this two-way table okay so looking at a frequency tree this one here we don't have an option we are given a frequency tree and we have to fill in this particular tree so we can't use this information and try and sort of draw a table we do need to fill in the frequency tree says each worker in a factory is either left-handed or right-handed 22 of the 45 workers are male 16 of the 34 right-handed workers are female complete the frequency tree well we'll start with that first line we are told here there are 45 people in total and that's drawn in the tree for us it says 22 of them are male so 22 go into the male Circle there we can tick that off it doesn't give us any more information as the 45 is already in but we know it adds up to 45 so we know the females have to be 23 22 and 23 will add up to 45 we can then go to the next piece of information we're only given one more 16 of the 34 right-handed workers are female that does give us two bits of information it tells us that from the right-handed workers 16 of them are female so that would be down here in the female right-handed workers so that deals with the 16 says of the 34 right-handed workers though so it does tell us that these two circles here both the right-handed workers has to add up to 34 so straight away we can work out the right-handed males we just need to take away 16 from 34 and that would leave us with 18 so 18 right-handed males and then we can fill in the remaining boxes because these two here have to add up to 22 at the moment we've got 18 in so that would have to be an extra fall and down here these have to add up to 23 at the moment we've got 16 down the bottom so that would have to be seven so there we go that is completing the frequency tree again you could be asked the probability from one of these it could say what's the probability if you pick a male that that person is Left-Handed or out of those people what's the probability of picking a left-handed person so you could be asked all the types of questions that we just talked about in terms of the two-way table but just from the frequency tree and of course if you were asked to write the probability that if you pick a male what's the probability they're left-handed that would be four out of 22 or if it said what's the probability you pick a left-handed male out of all the people that would be four out of 45 people so lots of different questions you you could be asked but there we go just an idea there for some of those questions okay so onto a pie chart now a pie chart does require you to use a protractor in order to draw these angles so we are going to switch and start using a protractor for this okay so when it comes to drawing a pie charts we need to make sure that all of the angles for the pie chart add up to 360 of course a pie chart is in the shape of a circle and angles in a circle we want to go up to 360 so if we look at what we have at the moment if we look in this table you can see if we add these up we have 11 + 17 which is 28 +8 which is 36 that's quite a nice number because that can nice and easily turn into 360 degrees so if I put the angle to the side so this is just the angle that each one will be represented by by we need to think about how we get to 360 so to do this now hopefully you can spot what we're going to have to do here to get from 36 to 360 but if you are not sure you can do 360 divided by the 36 and in this case it does come out as 10 so we need to multiply all the numbers in the table by 10 so 11 multiplied by 10 will give us an angle of 110° 17 multiplied by 10 will give us an angle of 170° and 8 multiplied by 10 will give us an angle of 80° obviously just double check those add them up make sure they do add up to 360 in this case they do so we are perfectly happy with that one we now want to get our protractor and line it up so that the Crosshair there is on the center of the circle we want to rotate the protractor so that zero is pointing up our starting line in this case the starting line is going right up to the top so I'm going to use the numbers on the the outside of the protractor going clockwise I'm just going to put a mark down here by the 110° I then want to pull the protractor out of the way and join that up from the center to that Mark there using a ruler so once you have joined that up it's going to look something like this and then we want to just label that so that particular section was to do with the burgers so we'll put burger and also you can put the degrees in there you can put the frequency in there it's up to you there but as long as we've labeled it now we can put the protractor back onto that Central spot and we want to rotate round to our new starting line the one that we've drawn so once we are around to that line which we've joined up with a ruler we can then draw draw the next angle on now in this case it's 170° it can be a little bit awkward here particularly if it's out the way or to the on the side of the page somewhere but we'll mark this on 170° is is just there there we go and then pull the protractor out the way join that up obviously using a roller to the center and then again we'll label that so that was for pi and that was 170° now of course we need to label the final section which is for hot dog so we can label hot dog and then put our 80° in there and a good thing to do at this point is just to check that final angle so pull the protractor back over put the central point back on the Crosshair and start from zero and measure it out make sure it's pretty close you can see the line goes up there it's just here it's pointing right on the 80 just here so I'm happy with that that final angle is 80° the pie chart is drawn we've labeled it up and there we go that is drawing a pie chart okay so looking at a frequency polygon now in a frequency polygon these are really nice and easy to plot it's obviously just remembering what you need to do for one of these so with a frequency polygon we are going to plot for these intervals here that you can see so for the first one we have 10 to 20 and we're going to plot that on the midpoint so to find the midpoint it might be nice numbers so I'd hope that you could spot that in between 10 and 20 is 15 but if you're not sure on finding those midpoints you can add those two numbers together at either side of the interval and then divide your answer by two so for example we would just do to the side 10 + 20 and then divide that answer by two and that's 30 divid by two which gives us a midpoint of 15 so we can fill in all of these midpoints so 25 35 45 55 and 65 and then we are just going to plot them against the frequency so obviously look at the axes we have our height down the bottom and the frequency going up the side so we're going to go to 15 which is in between 10 and 20 and then go up to seven obviously read the scale carefully you can see here between five and 10 where I'm trying to find seven there are 10 squares obviously with a gap of five that means that every square is 0.5 okay you can just do five divided by the 10 squares which is 0.5 or in other words that means every two squares goes up by one so I'm going to want to go four squares up from five on the 15 so 2 four obiously you can't see where I'm counting there but there we go that would be seven we can then just move along going across through every midpoint up to each number so I'm going to go to the 25 and then go up to 13 which is here and then we can go to 14 which is a nice one to find next as it's just going to be two squares above the last one so two squares above that would put me here we can then go to 12 for the next one which is is just here and then we have 16 and 18 16 not too bad to find two squares above 15 and then 18 on the last one which is going to be four squares down from 20 there we go and there is our frequency polygon now the way that you join a frequency polygon together we don't join anything to the Axis or to the origin we simply with a ruler and a pencil join these points together with nice straight lines so I would get my ruler join the first two with a nice straight line don't then immediately connect it to the next point you want to then obviously move your ruler so we are drawing perfectly straight lines between these two points between the next two points and then between the final two points and there you go that is your frequency polygon drawn plotting them on the midpoints so in this next question we're going to be having a look at a scatter graph it says here one of the points is an outlier write down the coordinates so an outlier is a point that lies away from what we would draw onto here which is the line of best fit so when you are drawing a line of best fit you look at the general trend of all of the points here we're going to be looking at these points here which do follow a trend and we're going to draw a line of best fit that goes around about through the points with around the same amount of points above and below so if I draw a few examples on here because there are a few different ways that we could draw a line of best fit so that there would be one example and that would be absolutely fine to use you can see that I have about four points below or to the right of the line and a few more just above or to the left of the line however you look at it I could also reang that and if I draw a second line on here I could draw a line which looks something like this possibly not as good but it's absolutely fine as well we are not computers we are not able to plot these absolutely perfectly we are just drawing it as best as we can so I would get my ruler and draw a line that pretty much goes nicely through the points with around about the same amount on either side of the line now you can see there is one point here which lies away from those points so this one here it's the furthest away from the line of best fit so that one there is the outlier and this question has asked us asked us to actually give the coordinates of that so with a ruler really carefully you can trace down to the axes to see what that coordinate is that lands perfectly between 9 and 11 so that is an X coord coordinate of 10 and going to the left you can see it lies nicely between 18 and 20 so that is a y coordinate of 19 so our answer for that one would be 10 and 19 says for all the other points write down the type of correlation now the correlation just means the relationship between two things so in this case it's asking for the correlation between the maximum temperature and the number of hours of sunshine now when it asks for a correl ation we're going to give one of two words so if the line is sloping upwards as one goes up the other goes up that is said to have a positive correlation if the line is sloping downwards as one goes up the other goes down that's said to have a negative correlation and of course if it didn't follow either of those you would say it had no correlation but that's quite rare for that to ever be asked in an exam because we do tend to use the graphs here that do have it one form of correlation so here the line is sloping upwards so we would say that that has a positive correlation you could also be asked in that particular type of question to describe the relationship between the two things so in this case if we had to describe what's going on here we would say something along the lines of as the number of hours of sunshine goes up the maximum temperature also goes up or you could say on days where there is a higher temperature there is more hours of sunshine so either of those descriptions would be good if it said to describe the relationship but here it said to write down the type of correlation so we had a positive correlation for that one of course it would be negative if it was sloping down so on the next question it says here on the same day in another British town the maximum temperature was 16.4 de C estimate the number of hours of sunshine in the town on this day so this word here is important where it says estimate because we are going to be using our line of best fit and as we talked about that line can be at slightly different angles depending on how you draw it so anyone's answer here could be slightly different although as long as we've drawn this line relatively accurately we will get very similar answers so 16.4 again I need to look at the scale here so 17 is in the middle so every Square we only have five squares between 16 and 17 so every Square goes up by two or02 so here 16.4 well we've got 16 on the line then we have 16.1 16.2 sorry 16.2 16.4 16.6 and 16.8 so 16.4 is that second line up so you would get your ruler draw across very carefully make sure you go across that line and show this on the graph and then we can go down really carefully and read that number now if my line had been slightly different look my answer could probably be on 13 maybe it' go just over 13 just a depending on the angle that I've drawn this one is just before 13 again I need to read the scale very carefully so we have in the middle 12 and that's one two 3 four it's just to the side there five squares along so it's five squares between 12 and 13 which means each one's 02 again so if I count along that's 12.2 12.4 12.6 12.8 you can see I'm reading that really accurately there and I'm giving the exact answer the that I've got off the graph so that would be 12.8 hours there we go and that would be our final answer now again you could be given one along the bottom and have to go up and across to read the maximum temperature so just read the question carefully and obviously apply that to the axes that has that label on so another variation of a scattergraph question you can see here it says Shan has the information about the height and weight of 10 rugby players and he's has to draw a scatter graph on the scatter graph you can see he has also drawn a line of best fit so it says here sha has plotted the points accurately so all of those X's that we can see they are all accurate points but write down two things that are wrong with his answer now straight away hopefully B especially based on the last question you're looking at this and thinking well the line of best fit should be following the general trend of the points it should be drawn somewhere like that so here you can see that all sha has done is he has connected the origin to the top of the graph and joined that together and that is not a line of best fit that is just going to be exactly the same no matter what points are plotted on the graph if we join it up like that so here we would say the line of best fit has not been plotted correctly is not plotted correctly there we go it's not plotted correctly and you can also back that up by drawing one on in a question like this you can plot the correct version and use that to back up your explanation for the next part there's only one other thing that could go wrong with the graph if the points are plotted correctly then it must be something to do with the axes so if we look at the axes we'll make sure that it's all scaled properly so let's just have a look it goes from 85 to 90 here so that goes up by five then it goes up to 95 100 105 so happy with those they look good this one goes from 140 to 160 so it goes up by 20 and then it goes to 170 so here we've got + 20 and here here that's only + 10 so the axes are not scaled correctly so again we could put the correct answer on and you could just write down here that it is not scaled correctly and you could say more to that you could say from 140 to 160 goes up by 20 and then the rest go up by 10 anything along those lines but as you can see with a scatter graph and any of these graph questions where it does ask you to see what's gone wrong quite of often it's to do with the scales on the axes or the way that the actual graph has been drawn but there we go just a little bit of a different scatter graph type question so moving on to fractions of an amount this question here is a slightly different one it does say four fifths of a number is 32 find the number now questions like this can be incorporated into larger questions but this is just a skill-based question looking at how to go about figuring this out so if four fifths of a number is 32 you could imagine this by using a bar model we can sort of visualize what this actually looks like so we don't want to get confused between fractions of an amount and this here which is more of a reverse fraction of an amount so here I've split my bar up into fifths I've got five pieces and this is saying that four fifths of the number is 32 well if we looked at four fifths of the number that would be only that much there of the bar just those four which would add up to 32 now if those four add up to 32 we can figure out what number has to go into each box as there's four of them we would divide by four and that would give us eight that goes into each box so if we were to now fill in eight into every box let's have a look what we have a total there of 40 so the number must have been 40 and that is how we go about finding a reverse fraction of an amount normally when we are finding a fraction of an amount quite a common way to remember how to find a fraction of an amount is to divide by the bottom and Times by the top but in this scenario here we're actually doing that in Reverse as it's actually given us the answer so actually we divided by four and then although I counted up the eights there I did 8 Time 5 because there were five of them which gave me my answer of 40 so a little bit different we were doing the opposite there because again we had been given the answer we weren't being asked to find four fths of 32 we were asked that 4 fths of 32 was the answer find that number so just think about these logically in terms of the way you'd approach it rather than remembering a trick when we are converting fractions decimals and percentages if we want to make any sort of comparison and in this one it's asking us to order them in size and to start with the smallest number so we are actually going to have to compare them we are going to want all of them in the same form now it's up to you which form you put them in typically it's quite nice to turn them into percentages or of course you could write them all as fractions if you do write them as fractions though so you need all the denominators to be the same which can cause a few issues with some questions but we'll have a look and have a think so here we could write them all as percentages percentages are quite nice to look at and think about which one's bigger or smaller so 3 over 10 we can convert that into a percentage in lots of different ways the easiest way is to make the bottom number 100 so we would times the top and bottom by 10 and that would give us 30 over 100 not forgetting that a percentage literally means per 100 this is 30 per 100 so that is 30% the next one the decimal to turn a decimal into a percentage you just multiply that by 100 so that would become 32 which is 32% for two fths again we want the denominator to be out of 100 we could times the top and bottom by 10 make it out of 10 first or you could times the top and bottom by 20 and we would get 40 over 100 which again is 40 per 100 which is 40% so 40% % and the last one again timesing that by 100 would give us 25% we can now put them into size order not forgetting it says write these numbers in size order so not our new converted forms we want to write the originals in size order so the smallest there is 25% that was the decimal 0.25 so that would go first you can tick them off as you go if that helps as well just to make sure you definitely write them all down the next one was 30% which was the 3 over 10 so that'll be our second and we can tick that off then we've got the 0.32 we've got the 35% and then we have the 40% which was two fifths so two fifths would be last that's our largest there and we've put them in size order okay so when finding a percentage of an amount we're going to take a similar approach with pretty much any of these questions I would always recommend finding 10% you might be more confident finding 1% but particularly without a calculate we just find 10% so 10% of 80 we divide it by 10 we get the answer 8 now this particular question does have a multiple of five in it so we're going to want to half that in order to get 5% as well and half of eight is four now this is quite a nice one because it is just 15% so you can add those two together and our answer for this question would be 12 but of course we could be given any form of percentage so for example if we were asked to find 35% you would want to find three of the 10% so times that by three and just one of the 5% so you can add those together that would help you find 35% there are other percentages you can go to without a calculator so things like 2.5% can be found from the 5% as well you can divide 5% by two so 2.5% here would be half of four which is two and of course from 10% you can also find things like 1% by dividing by 10 again now this is the number eight that we've got as 10% so if you divide that by 10 again you do get a decimal you get 0.8 but using these percentages you can pretty much find anything you could find something now like 23% because you could do three of the 1% and two of the 10% so you can build things up however you want but I would recommend starting with the 10% and then finding 1% from there or if it's a multiple of five sticking to the 5% now with percentages there's lots of different types of questions that you can have this one in particular is looking at a percentage increase it says asol is paid 1,500 per month and he's going to get a 3% increase in the amount of money he is paid work out how much money ASO will be paid per month after the increase so obviously an increase means he's going up in value he's getting a pay rise so we just need to be able to work out 3% now the most important percentage particularly when we are looking at non-calculator methods the most most important percentage to be able to work out is either 1% or 10% so if you go with 10% first 10% you just divide this by 10 which hops the decimal in which would be 150 you could have worked out 1% first as well which is dividing it by 100 but we can always get to 1% from 10% by dividing by 10 again so 1% would be 15 and hopefully you can see the pattern between those three numbers so for 3% we are going to want three of these 1% so timesing that by 3 take your time with that 15 add 15 is 30 add another 15 is 45 so asmol is getting a pay rise here of £45 so after the increase we will have 1,500 plus this new £45 that he's getting so £ 1,545 will be the amount that he gets after the pay rise now of course we can always have a percent percentage decrease as well so a percentage decrease just means that it's going to go down in value this could be a calculator or a non-calculator question but typically these percentage questions they're okay to do without a calculator we'll have a look as we go through and we'll think about whether this would definitely be a non-al or a calculat question now if it's going to go down by 32% and its starting value 675 we'll try and work out what that 32% is so again I'm going to start with 10 % so 10% of £675 again dividing by 10 to get 10% would give us 6750 so £67 and 50 at some point we're going to need a 1% as well because we're going to have to get to 32 so if we divide 675 by 100 which gets us 1% that would give us £675 so that is £675 now the numbers here are not awful but straight away it does stand out to me this is more likely to be a calculator style question but nevertheless we can do this without a calculator we just need to add up a few of these to get to 32% so we would have there quite a lot of numbers here if we did do it without a calculator because we've got three of the 67 50s of course you could multiply that by three if you like but I just think adding them up here isn't too awful we've also got the £675 and another 6 75 I probably would want to go beyond this many if I'm doing column addition because it would start to get a little bit messy but let's work this out so we have not n not 55 that adds up to 10 so carry the one then we've got 15 22 29 30 so then carry the three 7 14 21 27 33 plus the 3 36 so carry the three and then 6 12 18 plus the 3 is 21 so that adds up to £216 so actually once we've added these together that's not actually an awful number and to be fair we could actually have done this without a calculator so maybe this doesn't have to be a calculator question so going for this we've got 675 we need to subtract this £216 as it sale as the sale price is being reduced by 32% so we can do this as a subtraction typically when these come up it does involve some form of carrying the questions are structured in that way to force you into doing some carrying so we can't take away six from Five I'll have to borrow from this seven so that'll be 15 takeway six which is nine six takeway one which is five and six takeway two which is four so there we go the final cost here of the television will be £459 and that is the price after a £ 32% sale now when we are calculating percentage changes there is a calcul and a non-calculator method so we're going to have a look at a non-calculator method for this one but we'll also talk about what you would do if you had a calculator as well so it says here here Renee buys 5 kilograms of sweets to sell she pays £1 pound for the suets she puts all the suets into bags and they're going to be 250 g bags she's going to sell them all for 65p and she sells all the bags of sweets work out her percentage profit now percentage profit is a nice one to work out without a calculator because we know particularly with1 that we can find some relatively nice percentages of that so if we have 10 over here and just think about some of the percentages well 10% of 10 would be one pound just dividing it by 10 you could also find things like 5% 5% would be half of a pound that would be 50p or you could write that as 50 like that we can also multiply the 10% so maybe something like 20% which would be double that which would be 2 or you could times it by three you get something like 30% which would be 3 now if it's non-calculator it's going to be something that links nicely to the ones that I've drawn there but we'll have a look and see so if they buy 5 kilg of sweets but 250 g for the bags we're going to want to convert that 5 kg to see how many bags of sweets we've got so 5 kg is 5,000 G 1,000 G in a kilogram and we are going to put them in 250 g bags now I wouldn't really want to do 5,000 divided 250 so i' probably think about another way of doing that but we could actually divide it you when you are doing a division like this we can get rid of a zero top and bottom and it's 500 divided by 25 that's not too bad to do 25 fits into 50 twice and there is an additional zero there so it does it is actually 20 but you could count up in 250s you know four of them will fit into a th000 and you've got 5,000 say four times 5 which is also 20 so different ways that you could go about that of course you could do bus stop division as well 500 divided by 25 25 goes into 50 twice no remainder so zero so a couple of different ways that you could actually do that division without a calculator so if we've got 20 bags we're going to sell them all for $65 Pence then we obviously need to multiply that by 20 so however you want to line that up I'm going to do 65 * 20 this way there we go * 5 * 6 placeholder for the two 2 * 5 is 10 so carry the One 2 * 6 is 12 plus the 1 is 13 so adding that together we don't really need to because we've got the zeros above but that is 1,300 P we can convert that into pounds as well as our 10 was given to us in if you want you can write them both so that would be £13 so you can see here we bought the suet or Renee bought the suet for £10 she sold them for £13 now in terms of profits that means her 13 take away the 10 that she spent has left her with £3 in profit so if she has made3 in profit what is that as a percentage or what is her percentage profit well we got one of them at the start and it's this one here isn't it so they've made3 profit and that equated to 30% in terms of a percentage profit now if you are doing this with a calculator there is also another way of writing it so if you have a calculator and the numbers aren't quite as nice obviously we could have used a calculator here for all of the calculations along the way but you do the profit which is three over the original which was 10 we could write this as the profit or the change because this also works if you have a loss so a profit over the original and then times that by a 100 and the only reason you're timesing it by 100 is that will give you a decimal or a fraction when you do profit over original so it gives you 0.3 which is a decimal version of 30% when you times it by 100 it gives you 30% of course if you get a nice fraction so 3 over 10 you might be able to convert that into a percentage anyway using your conversion methods so there we go there is a calculator method as well for this topic so on to some algebra when we are solving equations now this particular equation does have a bracket involved as well so when you have any brackets involved or you see a bracket in a question it's always good to just expand it it's very unlikely that we're going to deal with this while the bracket's there now for this particular type of equation you can actually solve this without expanding the bracket but I would always encourage you just to expand the bracket first so that would be 3 m just multiplying by 3 3 * 4 gives you 12 and keeping the symbol the same so that equals 21 I'm not going to go into the method of how you solve this without expanding the bracket because typically it's not too difficult to expand the bracket there so I probably wouldn't look at using varying different methods but completely up to you if you if you can spot how to solve that without expanding the bracket so here we do the inverse so here it's 3 * a number takeway 12 is equal to 21 well what was the number before we subtracted 12 well to find that out we can add the 12 by doing the inverse so if it's a takeaway 12 we would add 12 to reverse that so three times that number would have equaled 21 + 12 which is 33 so another words before 12 was taken away and we got the answer 21 the answer would have been 33 so now we've got three times a number is equal to 33 that's three times a number the opposite of timesing by three is dividing by three so we'll divide both sides by three and we get that that number M must have been 33 divided by three which is 11 and of course in these scenarios you can sometimes get and quite often get noninteger Solutions so in other words you don't get a whole number for example it could have been something along the lines of 2 m is equal to and that's just imagine it's something relatively nice like 15 now that doesn't divide perfectly by two to give you a whole number so if it doesn't you can just write it as a division as a fraction so 15 divided by two you could leave as 15 over two and of course if it's a non-calculator question and you want to convert that you can do I know that 15 / 2 is 7.5 so you could write your answer as m is equal to 7.5 or 7 and a half however you choose to write your answer but it is fine to leave your answer as an improper fraction if this scenario does occur unless of course it says in the question solve it and give your answer as a decimal or something along those lines which is very rare so you can get non integer Solutions as well now when we have an x on both sides of the equation it's essentially the same as what we did in the last one we have a bracket involved here so before we talk about it let's straight away expand this bracket and rewrite this equation so we have 5x - 6 on the left on the right 3 * X is 3 x 3 * 1 is 3 and there is a takeaway in between so now we don't have any brackets to worry about all we need to think about is how do we make it look like the previous equation we want equations to look like this 3x + 2 is equal to 5 something along those lines now at the moment the only thing that's stopping it from looking like that is this 3x here on the right hand side so we need to get rid of the letters from one side of the equation now you don't have to get rid of them from the right or from the left you can do either so I always look to find the smallest list coefficient of x coefficient being the number in front of the letter so in this particular question three is smaller than five so all I'm going to do is I'm going to take away 3 X's from both sides of the equation so 5x take away 3x is 2X and nothing else changes other than the fact that we're removing these three X's from both sides we still have this minor 6 we still have the equal sign the 3x is gone and this is a really good one because you remember here this is actually a -3 so we need to pay attention to the symbol there because it's still equal tog -3 we've not got rid of a negative from the right we've just got rid of three X's so as it's equal tog -3 this obviously does make it a little bit trickier because we are dealing with negative numbers but this is a minus 6 here which we're going to get rid of just like before now it looks like what we wanted it to look like we can add six doing the inverse there of taking six away and that leaves us with 2x is equal to -3 add 6 is going to be positive3 and here's that scenario that we just talked about where we get an a number on the right there that doesn't perfectly divide by two so in this scenario we do need to divide by two because we don't want to know what 2x is equal to we want to know what 1 x is equal to and if we divide by two we can write that as a fraction 3 / 2 now obviously different ways that you could write your answer here we could write that as let's have a look three goes into two once with a remainder of one so you could write it as 1 and a half or you could just you might just know that 3 divid 2 is 1.5 and you can write it as a decimal so any of those answers is fine you wouldn't obviously give the variations you'd only give one of them so we could just say down here x is equal to 1.5 or any of the other variations but we wouldn't give more than one there we go X is equal to 1.5 so when we've got some inequalities typically is the same as equations now this inequality equality in particular is quite a nice one because although there are unknowns on both sides of the inequality there is only 14n on the left which does make it look a little bit strange in comparison to quite a lot of the ones that you'll see normally this one is a little bit different so and I rewrite this we've got 14n is greater than 11 n + 6 now you can if you like because you tend to deal with equations more so than you deal with inequalities you can if you like rewrite this as an equal sign until you've solved it I don't necessarily recommend doing this because what you don't want to do is get to the end of the question and leave the equal sign there but some people do like to change it for an equal sign just because you're more used to dealing with equations I would however advise to keep the symbol in there and just know that you can treat this in exactly the same way to an equation so here we have n's or coefficients of n on both sides so again I'm going to look for the smaller coefficient and that is 11 so we're going to take away 11n from both sides when we take away 11 n's from both sides we get 3 n is greater than six and then we don't want to know the value of 3 n we want 1 n so we'll divide by three and we're going to get n is greater than two and there we go that would be our final answer for Part B here it says on the number line below show the set of values for X which represent this in in equality now the problem here is that in the middle of the inequality we've got this plus three and we can only really represent this inequality if it's just X or any other letter in the middle but this has added three to both the numbers so to take this back to a normal inequality that just has X in the middle we're just going to subtract three from both the numbers as three has been added to both of them so -2 takeway 3 would be -5 and four takeway three is one so so the inequality that we can plot is there we just need to remember whether there are symbols on the inequality now on the left it had a symbol um without the equal sign on the right it has the equal sign so we need to keep that included in the inequality as well that's going to help us then to draw it on a number line so what we're going to do is put circles above any of the numbers so here it's neg five so we'd have a circle above neg five and for a joint inequality we're also going to put a circle above the other number we're then just going to connect them together and decide if either these circles need to be colored in and to figure out if they're colored in or not you just look at the inequality and if it has extra ink on the inequality like this one then we put extra ink inside the circle so for this one we're going to color in that circle on the right and that is our number line completed and if that's colored in all it means is that there was extra incon the inequality itself which just means it can be equal to that number as well now if you only have one inequality like the one above just here where we had n is greater than two you might be asked to plot that on a number line now if we were we would have put a circle I'm going to do it below but you'd put a circle above the two and then we would point it in the direction of what the either what the inequality means or if you actually look at the inequality itself n is greater than two if you put this line in to make it look like an arrow it points in a direction you need to point the line so we need to point the line to the right and that presents numbers greater than two and hopefully you can see here that that end of the inequality matches this one just here they look exactly the same they're pointing in the same direction and there you go that's your inequality drawn so there we go that is how you would draw that inequality and one where you have a joint inequality with numbers on both sides so when we are looking at error intervals there is a big variation of numbers that we can have to round or have been rounded in these types of questions and sometimes you are given given extra information in terms of how the inequality looks but this one here doesn't really give you anything it just says a number yse rounded to two significant figures the result is not .46 write down the error interval for y now with significant figures again significant just meaning important figures just meaning numbers so two important numbers we're going have a look here so the zero when it's at the start doesn't count as a significant figure it's more about the first whole numbers that come up so here the first whole number is in the first decimal place that's the four and the second is in the second decimal place so another thing you could say is that this has been rounded to two decimal places now if we think about a number line so 0.46 being in the middle of that number line the next number if we were looking at just two decimal places we'd have 0.47 just above it and 0.45 just below that when we are look at an error interval we're looking at what the boundaries are that could have rounded to 0.46 and the rules of rounding is that when we have a five or above it rounds up below that it rounds down so we need to figure out what that midpoint is there and it is linked to that number five we also have another one on the right and we can we kind of call these the lower bound and the upper bound lower being the lower number on the left and the upper being the upper on the right now it can help when you're finding these just to find these midpoints you can add zeros onto the end of these numbers and then you can sort of just read this part of the number here to think about what's in the middle so in between 450 and 460 is 455 obviously is not point that but that means it's 0455 that's 0.455 in between 460 and 470 is 465 so that's 0.465 now looking at those numbers there particularly if you've uh not seen these in a while you'll obviously know that this number does round to 0.46 okay it's a five after the five so it rounds up to 0.46 the problem is though this one on the right actually rounds to 0.47 so it wouldn't actually round to 0.46 now when we write our error interval we obviously have to include that when we write it so this is talking about the number y so we put a y in the middle we have both our arrows pointing to the left we put the ler bound on the left so 0.455 the upper bound on the right 0.465 and we obviously need to symbol put on the symbols there that it can be equal to the number on the left so this one on the left has the equal to symbol but the one on the right does not because that number does round to 0.47 so that means our number can be anywhere between 0.455 and 0.465 but it cannot include 0.465 because that would round up and there we go that's an error interval another topic which is very similar to error intervals is truncation now truncation is a very strange way of rounding it's just when you chop a number off so for example if I was to truncate let's go for this number here so 0.437 if I was to truncate that to two decimal places I'd get a very different answer than if I was to round it to two decimal places if I round it to two decimal places then we would chop after the second decimal apply the rules of rounding so seven after the line so they're three rounds up we'd get 0.44 but if I truncate the number truncate just means to chop it off and completely ignore any rules of rounding essentially this seven just disappears and we are left with 0.43 so truncation is a little bit strange particularly when the majority of the time we actually look at rounding so something slightly different here it says Kira used her calculator to work out the value of a number X she wrote down the first two digits and she wrote down 7.3 so if she only wrote down the first two digits it meant it means that whatever was after that she didn't even take into consideration it could have been that it was a nine after the three which we know in terms of rounding we would want to write that a 7.4 but if she's just written down the first two numbers it means that she has truncated the number even though this question doesn't say it's been truncated so when we are writing down the error interval we're going to write it down in the same way we've got an X with our same symbols as before we just need to write down what's the smallest and biggest it could have been now this one doesn't really help to write down any form of number line because we know the smallest number that it could have been is in fact just this one here 7.3 it couldn't have been any lower than 7.3 because if she wrote down the first two numbers and it was 7. two for example well she'd have written down 7.2 so the smallest it could have been is the actual number that we've been given so 7.3 of course it might have just said 7.3 on the calculator which in which case if she wrote down the first two digits it would say 7.3 now the number that it can't be is just the next number up so she couldn't have had 7.4 on the calculator because if she wrote down the first two digits he'd write down 7.4 so the very next number that goes up would be a four it can't be 7.4 and that would be the error interval so if she wrote down the first two and she wrote down 7.3 that number can be anywhere between 7.3 and 7.4 but of course it cannot include 7.4 because that would be written as 7.4 and there we go that's how we would approach it if we are truncating rather than rounding so when we are looking at sequences we are going to normally be looking at writing the nth term now a sequence can be given to you as um a pict picture or as a number sequence this one's given to us as a number sequence but of course they could draw a pattern so for five it could be like something like five dots and then the next one adds an additional amount of dots on now if you are given it as a picture you just count how many are in the picture and write it as a number and write your number sequence like we're doing here so the first thing we want to find is what's the difference between each number so from 5 to 11 that goes up by six and 11 to 17 also goes up by six always check at least two of them but once we've got two we're pretty confident there it goes up by six each time now if it goes up by six it means that our sequence is related to the six times table now the mathematical way of writing the six times table is you write six with and it could be any letter but it says here an expression in terms of n we do call it the nth term we tend to use this letter so we write 6n 6n means the six times table n is the position in the sequence so for the first number in the six times table we get the answer six we know from Times Table 6 * 1 is equal to 6 that's what 6 N means 6 * n 6 * 2 is equal to 12 and so on 6 * 3 is equal to 18 and that forms the 6 * table now this sequence is linked to the six times table but it's not exactly the six times table if we write those numbers above the sequence you can see 6 12 18 is definitely different to the sequence that we have been given the sequence that we've been given is smaller than the six times table so from six to five you would have to subtract one the same is from 12 to 11 the same is from 18 to 17 each number you are subtracting one so this sequence that we've been given is the six times table minus one and that there is our nth term now that is what the term or the nth term here actually means it means it's the six times table takeway one but you might have also applied a little trick and I know a lot of students do this they sort of go back from the first to see what would have been before five to do that you take away six 5 takeway 6 is Nega 1 and that negative 1 is obviously the number at the end there after 6n so it's fine to use that little trick if you are confident and you understand this topic well and if that's a little bit faster for you to find the number you can do that as well but I prefer to actually understand what I'm writing down but there we go that's how we go about finding the nth term now you might also be asked to use an nth term in a couple of different ways so we look at this one have to think about the other ways we could be asked to use it as well this one says write an expression in the first part we've just done that so we can find the nth term pretty quick this one goes up by four so that would be 4N the four times table is 4 8 12 this particular sequence is bigger than the four times table six is to bigger than four so it' be 4 n + 2 so that' be our nth term for the sequence says here the nth term of a different arithmetic sequence is 3 n + 5 so that means it's the three times table + 5 you can actually go about working that out so the three times table is 3 69 if you add five to each of those we would get 8 11 and 14 so you can see there it's quite quick to actually find what that sequence is but it says here is 108 a term in this sequence show how you get your answer now you can find and I'm going to use both the methods here that we could be asked in terms of a question it might say to us what is the 20th term in the sequence now if it did if you want to find the 20th term we can find the 20th term by doing three that's not a three let's write an actual three so 3 * 20 which is 60 and then add five so 3 * 20 60 + 5 that would give us 65 so the 20th term is 65 now we could keep going up in the sequence to see if we get to 108 so I could instead maybe try 30 I could go 3 * 30 which is 90 add five is 95 so that's the 30th term we're getting pretty close maybe go a slightly bit higher maybe 3 * 33 3 * 33 is 99 + 5 is 104 and I know each term in the sequence goes up by three because it's the three times table so I could add three or I could do 3 * 34 I'm starting to get more complex sums here so that would be 90 + 12 which would be 102 + 5 which is 107 and the next term is going to be 110 so you know here that 108 isn't in the sequence because you can see there it goes 107 and 110 so that's one method that you could use although you can see that's going to take quite a bit of time you need to know that method though anyway because if it said find the 20th term we very quickly do three times 20 and add five and you'd say the 20th term is 65 so you need to know that method anyway in case you asked that but if you are finding if a term is in the sequence you can use a method that we've already looked at you can make it as an equation so we can say does 3 n + 5 ever equal 108 to solve that like we've done before it says plus five so we'll take away five we get 3 n is equal to 103 and then you just need to see if that divides by three so we would do bus stop division or of course if you have a calculator you can just type it in three goes into 10 three times remainder one and then it goes into 13 when it goes up to 12 which is four and then we would have to carry over a remainder of one and as soon as we get this remainder here that we have to carry over because that fits in three times again remainer one and it carries on going so we do not get a whole number so here we would say no it's not in the sequence because it's somewhere between the 34th and 35th term which we can see here anyway because the 34th was 107 and the 35th was 110 so our answer we would obviously write in words no and then obviously depending on which method you used would be how you describe it on the left there we would say no because it's not a whole number the position is not a whole number or on the right there you would say say no because it's somewhere between the 34th and 35th term so with a Fibonacci Sequence a Fibonacci sequence is just where the number before is added to the current number to find the next number so to say here the rule to continue a Fibonacci sequence is the next term in the sequence is the sum of the two previous terms find the ninth number so the sum of the two prev so if we look at the two which has two terms in front of it 1 + 1 is equal to two and you can apply that that anywhere else so for the eight 3 + 5 is equal to 8 so all we have to do is take the two terms before add them together and it'll find the next one so we currently have six terms here so we need to find three more I'm going to put these in place so I don't forget which one I'm going to so 5 + 8 is 13 8 + 13 is 21 and then 13 + 21 is 34 so there we go the ninth term is going to be 30 4 for the next part here it says the first three terms of a different Fibonacci Sequence are a b and a + b show that the sixth term of this sequence is 3 a + 5B so show that means we absolutely need to show our working out so for the next term I'm going to do these two added together and if I'm going to show that I'm going to show some working out so a plus b plus b which would give us A+ 2 B for the next term we're going to now add together these two and again you could either write that down in terms of a long bit of working out or we could sort of just add them together you can see there we have two a's so 2 a and then you've got the 1B and the 2B so that's going to be 3B so adding those together that's on the fifth term at the moment we have one left to do and that's going to give us our sixth term which should match the next one so 2 a and 1 a is is 3 a and 2 B and 3B is 5 b so we get 3 a + 5 b which matches what we were given in the question so there we go some Fibonacci sequences with numbers and with algebra moving on to some angles we have angles in quadrilaterals it says here WXYZ is a quadrilateral which we can see so you can see just here that we have W X Y and Z that form this shape so we know that angles in a quadrilateral add up to 360 degrees so I'm going to put here 360 degrees it then also says x y v is a straight line and we can see that straight line as well x y v makes this straight line here and angles that are together on a straight line add up to 180° now on that straight line there you can see that there are three angles you've got this one here you've got this one here and this 147 now the only ones that add up to 180 are where they are together and form this almost semi circular shape on the straight line so this one over here is nothing to do with our 180 so if we want to find anything here we've got to just remember that and also remember that the angles in a quadrilateral add up to 360 well if these two together here add up to 180 we can find the angle marked a which is our first question so we would do 180 take away 147 and that leaves us with 33° so our first answer would be 33° and it says give a reason and our reason is that angles on a straight line add up to 180° and you have got to write the whole thing out so don't write it in any short hand angles on a straight line you can write sum to or add up to 180° and there we go that's our reason for the next part here it says angle zwx let find that Z to W tox makes this angle here with W in the middle and then you've got wxy where X is in the middle is this one here and it says that those angles are equal work out the size of angle zwx so one of those two angles well this is the point now where we are looking at the quadrilateral we know that angles in a quadrilateral add up to 360 we currently have two of the four angles in the quadrilateral and the other two are equal so if we add together the two that we've got 145 at 33 that adds up to 178° so what's left over to make that add up to 360 well 360 take away 178 will tell us what's left over going to have to borrow here so 10 takeway 8 is two 5 takeway 7 we can't do so 15 takeway 7 is 8 2 takeway one is one we have 182° left over that split between these two so that's 182° if they are equal we can just divide that by two so 182 divided by two of course if you're not using a calculator we would use bus stop for that so twos into 18 go nine twos into two go once so our answer would be 91° and there we go we got 33° for the first one angles on a straight line add up to 180 and for the second one 91° didn't ask for a reason there but our reasoning was that angles in a quadrilateral add up to 360° okay so for this question we have something a little bit more unique we still have angles in triangles and quadrilaterals but you can see here that we have a parallelogram involved now it says here AB CD is a parallelogram it says that e to D to C is a straight line and F is the point on a d so that b Fe is a straight line it then tells us some angles which we can see in the diagram and it says to show that ABF is equal to 70° now A to B to F is the angle made down here from going from A to B to F it makes this angle just here now where it says to show that it is equal to 70° that doesn't mean that we can put 70° in here and then work away from it we have to essentially find out what that angle is imagine it's just an unknown angle and as long as we find that that angle becomes 70 we will have actually shown it along the way so we'll work around this question based on it saying find the size of angle X now there are a few different ways of getting to this first of all knowing that opposite angles in a parallelogram are equal is going to help us out in this question obviously line by line it tells us these things for a reason so here where it says it's a parallelogram we know that the total angle down here with the B is equal to this total angle up here where the D is likewise the opposite angles where a is and where C is they are also opposite angles which are equal so that means this angle up here is going to be 75° now it says to give a reason with each stage of your working so you can either do sort of a bullet point sort of format to the side you can say that angle b a f is equal to 75° and you can say opposite angles in a parallelogram are equal now you can just label that on the diagram as well but I do like this way of writing it putting them to the side so opposite angles in a parallelogram are equal now there we go we found a missing angle and we have given a reason we just need to find something else you can already see that I now have one of the angles within the triangle so I'm sort of looking at this triangle here because I know that angles in a triangle add up to 180 so now that I've got one of them let's need to think can I find another now if we have a look we can actually see that that 35 is given to us as well now that is an opposite angle we have two straight lines that are intersecting here and we're given the angle on the other side and these angles are equal so this terminology here for those type of angles they are called vertically opposite angles so that is the angle AFB so AFB is equal to 35° and we would say that vertically opposite angles are equal there we go so vertically opposite angles are equal now we are pretty much there because we now have two angles within that triangle again the triangle that I highlighted earlier so this particular triangle here so when we have two of the angles within a triangle we can add those two together so 75 add 35 which add up to 110° we know that angles in a triangle are add up to 180 so if we take away the 110 that we already have there we end up with an angle of 70° so therefore x equals 70 and that's exactly what the question was asking us to show now we might want to add another reason there as well because it does say give a reason for each stage of your working so at this point we've got the 70 and the reason being there angles in a triangle add up to 180° and there we go that would be all that we'd need to write for that particular question so when we have angles in polygons sometimes we'll be given a full polygon sometimes we will have a regular polygon sometimes we will also have irregular polygons this question here is showing us a 12-sided regular polygon but it's only showing as part of it so you can almost imagine if it had 12 sides the full shape is going to continue round here we're going to have a nice big polygon there all the way around but it's only showed us part of that polygon now the part of the polygon it has showed us is just the two sides and it's been connected together so it has formed almost this triangle angular shape now as it is a regular polygon that means the sides of the polygon are all the same length so R to S this line here is the same length as this line here and that means that that triangle there is also an isoceles triangle and within an isoceles triangle hopefully we are aware of isoceles triangles the base angles are equal so here before I even start the question I know that these two angles are equal but this particular question says what up the size of angle s and if I put an X in that angle that's this angle just here so it's one of those two base angles now when it comes to a polygon we can find the size of the interior angle this one just here by using a formula so for the formula that we look at for working out the interior angle in a polygon we take the amount of sides which we tend to call n we take away two from the amount of sides which tells us how many triangles it's made of then we times that by 180 and divide it by the amount of sides now the formula can sometimes look a little bit complex so it does help to understand why the formula is made the way it is and if we think about a pentagon that has five sides if we split a pentagon up it forms three triangles of course just going from one of the points but it can never connect to and I'm going to join these up now it can never connect to the points that it's already connected to there's already always going to be two of those so it always makes two less triangles than the amount of sides here the shape has five sides and it makes three triangles so this part of the formula here is just looking at the amount of triangles that it makes we know that triangles add up to 180 in terms of their angles so we times it by 180 that tells us the total that they all add up to within that polygon and then to split that equally among all the the angles we divide it by how many angles there are so that's all the formula actually represents and understanding that formula can just help just to remember it so here if we have a 12-sided polygon we would do 12 take away two which tells us that there are 10 triangles and you can label that 10 triangles if it helps you to remember it we know that angles in a triangle add up to 180 so are 10 triangles multiplied by 180 gives us a total of 1,800 ° and that's the sum of all of the interior angles in a polygon now a 12 12-sided polygon of course has 12 sides so we would divide that by 12 and that would tell us the size of the Interior now without a calculator and of course this could be a calculator question but without a calculator we're obviously going to have to just do some division so 12 goes into 18 once remainder six it goes into 60 five times and then zero at the end so the interior angle that we have found just here is 150° and if we had the rest of the shape obviously if this carried on the full angle there would also be 150° but we don't need that one so here very similar now to our previous question we have a triangle involved with two missing angles but this time those two angles are equal so we can take this one angle that we have away from 180 that leaves us with 30 30° that 30° is the total of these two so to split that as they are equal we can do 30° divided by two and that tells us that each angle is going to be 15° and that would be our final answer so 15° would be the size of that angle St so there we go that is using the process of finding the interior angles if you don't remember the formula you can always remember a little pattern so a shape with three sides sides adds up to a total of 180° every additional side so for a quadrilateral we know that goes to 360 it actually just makes a pattern going up 180° each time so a five-sided shape you can add 180° again and that takes you to 540 I don't know why I've written 180 there let's change that so that is adding 180 and we get 540 and you can just keep following that pattern adding a 180 each time until you get to the amount of sides that you need then of course you can divide it by how many angles there are to split it up equally among all the angles there we go that is angles in polygons so when we have angles in parallel lines there are a few rules that we need to remember so we have alternate and corresponding angles and we also have co- interior angles now for the large majority of questions you can actually get by by using alternate angles but you do need to know all three of those rules so alternate angles are equal and they make this almost Zed looking shape okay so when you can spot one of those you can see that there's alterate angles in these places just here you also have corresponding angles now corresponding angles occur when you have a line which is more extended through the two so if this line was extended a little bit you actually have these corresponding angles that occur that make this almost F shape where you get these angles here along with that 142 and you also have co- interior angles as well which make an almost C shape and co- interior angles are inside the two lines just like so and they add up to 180° so there's quite a lot of things that we can look at in this diagram we don't have to use all of those rules but when we have parallel lines involved we are going to be using at least one of them so it says here we've got the parallel lines it says we have an isoceles triangle you can see the isoceles triangle is here within the shape just there as indicated by those lines there showing it's an isoceles so here then we can start by just looking at the shape and thinking is there any angle I can find well we've got 142 that's given to us up here and that makes a straight line onto there and we know that angles on a straight line add up to 180° so we can find this one by doing 1880 take away 142 and that leaves us with 38° so that'll be 38° in the top of that isoceles triangle now it does say in the question give a reason for each stage of your working so make sure you read the question carefully because at this point now we would want to say angles on a straight line equal 180 so angles on a straight line add up to 180° and there we go that's our first re reason we can then think about how to find somewhere else and there two specific directions that we could go in here we could either find the base angles in the isoceles triangle so we know that these are the base angles here underneath those lines that indicate the same length so we could find those two or we could look at the parallel lines and either go this way and hook in and find that that is 38 down there because that's an alternate angle and Alternate angles are equal or we could actually use this 142 and we could go that way the only problem with this way is it is two angles and it's split in the diagram so it's not the nicest to look at but we could go that way as well and write that that full angle is 142 now I'm going to go the other way so I'm going to use alternate angles I'm going to go down this way and I'm going to say that that angle down here is 38° on my rational behind that I'll give my reason I'm going to label it on the diagram this time so I'm going to say here alternate Ang angles are equal so there we go alternate angles are equal there we go so we're almost done we just need to obviously find now that missing angle in the triangle which we could have done just before so we'll find these two angles here by finding both of them obviously we'll find that one there which is going to help us get to that X so to find the base angle in an isoceles we know that angles in a triangle add up to 180 so we do 180 take away the top angle of 38 and once we've got the answer to that we can divide it by two because base angles in an isoceles are equal now we know that that is going to Total 142 on the top because we've already done 180 takeway 38 or 180 takeway 142 in the first step so that's going to be 142 divided by two which is 71° so both of these angles in the base of the triangle are 71° and again we would want to write our reasoning for that I'm going to label it I'm just going to label it like this by drawing an arrow and I'm just going to put base angles in an isoceles are equal so base angles in and isoceles are equal there we go right we've got one more step and the final step here is quite a nice one we just have three angles there that are on a straight line so the straight line that I'm looking at is this straight line here along the bottom we have two of the three angles we've got 71 38 and our missing angle so I can do 71 plus 38 which is 109 and then take that away from 180 and we have already said in the first step that angles on a straight line add up to 180 so there we go that is equal to 71° and you'll probably notice 71 is the same as the angle in the isoceles triangle in the base of the triangle and that's because as well you could or you might have already spotted that this angle here is another alternate angle as well pointing the other direction to the other but that is 71 degrees as well so we could have in the final step there said that that was 71 because it was an alternate angle so lots of different ways of moving around this shape but the important part there is we got to the answer and for every step we give our reasoning based on the actual steps that we took through the question so when it comes to a reverse mean we have questions that look very similar to this it says that there are 10 boys and 20 girls in a class and the class has a test and then it gives us this sort of information where it gives us a mean so it says the Mean Mark for all the class is 60 the Mean Mark for the girls is 54 work out the Mean Mark for the boys so when we are told a mean this is what a reverse mean is we are going to essentially do the reverse of what we normally do when we are working out a mean so when we normally work out a mean we get the total we divide by how many there are and it gives us the mean so for this question here where it says the Mean Mark for the whole class is 60 well let's see what pieces of information we have so we take the total score for the whole class which is currently not been given to us so let's just call that X we divide it by how many there are now in this case it's the whole class so that's the 10 boys and the 20 girls so in total that's 30 people so that's the calculation we would to work out the mean and it's telling us the answer to that is 60 so what we can actually do is we can reverse the process here to figure out what the total is So if something divided by 30 is 60 well if we multiply by 30 that will tell us what that total must have been so 3 * 6 is 18 add the two zeros so that's 1,800 so in other words the total score for the whole class is 1,800 we can take that approach with the next line as well cuz it says the Mean Mark for the girls is 54 now again we don't know the total but something divided by the and it says that there's 20 girls so something divided by 20 that has to equal 54 again we can take the same approach now to work out the total so timesing by 20 there we go and we can do that relatively easily as well timesing by 20 we just need to Times by two and add the zero so 50 4 * 2 is 108 so 108 add the 0er 1080 so that is the total Mark for the girls so now we have the totals if we imagine we'd been given that in the first place if it had just said the total score for the whole class is 1,800 the total score for the girls is 1 180 work out the means Mark for the boys well to start with we'd need to take those away from each other to see what the actual total for the boys is so 1, 800 take away 1,80 arguably you might not need to do column subtraction for that it's not too awful to work this out so n takeway n is n we will have to borrow for that so 10 takeway eight is two seven takeway n is seven and one takeway one is zero so the total score for the boys is 720 and as you're going along you can label these things if it helps as well we had the whole class at the top there 1,800 we might have labeled the 1,8 the girls and then we could label this the boys so to work out the mean we already know as we've discussed it we know in this section here we take the total divide by how many there are well in this case we know the total 720 we know that there are 10 boys so we would divide by 10 and that would give us a Mean Mark of 72 and there we go that's our final answer for that one and again just remembering that you multiply the mean that you are given by how many there are essentially in these questions if you look at the first one we took the mean of 60 and times it by 30 and for the second one we took the mean of 54 and we timesed it by 20 so all we really did was take the mean times it by how many there were to find the total okay so when we're looking at averages from a table where we have grouped frequency like the one that you can see we are asked to work out an estimate for the mean now if it wasn't group frequency we could actually work out an actual mean if we knew the actual amounts but here they've been grouped so this is about some weekly earnings now here you can see with the first group is if they earned 150 to 250 and there was one of those people below that you've got 250 to 350 and there were 11 of those people so the reason we would be asked to estimate this is because we don't know their actual weekly earnings they've been put into groups but we can make an estimate and a good mathematical estimate here would just be to take the midpoints of those numbers so that's what we do so in between 150 and 250 that is 200 between 250 and 350 we have 300 between 350 and 450 we have 400 then we've got 500 and then we have got 600 so for the one lot of the 200 that's a total of 200 for that Row for the 11 lots of 300 Well we'd have to count up 11 300s but there is a quicker way of doing this all we have to do is mult multiply how many there are by that amount that we're using so one lot of 200 gives us a total of 200 11 lots of 300 gives us 330 0 five lots of 400 5 * 4 is 20 add the two zeros is 2,000 zero lots of 500 would give us zero and three lots of 600 well 3 * 6 is 18 and two Zer there so that is the total earnings now or estimated earnings of everybody in this table now in order to find the mean like we've discussed before you take the total of everybody and divide by how many there are so we would need to work out the total of Everybody by adding all of that together so if we do that to start with we've got 1,8 it would probably be easier particularly if we're not using a calculator here to either add pairs of them together or to put it all into one big column addition so we could pair these up that would add up to 3,500 and then you've got the 2,000 and the 1,800 which adds up to 3,800 that's a little bit easier then just to quickly add them together rather than having one big column addition which you could make a mistake on so 0 0 that is 13 and then 367 so 7,000 and 300 now we've got the total we just need to know what we going to divide that by well if it's to do with how many people there are then we would divide by this total frequency so if we add those up we get 127 that adds up to 20 so there is 20 so there we go the calculation we need to do is 7,300 ided by 20 now of course this can be a calculator question is quite a common question to be on a calculator paper but it's not exclusive to a calculator paper so we might have to do this without one so here we can cancel off a zero top and bottom and we just have to do 730 divided by two so it's not awful to do because we can set that up in a bus stop 730 divide by two two goes into 7 three times remainder 1 it goes into 13 up to 12 so six times remainder one and it goes into 10 five times so £365 would be our answer and that is our estimated mean for the earnings here so £365 now when you get your answer do check to make sure it makes sense if we look £365 would be in this category here it's somewhere in the middle it's not the most popular it's not the modal category that would be this one with the 11 in between 250 and 350 but it is somewhere in the middle relatively close to that category only being 15 pounds into the next category so there we go obviously as well you've got some people here there's three people here who are earning a lot more than everybody else so that is going to skew the mean a little bit as well because you've got those higher values in that final box there there we go that's how we work out an estimate for the mean get the total of each row using the midpoints and then do the total divided by the total free frequency okay so looking at a stem and leaf graph now here it says the table shows the heights of a group of students in year nine and the stem Leaf shows the heights of some students in year 12 it ask us to compare the distribution of heights for the year so when we are looking at this there are a couple of averages that we can look at from a stem and leaf now over here in the table we are given the median and then we are also given the least and greatest height now the median is already in average that's given to us so we definitely want to going to want to compare the median when we are looking at this when you think about the least and the greatest height the only thing that you can find from those two is the range so the range is obviously the biggest takeway the smallest so from the table we could say 170 minus 150 and the range is equal to 20 so if we are looking at these two averages they are the two that we can use to then look at this stem and leaf so looking at the stem and leaf we always need to look at the key first every stem and leaf whether you're reading one or drawing one needs a key and this tells us here that the 15 line 8 represents 158 CM so if we want to find the range from the stem and leaf which is probably the nicest one to find here we will find the largest and the smallest value now the smallest or the largest it doesn't matter which one we find first so we'll go for the largest that one down there is 182 so we've got 182 the smallest is up here at the start 158 so to find the range then we would subtract those away from one another so if we take those away from each other we get 24 there we go so straight away you can see the range in the stem and leaf which is the students in year 12 they have a larger range than the students in year 9 in the table and that would be our first comparison so obviously we want to mention the context here we don't just want to say that range is bigger than that range or 24 is bigger than 20 we want to say the range of heights in year 12 there we go in year 12 is greater than in year 9 there you go or you could say that the other way around there we go greater than year nine of course you could say the range of heights in year nine is smaller than the Heights in year 12 so that's number one we've made part of a comp comparison now we want to have a look at the median now the median a little bit more Awkward to find from a stem and leaf but essentially all we need to do is cross off from either side until we get to the middle you can apply a statistical method as well you can count up how many there are add one half it to find the middle and stuff like that but I think it's easier with a stem and leaf just to cross off some numbers either side from the start and then the same amount from the end working backwards until we get to the middle so I'll do three at a time until I can see that I'm getting close to the middle I'm getting pretty close now so I'll just do one more from the top one more from the end and that's my middle number or the median right there so that median is 168 and we know the median from the table from the year n students is 165 so our second comparison we would say the median height in year 12 there we go is greater than in year 9 so is greater than in year n there we go there's our two comparisons we have mentioned both of the averages and we have mentioned the context that it is about the heights so there we go that's how we would make a comparison and that's how we would also find a couple of the averages from a stem Leaf diagram okay so look at sampling and bias now here we are given a table it says that Hannah is playing a day trip for 190 students she has a sample of 30 students where they want to go so we have in the table 10 students that want to go to the theme park and then we have the theater Sports Center and the seaside it says work out how many of the 195 students you think will want to go to the theme park well if we look at the sample you can see that in terms of the theme park we have 10 people that want to go now we know that that is also out of 30 students so as a fraction that would be 10 out of 30 now if we are looking look at 195 students we are going to assume this proportion or this sample represents the rest of the people so here we would assume that the fraction percentage or or proportion of the 195 will be the same now either we can have a look and think can we make this fraction out of 195 so can you times 30 by anything easily to get to 195 if the answer is no then more than likely we could have simplified this fraction 10 over 30 does simplify so it goes down down to 1/3 you can divide the top and bottom by 10 so writing that in just to show what we've done we've divided the top and bottom by 10 now we can get three to become 195 otherwise we're not going to be able to actually do this but we need to figure that out so the best thing we can do is actually do a bus stop division how many times does three fit into 195 while it fits into 19 six times remain to one and then five times into 15 so that would be 65 out of 195 so there we go our answer for this would be 65 students as it has gone to 65 out of that 195 now our assumption here that we have made is we are assuming that the sample represents the rest of the students so we go that it is a fair sample that represents the rest of the students so I won't write that down there so we've mentioned that a few times there we go that's how we will approach these questions where we have a sample involved so in this question we're going to be having a look at some functional maths where it involves the area of a trapezium now with a trapezium and we are told here that this shape is a trapezium we can always work out the area of a trapezium as long as we know the formula which we'll discuss in just a second but it says here John is going to paint the floor each 5 liter tin of paint cost £16.99 and one liter of paint covers an area of 2 m squared John has 160 to spend has he got enough money and he must show how you get your answer so if he's going to cover the floor with this paint we are going to be looking at the area of the floor how much paint he's going to need to use to cover that area so to start with what we're going to think about is how to find the area of this trapezium now it's told us it's a trapezium and these sides here are those parallel sides so we're going to be working out the area based on those so the formula for the area of a trapezium is a plus b which are the two parallel sides you need to divide that by two and then times it by the height or the distance between the parallel sides when you can that slightly differently as well you can say half of a plus btimes the height so here to work that out then we will do 10 + 16 which is 26 / 2 is 13 I'm going to write that down straight away and then times that by the height which is seven so we need to work out 13 * 7 we are going to be using a calculator for this question as well so 13 * 7 comes out as 91 so the area there is 91 the units were meters and this is an area so me squared so there we go that is the area of the floor now it says here each 5 L tin of paint cost 1699 and one liter of paint covers 2 m squared well if we figure out how many of those twos fit into 91 then we'll know how many liters we need and then we can figure out how many tins we're going to need so for this step here where it says one liter of paint covers 2 m squared we'll take the 91 M squar and divide that by two and see therefore how many liters we need now when we do 91 ided by 2 we do get a decimal we get 45.5 now that means we need a decimal amount of liters uh which is fine at the moment we can stick with that but we may need to adjust this to take into account the fact we might have to buy a little bit more paint if it doesn't fit perfectly into that 45 . 5 so here that's 45.5 L and we know that one liter of paint covers 2 m squar so we're going to need 45.5 L now it says in the question here that a 5 L tin costs £16.99 so how many of those tins are we going to have to buy well we could count up in fives or we could do the 45.5 L that we need divided by 5 so if we type that in 45.5 divided 5 it tells us that we need 9.1 tins now you can't go into a shop and ask for nine tins and a little bit if you need a little bit more you have to just buy an extra tin so in this particular type of scenario we would round that up even though it doesn't follow the rules of mathematical rounding this is a reall life functional scenario so we would not be going in and buying nine tins and then not having enough paint we would just go in and we would buy 10 tins and that would just cost us a little bit more so there we go 10 tins now obviously the question was to figure out whether they had enough money they had 160 to spend each tin cost 16.99 we're going to buy 10 of them so 10 multipli by the £699 that's quite a nice one we don't need to use a calculator for that so 10 * 1696 16.99 is £1 16990 there you go so £169 90 that's the total cost John only has £60 to spend so does he have enough so we would say no John does not have enough or no John does not there we go it does say You must show how you got your working or how you got your answer we've shown all of our working out we've made it very clear but it's £69 90 that he needs so everything is backed up and we have shown how we got our answer okay so this question here is the area of a compound shape compound shape just being where two shapes have been put together and I've also s obviously highlighted here that we are going to be using Pythagoras Theorem it doesn't make that very clear in the question but quite often when we have triangles involved somewhere within a question we just need to have Pythagoras in the back of our mind just thinking that we might actually need to use it somewhere particularly when we are looking at lengths so here it says work out the area of this shape and it says it's a pentagon it has got five sides now there's no formula for working out the area of a pentagon that we would use so we need to think about how we can break this up into two different shapes or even more now there are two ways that you could break it up you could split this down the middle look and it makes two trapeziums the problem is we don't know the length of this side here so let's have a thinking if there is another way that we could do it well we could split it up this way and again that gives us a rectangle on the bottom that we can work out the area of but the triangle on the top we are going to struggle because we don't know this height here and that's where Pythagoras is going to be involved but let's start by working out the area of the rectangle so we have a length of four and eight so we would do 4 * 8 which is 32 cm squared we can then think about how we are going to get that height there the area of a triangle is the base times the height divid two or half the base times the height and we know that the base is 8 cm because the base is the same as this length down here so if we think about the triangle and specifically the triangle I'm going to be looking at is this one here on the left we could look at the one on the right but I'm going to look at that triangle there and if I draw that to the side there we go we can see it's a right angle triangle we have a hypotenuse or the longest side there is 5 cm and the base all the way along long is 8 cm so halfway along which is what we're looking at just to here will be 4 cm so 5 cm 4 cm and we're going to work out this height so Pythagoras Theorem is a 2 + b^ 2 equals c^2 C being the length of the hypotenuse which we know in this question so we will do a takeaway instead so we are going to do 5^ 2 takeway 4^ SAR which is 25 take take away 16 which is 9 and then we square root our answer and that tells us our length so three CM so there we go the height of that triangle which we can label over here on the diagram now that we know it we know that that is now 3 cm we can now use that to work out the area of that triangle on top so if this is 3 cm to work out the area of a triangle base time height ID two well the base is eight the height is three and then we have going to divide it by two so that comes out as 12 so that' be 12 CM SAR that is the area of the triangle so now we have the area of the triangle the area of the rectangle and we can add those both together so 12 + 32 gives us a total area of 44 cm squared and there we go there is our final answer that is work out the area of that Pentagon so on to some surface area of prisms now of course we can have the surface area of lots of different types of prisms you could have a triangular prism um in this case it is a cuboid but we can have lots of different types there also the surface area of a cylinder you could even look at the surface area of a cone or a sphere as well so for this one the surface area of a um cuboid it's not going to tell you necessarily in a question to work out the surface area this one here is more of a functional question it says a sofa has six identical cushions and each cushion is this cuboid the cushions are covered with a protective spray and the spray is in cans and the label on each can has the information that the spray in the can covers 4 M squared work out how many cans are needed to cover the six cushions with protective spray so this is a more advanced question than just working out the surface area but to work out the surface area which is the key bit that we need to do in this question we just need to look at the different surfaces now with a cuboid we have three that we could look at we've got this rectangle on the front here which is the same as this re angle on the back we have the rectangle on the side which is the same as the rectangle on this side and of course finishing that we also have the rectangle on the top which is the same as this rectangle here on the bottom so what you can do is you can label those three up we've got the one on the front we got the one on the side and we've got the one on the top and I'm just going to work them out to the side so for number one we have a length of 80 and a height of 18 I'm going to be doing this on the calculator so 80 * 18 let's actually write that in properly there we go 80 * 18 that gives us 1,440 cm squared onto the one on the side we have a length here of 95 and the height we already know is 18 because that's on the other side of the cubid so that one there 95 ultip by 18 and 95 * 18 comes out as 1,710 we're almost there we've just got that last one to do so the one on the top we have this length of 80 and we've got this length here which is 95 so for that shape number three 95 multiplied by 80 and 95 * 80 is 7,600 cm squared now at this point we have only worked out the surface area of one of each face so you could either times them all by two and then add it all together what I'm going to do is I'm going to add them all together so let's add them up there we go which is 10,750 and then multiply that answer by two to take into account that we do have two of each of those faces and that gives us a total surface area of 21,500 cm squared so if the question just said to work out the surface area that would be plenty and we would be finished but this obviously does want us to work out how many of these tins of spray are going to be needed so we have to go back to the question because it does say up here there are six identical cushions so looking at that we're going to need to multiply this surface area by six that will give us the surface area of the six cushions so timesing that by six gives us a total of 129,000 cm squ and now we can go down and read the question right at the end here it says that the spray in the can covers 4 M squared so at the moment this is in centim squar so we need to think about our conversion between cm squar and meter squar now to get from centim to meters not centim squar to meter squar but to get from centimet to meters you divide by 100 and likewise going back the other way to get from meters to centim you times by 100 now when we are looking at centime squared to me squared all you do is add squared onto that conversion so we're going to divide by 100 squar or Times by 100 squar now in this question we are going from cm squ to Meer squar so we need to divide that by 100 squar we can do that on the calculator and that gives us 12.9 so that's 12.9 m squ now obviously it says the spray can can cover 4 M squ so you can either divide that by four or you can count up in fours if we go 4 8 12 that would be three cans but it wouldn't quite cover it completely so we would have to get a fourth can as well so you could divide by four if you wanted to obviously you want to show you working so if we divide it by four it gives us 3225 225 cans we can't buy 3225 cans so we will have to buy four cans and round that up to make sure that we cover it obviously we'll have some left over but we will have to buy just over to make sure that we cover it so there we go that is how you work out the surface area we've already gone through area of a triangle so if it was a triangular prism you would very much take the same approach but obviously using half base times height for the area of any triangular faces so there we go that involved a lot there with some conversions and that functional aspect of seeing how many tins would cover the shape okay so in this question we are looking at volume now here we've got a diagram that shows a fish tank and it is in the shape of a cuboid says the dimensions of the tank are 50x 32x 20 which we can see in is 3/4 full of water and sand says the ratio of the volume of water to the volume of sand is 5 to one work out the number of liters of water in the tank so we've got a lot of maths going involved in this question we have to know how to do the volume of a cuboid which is obviously the key skill here but then we also have fractions and ratios and also the issue there that it's talking about liters and everything that's given to us in centimeters so we've got a lot to get through in this question but we're going to start by working out the volume of the cuboid now to work out the volume of a cuboid it's a really nice process all you do is multiply the three dimensions together so we have 50 multipli by 32 multiplied by 20 so if we type this into the calculator we get 32,000 and that is going to be cm cubed this time as we are looking at volume now we could have done this a slightly different way but I'm going to take the approach of working out the overall volume because it now says that it is 3/4 full of water and sand well we can work out 3/4 of 32,000 to find out what the volume is just up to to this point here so to work out three quarters we could first work out one quarter by dividing this by four so if we divide that by four we get that the one qu is equal to 8,000 cm cubed to get from one quarter to then three quarters we just need to multiply that by three so multiplying that by three is 24,000 cm cubed so we now know know what the volume is up to that point we can then start to think about this part of the question where it says the ratio of water to sand is in the ratio 5 to one so sharing in a ratio we want to know how many parts there are in total so 5 + 1 is equal to six parts so we can split this up this 24,000 into six parts so that we can get it in the ratio 5: 1 so 24,000 if we split that into six just dividing by six that gives us 4,000 so each part in that ratio is worth 4,000 and the ratio is 5 to 1 so if we times them both by 4,000 we're going to get some quite large numbers here when we do this in terms of fitting it below the ratio but that's going to give us 20,000 in place of the five and 4,000 in place of the one as we are just multiplying it by one so 20,000 to 4,000 and again that's the volume in cm cubed so in other words the sand here is 4,000 cm cubed and the water just above that is 20,000 cm cubed there we go so it says in the final lines here work out the number of liters of water in the tank well we know the amount of water in the tank in terms of cm cubed is 20,000 so we just need to know what the conversion is between centimet cubed and lers so this is one that you do need to know it's ,000 cm cubed is equal to 1 liter so if 1,000 cm cubed is equal to 1 liter we would want to divide that by a th000 to see how many liters it is so 20,000 divided by 1,000 is quite a nice one that just comes out as 20 lers so there we go our final answer for that is 20 lers now there are other ways that you could have gone about this as well you could have actually found the height in terms of a fraction you could have worked out 3/4 of 32 that would have given you the height of this part here then you could have split the height into the ratio 5:1 just to work out and I'm kind of overlapping with the lines here just to work out the height of the water just there and then you could have worked out the volume of the cuboid there which is in the in the sort of water shape inside the cuboid or or the actual just the water which is in the shape of a cuboid so you could have done it that way as well I do prefer just working out the volume here particularly as not all questions are you know in the same um functional sense as this one you can just have a cuboid that you have to work out the volume of so it's good for us to practice just doing it that way as well but there we go that's how we would approach a sort of more difficult problem here where we do have this functional aspect and all that fractions and ratio work in between now with a distance time graph you can have lots of different types of questions you can have to complete the graph you can have to talk about aspects of the graph in terms of flat lines and what they mean and all that all that sort of stuff in between but this question here says between which two lines does the car travel at its greatest speed and give a reason for your answer so we'll talk about that but we'll talk about the whole journey because that'll explain other questions as well so you have the car leaving here where we start at a zero distance and goes up at quite a fast speed there and then it slows down so it slows down for this part of the journey here and then it stops and it just goes along here okay so the distance doesn't change in that part so you got three parts here you've got the fastest part of the journey where it's going up much faster you can see the gradient of that line or the steepness of the line is the highest then it slows down for this part here you can see the gradient changes it's not as Steep and then it doesn't go any further the distance for a period of time here where it has stopped so that's how we would describe the three aspects of the journey so in terms of between which two times does it go at its greatest speed well that's that first part of the line here which we can trace down and you can see it's between 0o and 20 so in terms of our answer here we would say between 0 to 20 as the line is the steepest or as the line has the greatest gradient at that point okay as the line is steeper here okay it does actually ask us in Part B to work out the speed so we are actually going to work out the speed of that point there and there are a couple of ways that we could go about this so that is a a time of only 20 seconds in that particular point and that is going up to we're going to have to count this very carefully so look looks like that's 300 there halfway in between and then it's five squares to get to 400 so there's a difference of a 100 if we split that into five that means each Square goes up by 20 so that would be 320 340 360 so that's 360 M that it goes in 20 seconds so if it goes 360 m in 20 seconds well there's a couple of way that we could give a speed it doesn't say the units that we have to give a speed we could give it in either meters per second or we could give it in meters per minute it doesn't actually say so it's completely up to you at this point if we want to find it in 1 second which would be a speed in meters per second then we would just need to divide both of these by 20 as that is in 20 seconds at the moment so we could do 360 divided by 20 and we can obviously do that to the side if if you want to work that out you can do 36 / 2 instead which does make it a little bit easier but that comes out as 18 m/s so 18 m in 1 second would be written as 18 m per second and that' be our final answer for that one of course you can use a speed distance time formula as well speed is equal to distance divided by time the distance was 360 we divided that by 20 and it's giving us our answer of 18 m/s there we go 80 m per second if you wanted it or wanted to work out in a speed per minute for example you could have multiplied them both by three that would have got you to 60 seconds which would then got you a speed in meters per minute that would be fine as well as it doesn't give us any units that we have to follow there we go there is looking at a distance time graph so when looking at linear graphs we may or may not be given a table of values that we have to fill in as well but this one here doesn't give us a table table and it says on the grid draw the graph of y is = to 3x - 2 for values of x from -2 to 4 now it may not even give you this so it usually does but where it says values of X it might not give you that and you have to find it from the graph over here we can see the graph goes from -2 to 4 more often than not it does give it to us though but if it doesn't then we know where to find it now for this I would always usually advise drawing a table of values so we know our our values of X we know it has to go from -2 to four so we can draw all the numbers 1 2 3 and four we need to find the values of Y then we'll know the coordinates and where we can plot them now this equation here is just a code that tells us how to find y it says that Y is equal to 3 X's takeway two well if we don't have a calculator especially for this I would always start over here with the positive number just it's easier to put the positive values into that equation so 3 x means 3 * X well X is 4 so we would do 3 * 4 and then take away 2 3 * 4 is 12 takeway 2 gives us 10 and then we would just follow the same process for each number just replacing the four here with our new X number so for the next one we would do exactly the same to find this position here but we would do 3 * 3 and then take away two well 3 * 3 is 9 takeway 2 is 7 we can then do the same for the next one replacing the three again but this time we are looking at where the x value is two so for this one here 3 * 2 take away two or 3 * 2 is 6 takeway two is four now once you have done a few values for a linear graph where it is going in a straight line we can actually spot a pattern and you can see here that it is going down in threes so if it's going down in threes we don't need to continue to doing our substitution we can just start subtracting three so four takeway 3 is 1 takeway 3 is -2 takeway 3 is5 takeway 3 is8 so from there we can now just plot the points of course if you did start on the left you could follow the same pattern as well you could spot that it's going up in threes this way and just continue the pattern so if you can just continue the pattern we can then look at plotting the graph so plotting this graph we have -2 and -8 -1 and -5 0 and -2 and you might even be able to start spotting a pattern in terms of the coordinates so the next one we have 1 one 2 4 3 7 and four and 10 there we go once we have plotted all the points you would get a ruler and a pencil and join these up using a perfect straight line and that would be your graph drawn and that question finished now a few things to take note of you can also see that this number at the end this -2 is linked to this point here that's called the Y intercept so where it's where it crosses the y- axis that's going to help us if we're working the other way around and also this pattern of plus 3s happens to be the number in front of X now that is called the gradient it shows us how to draw these points as well you can go one across and three up and you can see there is a little staircase pattern that goes up by three each time so we're going to have a look at a question like that as well but that would be how we would go about plotting a graph using a table of values now when we are approaching one of these questions essentially in reverse because this has drawn the line for us and it says to find an equation for L now when we are finding the equation of a line it's important to take note that all the equations of lines are in the form y isal to mx + C now like we did discussed very quickly on the last one C in this instance here is our Y intercept and the Y intercept can be found really easily you can see it's just down here where the line crosses through -6 so finding the first part of the equation we know that Y is going to equal something something Min - 6 and that right there is our Y intercept the only other thing we have to find is the gradient now the gradient we can find in one of two ways we can either find two coordinates and calculate the gradient between them or we can look at the pattern between those two coordinates so if I pick two nice whole number coordinates you can see that there's one there one here and one there we're only going to choose two of them so I'm going to select this one here it doesn't matter which two we choose and this one here now that coordinate there is two across zero up so 2 0 and this coordinate here is one across 3 down so 1 3 to find the gradient between those two points you do the change in the y coordinate divided by the change in the x coordinate which you can also call the rise over the run so if we have a look we have a triangle here where our rise goes from -3 to zero so that's a rise of three and the Run goes from one to two so that's a run of one so the change in y is three the change in X is 1 and 3 div 1 is 3 and that is our value of M so this value here is our gradient so this would be 3x minus 6 so there we go m is the gradient and we use the formula the change in y over change in X to work that out so our equation to finish this question is y is equal to 3x - 6 now although I did that really nicely using this triangle here that only had a run of one you could actually have used the other coordinate and had a slightly larger triangle there and that's where the formula really does help so for this we would now have a run of two and this would have a rise of six so if I did use those two coordinates I would still have done the change in Y6 the change in X2 and 6 divid two is still three so it doesn't matter which ones we choose as long as we use whole number coordinates we wouldn't want to use one here where we can't quite tell what the rise the runn is from that point so we do want to use those nice whole numbers to find it there we go that's how we go about finding the equation of a line from a diagram so we can also have some more complex problems where we are looking at gradient of a line here it says a is the point with coordinates 59 and B is the point with coordinates D15 the gradient of the line AB is three work out the value of D there's a couple of ways we could think about this but let's try and think about what this might look like so if we imagine this is coordinate a and this has the coordinate 5 9 now we know the gradient is three so we know it's going to be sloping upwards and it's going to get to this coordinate B now I know B is going to be up here because the y coordinate is 15 and the y-coordinate was nine on the previous one so we know we're going up to get to point B so this here has the coordinate D 15 and we know from the last question to find the gradient we would find the rise and the Run or the change in y over change in X and that would give us the answer here of three so can we work out either of those well it goes from five along to D so we don't know what the run is but we can work out the rise or the change in Y which goes from 9 to 15 to get from 9 up to 15 that would be a rise of six we don't know what the run is so let's call that X for now but we do know what the gradient is and we know that if we do 6 ided x whatever that X number is that we would get the answer three as it's told us the gradient is three now that's a relatively nice problem to solve if you look at that 6 divided something is equal to 3 well we know that X will have to equal two 6 / two is equal to three so we know the distance there must be two now The Not So Nice part of this question is that kind of feels like you found the answer at that point but not quite because that is the run the run here is equal to two so if we get rid of the X and we replace that with two we now know the distance that it goes across and it started on five it goes two across to get to D so to get from five over to D that is a distance of two across so we know that D will have to equal 5 + 2 which is equal to 7 so the coordinate up there would be 715 but the value of d is seven so we go we could write D is equal to seven or we could just write seven but we have found that using that idea behind the rise change in y over change in X or the rise over run now when we are looking at Transformations there are four main Transformations that we need to look at this one here says describe the single transformation that Maps shape a onto shape B now this is a translation which just means to move a shape and when we are moving a shape on a coord grid like this we write it using a vector so here we're going to have a look at what the vector is and how we write it so we need to pick a point on the shape so to get from A to B we need to also pick the same point so in order for that shape to move we don't want to be matching the top right there with say the bottom left because that's not going to give us an accurate movement if we matched it up to the bottom left actually the shape would be down here so we do need to match up the perfect points but from there all we actually have to do is count how many squares where's across and down it goes now we could go down and across or across and down we're going to get the same answer either way I'm going to go across so 1 2 3 four jumps check the scale that is going up by one each time so I'm happy that that is a jump of four across and it's going down by 1 2 3 now when we go down we do say minus three as you can see on the axis here positive numbers go up and negative numbers go down the same on the x- axis positive numbers go to the right and negative numbers go to the left so we tend to just give the same symbol in front of the number in terms of the direction we're going in so down is negative and left is also negative so as a column Vector in terms of the movement the top number goes uh is the X movement so the left and the right so we have gone four across and the bottom number on a vector is the Y movement or up up and down we've gone down by three so that Vector there inwards means four right and it means three down and of course if we changed the four to a negative four that would mean four left and if we changed the negative3 to a positive three that would mean three up so now we have the vector the description here we would have to say is a translation which is the mathematical word for moving so a translation ideally then say by the vector and then write the actual Vector so four and negative3 so that would be a full description there of what's happened and of course you could be asked to describe that the other way around so from B to a and that would just be obviously slightly different because we would be going left and up which would change the symbols to -4 and positive3 okay so when it comes to a rotation you are allowed to use tracing paper for these questions so for a question like this where we are having to describe a rotation it's a little bit trickier than if we are to do the rotation ourself so typically a question that's asking us to do it oursel will tell us the point of rotation in which case we can put the tracing paper over the top we can trace the shape we can put our pencil onto the point of rotation such as the origin and then we can spin the paper round and obviously it will tell us whether to go 90° clockwise or anticlockwise or essentially 180° now with this type of question it's asking us to actually describe what's going on so we'll do the same First Step we're trying to look from P to Q so here if we were to rotate the triangle and hopefully you can see it's going 90° anticlockwise but if we rotate it it means we're going to have to find the point of rotation so here if we use the origin and we go 90 degrees anticlockwise you can see that it doesn't land on the shape so what you have to do is you have to pull it back you have to readjust your pencil put it into a different position till you find the point and once you found the correct Point you'll notice that when you do your rotation it does land over perfectly on top of the shape so this one here it went 90° anticlockwise and you can see that my pencil is on the point 01 so the way we would describe this we would just say it's a rotation 90° anticlockwise about the point Z 01 okay so in this one we have a reflection so it says Kyle reflects triangle a in the axis to get triangle B he then reflects triangle B in the line Y is equal to X to get Triangle C Amy reflects triangle a in the line yal X to get triangle D she is then going to reflect in the x-axis to get triangle e okay there's a lot of descriptions here Amy says that triangle e should be in the same position as Triangle C is Amy correct correct well without actually going about doing these Reflections that's very hard to figure out so we do need to actually go about doing this so we'll start with Kyle we'll do that in one color so Kyle is going to reflect in the x-axis to start with now the x-axis is this line here so when we reflect that we're going to count each point and make sure we do that perfectly so one square away is down there for that first point and that's three squares away for the next point and you can see then that that's going to line up and it has reflected and that is now triangle B he then reflects triangle B in the line Y is equal to X now Y is equal to X is a diagonal line that goes perfectly through all of the same coordinates so 1 1 2 2 3 3 4 4 55 Etc and that is our reflection line Y = X now reflecting in one of these lines is not very nice to do what I would normally recommend doing is actually rotating the paper and front of view so that the line is pointing upwards because in this particular uh reflection we are going to have to count diagonally along the squares so that's one two diagonals along and a half we want to go the same distance away so the half first and then an extra two diagonals away and gets us to there we're going to do the same with the other points I'm going to go for this bottom left one so that's one 2 three diagonals so one two three diagonals gets us to here obviously we've drawn all over the diagram now so I'm going to get rid of some of these just so it's going to help me to finish off this last one and the last one is sort of overlapping the shape which makes it even worse so I'll do it in a different color for this one let's go one two three and a half and we'll do the half first and then an extra three so one two three away so that point gets us up to to there now that is reflecting diagonally in a line yal X which isn't very nice to do at all some people don't like the way it looks it doesn't quite look right but just trust the process you are reflecting per the perfect distance away at the perfect 90° angle to the reflection line but to us it's almost a 45 Dee angle which is very strange um for us to actually visualize there we go that is Triangle C so that's the first one done we are now going to look at the next one where Amy is reflecting so Amy reflects triangle a in the line yal X to start with that's much nicer than that last one because it's already connected to the line yals X so at least one of those points isn't going to move anywhere we've got this point here to reflect half a square away so half a square away gets us to there and then this point which is one and a half diagonals away one and a half diagonals away gets us to there so there we go that is the one in the line yal X and we can join that up and and we're going to call that triangle D there we go and then it says she's then going to reflect triangle D in the x-axis to get triangle e so again the x-axis is down here and we're going to reflect that in the x-axis so we'll start with that bottom point which is three away one two three gets us to here the next one's four away so that gets us to there and then the top left point on that triangle is also four away which gets us to here so there we go it's very close to Triangle B but triangle e is not in the same position so we have shown how we've got our answer because we have done all the reflections on the diagram but we just need to answer is Amy correct so we would say no Amy is not correct there we go so no am is not correct and we've shown all of our Reflections on the diagram including one on a diagonal line now looking at an enlargement now typically exam questions tend to be a little bit easy easier than this one when we are looking at enlargements but this is quite an interesting one when we have a scale factor of a third typically exam questions just sort of ask you to expand or enlarge the shape by a factor of two sometimes not even from a particular point so they can be a lot easier than these ones but if we can do this one then we can both enlarge by a integer scale factor and we can work in Reverse as well so this one says to enlarge it by a scale factor of a third from the center 01 one which is just down here now to do an enlargement all we actually have to do is count the vector to a particular point so to get to that point there thinking about vectors again we go three across and three up now if we are going to do a scale factor version of that all we have to do is apply the scale factor to that to that movement so if it is 3 three to get to that point while oneir of that so if we do a third of that we would get one 1 one so our new shape what we need to do from that Center of enlargement that we've already got we'll get rid of this we just need to do the vector one one so one across one up and that takes us to here then we just have to draw the shape in one3 of the size so you can see if you look at this shape it goes up by six while 1/3 of that is two so we'd go two up and I might do this in a different color just to differentiate that from that Vector so we would go two up the line along the top there is three along so we would go one along the same for here three along so we'll do a third of that which is one and then we can join it up and there we go there's our new shape now you can see from this as well this is why I like this question we could also think about this in Reverse so if that was the shape that was given to us the one that we have drawn in it might have said enlarge the shape by a scale factor of three from the point and it's basically just the reverse of what we've done so so if we pick a point we go across one and one that is the vector one and one if we want to do a scale factor of three we would just multiply them all by three actually I better better not write it like that because it kind of looks like a power the way that I've written it but I'll write it more like this so timesing by a third and here we are going to be timesing by three so when we Times by three our Vector is going to be 3 three and we already knew that so going back to our original position we would go three along three up and then we would draw the shape in three times the size again just counting the sides this is a length of two timesing that by three gives us a length of six so this is a really good question for looking at it in both ways but it is quite rare for a fractional scale factor to come up on one of these ones but you can get some nice questions where it does just ask you just in larger shape and put it anywhere on a grid but there you go this is a great question to have a look at just for thinking about how you would approach it using a vector now we've already looked at a little bit of sharing in a ratio but ratios are a standalone topic as well that can come up just as a ratio question this is a really good one it says the perimeter of a right angle triangle is 72 cm and the lengths of its sides are in the ratio 3 45 work out the area of the triangle now as it is a right angle triangle if we were to draw a sketch of a right angle triangle which is going to help a lot with this question we know that the area of a triangle is the base times the height so essentially we want to know what's this length here and what's this length here and for working out the area of the triangle that hypotenuse there or that longest side doesn't really interest us so we'll have a look and we'll share this in the ratio 345 we should already hopefully remember from earlier as well we've got a total there so that adds up to a total of 12 parts so to figure this out we'll start by doing 72 / 12 12 which tells us each part is worth six well ratio is 3: 4 to 5 each of those numbers we will need to multiply by 12 Sorry by six to get to each part so 3 * 6 is 18 4 * 6 is 24 and 5 * 6 is 30 now we know the longest side there is going to be our hypotenuse that's the one we don't need but we don't actually know where 18 and 24 go now because of the way we work out the area of a triangle it doesn't actually really matter now because of the way that I've drawn it the base looks bigger so I'll put 24 there and 18 here but essentially we are just going to multiply those numbers and then divide our answer by two so it doesn't really matter what order we put them down in so to work out the area of a triangle we'll do 18 * 24 then we're going to divide our answer by two and that's our area of the triangle so let's work this out on a calculator we 18 * 24 that gives us a total of 432 that we need to divide by two so dividing that by two will give us a total area here of 216 cm is our units squared as its area so 216 cm squared these questions where we have fractions percentages and ratios are relatively common questions we already saw one earlier where we did have fractions and ratios involved D but this has all three says Daniel bakes 420 cakes he bakes only vanilla banana lemon and chocolate 27th of the cakes of vanilla 35% of the cakes of banana and then lemon and chocolate are in the ratio four to five look at the number of lemon cakes Daniel bakes well to start with here it does give us the two7 our vanilla so straight away we can ignore the other two we can kind of work through these in stages we've got one two three things to work out and the lemon cake that we're looking for are in that very last piece of information so we will get there but we'll start by working out 27ths so starting with number one there we could work out 17th to start with 17th of 420 that's a relatively nice one to work out because 42 divides by 7 that's six so that would be 60 okay just dividing by seven we want 27ths so to get from 17th to 27th we just need to multiply by two so 27th is going to be 120 and that is vanilla so we've got 120 vanilla cakes The Next Step we've got the percentage we've got the 35% so we'll label this over here so step two 35% well if we are going to work at a percentage of an amount we'll always start with 10% or 1% but I like to start with 10% so 10% of 420 would be 42 cakes we are going to want to get to 35% so getting 5% is going to be helpful as well so half of 42 is 21 and now we can use that to get to 35% so we're going to want three of the 10% one of the 5% and that's going to get us to 35% so it doesn't matter how you work that out you can do multiplication I'm just going to add three of those together so 42 42 42 and the 21 that gives me 35% in total so 2 4 6 and 7 4 8 124 so that's 147 and there we go so that's 147 and that was for the banana cakes there we go right so we're almost done we now know how many vanilla and banana cakes there are but now it says the ratio of the number of lemon cakes to the number of chocolate cakes is four to five that's a little bit of a different explanation here because previously he said 27th of the cakes 35% of the cakes so that was referring to the total but here in this third step it's only referring to the lemon and the chocolate cakes it's not referring to that as a ratio of 420 it's ratio of the lemon to Chocolate so we need to actually work out how many lemon and chocolate cakes there are in total so we're going to have to add together what we've already worked out 120 and 147 that gives us a total of 267 cakes we've got 420 in total so taking away what we already have so taking away 267 let's see what that gives us we're going to have to do some borrowing so 10 takeway 7 is three borrow again 11 takeway 6 is five 3 takeway 2 is 1 so we have 153 cakes and they are going to need to be split in the ratio four to five so again thinking about sharing the ratio that is nine Parts in total so 153 divided by 9 let's work that out so 9 goes into 15 once remainder six so 9 goes into 63 seven times which means each part in the ratio is worth 17 so from here we've got the ratio four to five each part we've just worked out is 17 so 17 * 4 would give us 68 and 17 * 5 would give us 85 now of course we only need one of those because it says work out the number of lemon cakes that was the ratio of lemon to Chocolate so lemon is our number on the left and our final answer is 68 so there we go a lot to workout there we had a fraction of an amount to start with a percentage of an amount and then what was left over was split in the ratio four to five so you might need to read through that one again but a lot of the worded questions like this are very similar where we have to work out a ratio a fraction and a percentage in order to solve a problem okay so when're looking at combining ratios it doesn't necessarily say in the question that you need to combine the ratios sometimes it can sometimes it can say two separate ratios and ask you to write a three-part Ratio or sometimes in a question it might just give you two different ratios and expect to know or you to know that that's something that you're going to need to do so this one here says the number of houses and the number of flats are in the ratio 7 to4 and then it says the number of flats and the number of Bungalows are in the ratio 8 to five it then says there are 50 bungalows in the village how many houses are there now there are a couple of different ways that you could use to work this and you can actually just about get there without actually combining these ratios but whenever we see something like this where we have two ratios where we have the same thing in both in this case we have flats in both we are going to take an approach where we combine those ratios now the way that I approach them is I tend to write the three things out so we've got houses to Flats and then the other one is Flats to Bungalows so we can put the Bungalows after the flats there but because these two separate ratios we can't combine them straight away now the scenario where we could combine them straight away is if this number here for the flat and this number here for the flats in the other ratio if they were the same we could combine these straight away but in this case they're not they are different so houses to Flats it says is 7 to four and then Flats to Bungalows is 8 to 5 now all you need to do is look at these two middle numbers so we have a four and an eight and think to yourself what can we do to make that number the same now it might be that you have to change both of the ratios but in this case we can actually make the four become an eight by multiplying that ratio by two so if we do multiply this top ratio by two the four would become an eight and these seven would become 14 now that all the numbers are the same they can all the numbers in the middle are the same they can actually just be combined so we can just write 14 to 8 to 5 and that's our new three-part ratio so from here we could you know if we were told the total we could actually share in that ratio now if we were told the difference between some of them we could obviously find the difference between them now but this question here says that there are 50 bungalows in the village which is this number at the end which is a five so to get from five to this new amount of 50 we just need to think what we have to multiply by now hopefully you can spot that that is timesing by 10 but if you're not sure you can always do 50 divided by five which tells us that we've multiplied by 10 so as with all ratios we're now just going to multiply each of the numbers by 10 and we get 140 80 and 50 we can now answer any question so this particular question says how many houses are there in the village well houses is the number at the end here so that would be 140 so again just combining the ratios there by making sure that the overlap number in this case which was the flats the thing which is in both of the ratios making that number the same and then essentially just sort of squeezing that ratio together and putting them as a three part ratio so there we go that was a bit of combining ratios okay so when it comes to direct proportion all we need to remember with these types of questions is that as one goes up the other goes up as well or as one goes down the other goes down and that's just what in things in direct proportion mean so here it says we're building a wall which is 5 m long and the complete wall is going to be 8 m long and wants to know how many bricks we need so it's going to go up proportionally in the same way that we used the bricks for the first 5 MERS so we can write this out thinking about this in a sort of table way although we don't need to draw a table we can say that 5 m is going to need 300 bricks so in order to figure out how many bricks we're going to need for 8 m or for anything else we want to go down to how many bricks we're going to need for 1 meter so to find the amount of bricks we need for 1 meter we can divide this by five if we divide the 300 Bricks by five on the other side as well that's going to tell us how many bricks we'll need so 300 divided by 5 you can do bus stop division to the side if it helps but that's going to be 60 Bricks now because we need that now 60 bricks per meter we can pretty much answer any question for however many meters the question is asking about we need to read this carefully though because it says how many more bricks does Jack need so we don't necessarily need to jump now down to 8 m because we only want to know how many more that we need so to get from 5 to 8 that's going to be an additional 3 m so we only really want to know how many bricks we're going to need for 3 m to get from 1 meter to 3 m we multiply by three so the exact same on the other side to get from 60 bricks to our new amount of bricks we're also going to multiply by three so 60 multipli by three gives us 180 so our final answer would be 180 bricks so you can see there all I've done is I've made a comparison to the side where I have done one the same to one side as I have done to the other so as the meters went down the amount of bricks is obviously going to go down it's quite logical and then as the amount of meters goes back up the same the bricks go up in the same proportion so that's going to be really helpful because when we look at inverse proportion we're going to do something very similar to this but in a slightly different in a slightly different way because with inverse proportion that's the opposite it means as one goes up the other goes down but that's what we're going to have a look at now so when it comes to inverse proportion when we are looking at these we can take exactly the same approach but it's understanding that this is about inverse proportion it says here it would take 120 Minutes to fill a swimming pool using water from five Taps how many minutes will it take to fill the pool if only three Taps are used well hopefully it seems quite logical here that that if we were to use less Taps to fill up a pool then the amount of time it takes to fill it up is going to go up so you can see there straight away that as one goes down the other goes up and that's what makes it inverse proportion so we can do the same thing with a kind of almost table we can say here and I'll put some titles for this one this time I'll say here over here we'll put the minutes and over here we'll put the Taps so at the moment we have 120 minutes and that is for five Taps now just like before we want to figure out how long it would take for One Tap so over here again we would just divide by five and that's going to take us down to one tap but we don't want to divide the minutes by five because if we're using five times the the amount of less taps or five times less Taps then the time is going to be five times more or in other words we're going to have to actually multiply by five on this side and 120 * 5 is going to be 600 minutes and if you think think about that logically it makes a lot of sense if we use five Taps that takes 120 well it would take five times longer if we are only using one tap so it makes sense that as we divide down to one tap on the one side we would multiply by that same number on the other side for the amount of minutes as well as that then if we're going to go to three Taps because it does say how long will it take if only three of the Taps are used well to get to three on the right hand side we're now going to multiply by three so on the other side it's going to be three times faster or in other words we're going to divide by three so that will give us an answer of 200 minutes so our final answer for that question is 200 and hopefully you can see the link between what we've done on either side of the table and it's quite logical there that as one goes up the other goes down now of course any of these questions we are assuming in our working that the Taps are running at the same rate and that leads us into Part B Because if one of the Taps was running faster than the other other then obviously the time could change it could be more or less our working out could be completely wrong if we turn off the best tap the one that fills up with the most water in the fastest amount of time then obviously it's going to make a big impact so in terms of our assumption there we would want to say something about the rate of the water so we could say we assume the Taps run at the same rate there we go and that same rate can be used in quite a lot of these questions to do with inverse proportion particularly when it's talking about workers or Builders and how long it takes them to actually complete a job so we do assume in those scenarios that the workers are working at the same rate as well so there we go there is a bit of inverse proportion okay so when we are looking at Best Buys here's another example of where things are in direct proportion so here milk sold in two sizes of bottle so four pint PS cost 1 18 and six pints cost 174 now these are two different deals but the amount of pints in the bottle is proportional to the price so in other words it means that in the first option there we are paying a certain amount for one point obviously we're buying four of them but the price for one point is the same for all the four that's in there the same as the six points each point cost the same but it's broken up into six points for this one so when we are looking at which is the best value for money we want to know which we are paying the least for for that one point so for the one on the right if we start with that one we could actually just divide that by six there are two different ways of doing this and we will talk about both but we could do 174 divided by six and that's going to tell us the cost for one point typically these are calculator questions there is a non-calculator method as well but we'll do the calculator method to start with so we'll just type that in and we get 0.2 n which is 29 P per pint for the one on the left the four pints will take the same approach this will be1 18 P divided by 4 again just typing that in on the calculator 1 18 divided by 4 it comes out as 0.295 so you can see straight away that these are relatively close to one another they typically will be on these questions it's also getting us to have a look and see if we understand decimals so you could balance them both out with zeros at the end of a decimal if you need to so that they both have three decimal places or you might just be able to recognize here that this one on the right has a zero at the end and the one on the left there has a five at the end so the six points is better value and we would say six points is better value and that finishes it off and we have shown all of our working like the question has asked there so we've got clear working that shows why it's better value so there we go the other method that you could take to do this as well you could actually look at which is the lowest common multiple so here we've got four points and six points the lowest common multiple of four and six is 12 so you could say well if we buy 12 points of this that's multiplying it by three three lots of four or if we buy 12 pints of this one which would be multiplying it by two which one do we get for the cheaper price and that's something you can do quite nicely without the calculator because you can just add them up using column addition particularly when it's only three and two so there we go two different methods that you could do both absolutely fine but we are then looking at the better value by either comparing the cost of one unit or the same amount of both in a different form like here where we could have 12 points now when looking at exchange rates it's really important that we just understand how to move between one currency to another this is quite a long question because it has two different exchange rates in so we just need to match them up carefully but it says Gina pays or finds out the price of a CD box set in three different countries or a CD box set in three different countries the price is £98 in the UK $134.99 in the USA and € 139.99 EOS in Germany it says the exchange rates are and then we've got the pounds to dollars and we've got the Euros to pounds and we've got the exchange rate here shown in two different ways one where it's one pound to the currency in dollars and the other one where we have the euro currency as one and the exchange rate into pounds so we need to think logically about these exchange rates because typically with exchange rates you just multiply by the exchange rate to go abroad and divide to come home but this here has given us an exchange rate as if we were already abroad so that's the exchange rate you'd get if you were actually in Germany you would have the exchange rate coming back to pounds so here when we are looking at which is the cheapest price for the box set we actually need to put them all into the same currency typically in these questions we can move them all back into pounds so when we are looking to move them back into pounds let's start with one we'll start with this one here and we'll change the the uh dollars amount first so you can see in the actual exchange rate that the pound number not the value but the number itself is smaller than the dollar number so if it's smaller than the dollar number we know if we take 13499 we only have two options we can either multiply by 1.43 or we can divide by 1.43 now you can type them both in on a calculator but you will see the one that gives us the logical answer that takes us back to a smaller number in pounds is if we divide by 1.43 so if we do that divide by 1.43 that takes us back to Ina £94 and 39 and a bit Pence I'll put it to three decimal PL places 398 it's very unlikely that these are going to come out to four decimal places to the same value they could but it's very unlikely so I'm going to leave it like that because we should be able to make a comparison you can see that that is cheaper than in the UK so we already have one option that might be the better one now for the next one we are given Euros as one and then pounds as the smaller number again this time so we know that the pound number is going to be smaller than the Euro number again they have the same value but the actual number itself is smaller so here when we are actually going from our 139 99 Euros down to pounds this time if we divided by the exchange rate we'd actually get a larger number so because it's the sort of exchange rate that we would get when we were abroad we kind of have to do it in the opposite way so here we do have to multiply by the 0.73 that is going to give us the smaller number that we are looking for so 13999 time 0.73 and that gives us2 and I'll keep it to three decimal places we've got one nine and this sort of rounds to a three it say seven after the two so there we go um we didn't actually really need to write it to as accurate amount of decimal places because you can see here it's very clear which is the cheapest we have this one in the USA which has given us a 94 um price compared to the £98 in the UK and12 in Germany so in terms of buying the box set we would want to buy the box set from the USA and we've shown how we got our answer because it's all up here and it's all nice and clear in terms of our working out it says here though that Gina lives in the UK why might your answer to a not be the best country for Gina to buy the box set from this is an interesting question because there not really any maths that we need to do for this it's more just a logical question in terms of why it might not be the best idea if we want a box set that's only you know £370 36070 different to buy from the USA why might it not be the best place to buy it from well if we are living in the UK and we want to buy a box set that's going to cost money to be shipped over to the UK and it might be that that postage price is pushing it over the top of the price in the UK it might be that it's free but it might be that that postage does actually cost money it does say why it might not be the best place so you could give any options here that are logical and reasonable I think the best option in this scenario though we would just say that the postage prices might make it more expensive okay so there we go there is a question looking at exchange rates okay so moving on to some recipes says here Dion needs 50 g of sugar to make 15 biscuits she also needs three times as as much flour flour as sugar and two times as much butter as sugar Dion is going to make 60 Biscuit to work out the amount of flour she needs well it says here in this first part we need three times as much flour as sugar so if we need 50 gr of sugar to make 15 biscuits how many grams of sugar are we going to need to make 60 biscuits because straight away we obviously have different amounts there well if we go up in 15s you've got 15 30 45 60 so to make 60 biscuits we're going to need to use four lots of that amount of sugar so the first thing to do is 50 g multiplied by four which would give us 200 g of uh sugar for the 15 biscuits so we need 200 gam of sugar and it says obviously just here we need three times as much flour as sugar so we'll take the 200 g of sugar we need three times as much so multiplying that by three and that gives us 600 G of flour so there we go that's our final answer for that one 600 G for the next part it says Dion has to buy all the butter she needs to make 60 biscuits she buys the butter in 250 gram packs how many packs of butter does Dion need to buy well up here it says that we need two times as much butter as sugar so if we needed 200 g of sugar and we need two times as much butter as sugar we would times that by two and that comes out as 400 G of butter that we are going to need but of course we have to buy this in 250 g packs so if you were using a calculator you could do 400 / by 250 and it's going to come out as 1 point something but thinking logically here if we buy the first pack of butter we'll have 250 g if we add another pack of butter because obviously we don't have enough we get to 500 G now that's enough butter now to to actually make this obviously we have 100 gram more than we need but we're not going to be able to make it with only 250 gram so we will have to buy two packs here and there we go that would be our final answer two packs of butter and that's going to allow us to make this recipe so there we go that is a little bit of recipes but typically with recipe questions you are looking to see how how much is needed to make one and we can apply a direct approach or a direct proportion approach in order to do that so here if this 15 biscuits at the top needed 50 gram of sugar well what we could have done is we could have done 50 g divided it by 15 and that would have told us how many grams we needed for one biscuit from there we could have multiplied that by 60 because we were making 60 biscuits that would have also got us to that 200 g of sugar so we can also take that approach but this was quite a nice one as it did tell you exactly what proportion you needed and to get from 15 to 60 was also quite a nice version because it was just a multiple of four but if it wasn't one that you could jump to straight away for example if we had to go from 15 biscuits down to and this said 70 biscuits that would have been a lot more complicated for us to do so we would have probably taken that approach there by doing 50 g divided by 15 to see how much we needed for one biscuit and then just multiplied it to whichever one we needed but there we go one variation of a question looking at recipes now we've already looked at a bit of Pythagoras Theorem within this video but this is an interesting question here as we move more into Pythagoras and trigonometry so it says triangle ABC has a perimeter of 20 it says ab ab is 7 BC is four and by calculation deduce whether triangle ABC is a right angled triangle now if we actually draw a triangle and we label it ABC so we imagine a right angled triangle we don't really know where the numbers are just yet but we can call it a b and c now here it says A to B is seven so that's actually a longer length than the length of four but we don't actually know which is the longest length on this triangle so before we put any of these letters on and just start putting the numbers anywhere let's think about what the other length is as well because it does say here that the perimeter which is all of the lengths on the outside added together is 20 well if we do 7 + 4 that equals 11 and therefore 20 takeway 11 would tell us that the final side that's not been given to us is 9 cm now that is the longest side so we would say this is 9 cm we could say this one's 4 cm just because of the way that I've drawn it it does look like it would be the the shorter of the sides and this one would be seven so it says a to b is 7 cenm so and then and B to C is 4 cm so we'd have to have this one as B this is a and this is C just based on the way I've drawn it of course you could draw this in lots of different ways you could put the seven on the left the four on the bottom but that matches up with what they've given us in the question so it says by calculation deduce whether triangle ABC is a right angled Triangle Well to know if something is a right angle triangle we can use Pythagoras Theorem because we know with Pythagoras Theorem that a 2 + b^ 2 is equal to c^2 and we have been given the values of the shorter sides so because we've been given those we can have a look and see well does 4^ 2 + 7 s does that equal c^ 2 or 9 s is it equal to 9 s now if we test this out we'll see what we get well 4 SAR is 16 7 SAR is 49 and 9 squar is 81 so what's 16 + 49 well add the 6 is 55 add the 10 is 65 so does 65 equal 81 absolutely not they are not equal so we would say that that is not a right angle triangle so to say deduce whether it's a right angled triangle so we would want to give a description here to say no it's not a right angle triangle because that is not equal you could also have said in other words like when if we were working out the length of this one here we could have actually done the square root of 65 and see does that equal 80 or so does it equal and of which case it doesn't Okay so two different ways that we can obviously show that final step there but that was using a little bit of Pythagoras in a slightly different way now this has some Pythagoras Theorem which is almost hidden within the question it tells us here that we have a trapezium A square has the same perimeter as this trapezium work out the area of the square give your answer to three significant figures well if we knew all of these lengths we know A square has equal side lengths so for example if they all added up to 40 we would would know that all of these were 10 and then we could obviously work out the area of that square by doing base times height or length time width we don't actually know what the perimeter is though because we are missing this side here so we have a missing side on a trapezium so this is similar to a question that we've looked at before but we have to split this up to find a right angle triangle now obviously doesn't say in the question that we're going to be using Pythagoras anywhere this is one of those questions where you just have to spot that oh actually there's a right angle triangle that's connected here that I could perhaps use Pythagoras on now we know that this five here is the same as the height over there because that's now just a rectangle and a triangle put together so that is 5 cm and for this length down here that's going to be the difference between the top of the rectangle which is seven and the full length along the bottom here which is nine so the difference between those is going to be 2 cm and now we have a right angle triangle that we can use py for so now we can use a 2 + b 2 is equal to c^ 2 We've Got A or B we can use in either order so 2^ 2 + 5^ S which is 4 + 25 which is 29 and the final step now we are actually going to have to square root this this time so the square root of 25 this would be a calculator question so we're going to type it in the square root of 29 sorry is 5 3 8 and a lot more decimals so 5 1 6 4807 now at this point in this type of question you might think that you've made a mistake there because that's going to be quite a horrible number that now to add up and get a perimeter for but it does give us a massive hint in the question that we are going to get a quite a long decimal it says just here give your answer correct to three significant figures that gives us a massive hint that there are going to be some long decimals involved so now we can actually work out the perimeter of this shape which is going to be the same as the perimeter of the square so if we add them all together five + 7 + 9 plus this number 53851 164 807 let's add those all together on the calculator so add five add seven and add nine that gives us an answer of 26385 1 16 481 so that is the perimeter of the square the square has four equal side lengths so we can actually divide that by four that perimeter which we can write up here 26 3 85 and some more decimals we can divide it by four that'll tell us each of the side lengths as the square has equal side lengths so divide by four and we get a side length of 6.59 six and some more decimals but I'm just going to write a few of them down because to find the area of a square we're just going to times those two numbers together or in other words I could actually just put squared with it on my calculator rather than typing it all in again so I'm actually just going to press answer squared but you could just do obviously write it write the whole number down and make sure you times them both together so I'm going to do 6 596 and the rest of the decimals and I am just going to square that so I'm just going to press the x squ button on my calculator and that gives me an area of 43511 05 there are some more decimals but I've no in the question it only wants my answer to three significant figures so if it want my answer to three significant figures I can go one two three chop it off after that and then just apply my rules of rounding so I didn't need to write all of those decimals out so it's a one after the line so that's going to be 43.5 and the answer there is in cm squared so again there you go using a bit of Pythagoras but also involved in a problem there where we had to find the overall perimeter we had to split that then into the perimeter of a square and work out the area of that square so a lot maths going on in that question that be a good problem there involving some Pythagoras okay so when it comes to trigonometry obviously with trigonometry there are different ways that you can learn this now a lot of people people do like to do is use a formula triangle so regardless of which method you use you will be okay using a formula triangle if you know the actual trigonometric ratios but for the purpose of everyone here I will use a soaka TOA formula triangle there we go just writing them all out so we have soaka and TOA so deter to determine which one we're going to use we need to label the sides we know already from Pythagoras that this side here is is called the hypotenuse opposite the angle in the question is called the opposite and the other side is called the adjacent you don't need to label up all three sides you only actually need to label the side that you've been given and the side that you're looking for now this question here is asking us to work out the length of ab and the length of ab is this length just here on the left of the triangle so we have that as the opposite we are given the hypotenuse so actually we didn't need to label the adjacent we've got the two sides in the question o and H so we look at our formula triangles we can see that the what the O and the H is in the sign triangle so we don't need this one and we don't need this one so as we are looking for the opposite if you're using a formula triangle you cover that up and we know there that it is s * H now s stands for sign and we put the angle with that in fact when you press it on the calculator most calculators will open a bracket for you to remind you to put the angle in with that so when we put 38 in we also need to close the bracket and then we're going to multiply that by the H which in this case is the hypotenuse so we just need to type in sin 38 multipli by 16 there we go and we have our answer and this one comes out as 9 85 05 a few more decimals after that but the question does say to give you answer to two decimal places so again we'll chop this off after the second decimal and we get an answer here of 9.85 as it say zero afterwards so 9.85 and the question's in centimeters so that would be centimeters now obviously there are six different variations that you could have for working out a side length in this question if it had a asked us to work out the adjacent well in that case we wouldn't have needed the opposite and to to work out the adjacent now we would have a and H and A and H is in our C triangle so instead we would have used this formula up here we would have been looking for the adjacent so in that case we would have still covered up the top there and it would have still been a multiplication if in either of those scenarios though we were looking for the hypotenuse then we would have had to have done a divide the good thing about the formula triangles it does show you which goes on the top and which goes on the bottom of your divide so we go lots of different variations of trigonometry there if you're not sure about all of them obviously you can check out the full video on that but that is just a reminder there for one version of a trigonometry question where we are looking at side lengths now when we looking at Angles we still still use a very similar process which is do a slightly different button on the calculator which allows us to find an angle so I'm still going to write down my Soaker TOA triangles so s o h c ah t o a and I'm still now just going to label up the triangle so in this particular question where we are looking for an angle and it has been highlighted for us it's the angle a b c we're going to label up the sides so this is our hypotenuse opposite the angles not given to us and we're not looking for it so we don't need to label the opposite we're just going to label the adjacent now for the adjacent and hypotenuse that is in our Center triangle there a and H and we are not looking for the a or the H this time we are looking for the angle so we cover up the C and it tells us that we need to do a / H now looking at the numbers we've been given a is seven and H is 11 but of course if we type that into the calculator that's just going to give us a decimal so in order for the calculator to know that we are looking for an angle we do the inverse operation or in terms of the reverse this time we're looking for the angle rather than the side length so all you have to do and you should be hopefully be able to see it on a calculator just above the C button there is a little cos minus one and to get that you normally have to press shift or some form a button that allows you to access that so once you've done that you will have cos minus one on the calculator and it will open up a bracket to remind you to put something in there so we're going to put the fraction in you can write that as a divide as well and then close the bracket so you can either write it as a fraction or you can put 7 divided by 11 but if I press shift cause and then put the fraction in 7 over 11 close the bracket and we get an angle now which has come out as 50.4 78 and a few more I'm not going to write them all because this question does ask us to round it to one decimal place so if I cut it after the first decimal this time we have a seven after that first decimal place which is going to round it up so it is going to go to 50.5 and this question is asking for an angle so we would give that as 50.5 Dees so our final answer would be 50.5 de and of course there are three variations of this question where you could be given either the opposite and the hypotenuse in which case you'd use S the adjacent and the hypotenuse where we use cos and you could be given the adjacent and the opposite in which case we would use tan now moving on to some probability probability from a table is always quite a nice one probabilities have to add up to one one meaning absolutely certain and anything between zero and one being somewhere in between so here it tells us that we have some cubes it tells us that there are blue red and yellow cubes and we can see that in the table it tells us the probability of blue is 0.2 it says the number of red cubes in the box is the same as the number of yellow cubes in the box so we know here that these two are going to be the same so to find out what we're missing because we know these have to add up to one we would do one take away 0.2 and that leaves us with 0 .8 which is going to be split between those two boxes as they are equal that makes it nice for us to do we can just divide it by two of course that could be split in a ratio it's not in this case but it could be split in a ratio as well for example it might say it's in the ratio one to three in which case you'd have to share in a ratio like you normally would we'd add them together you'd have four parts so we divide it by four and then split it by one and three or multiply it by one and three so here both of these probabilities are 0.4 so completing the table there quite nice for us to do now just because this is written in decimals don't forget that these actually are just percentages or fractions as well so 0.2 is the same as 20% or you could write it as 2 over 10 or 1 over 5 we're writing as 2 over 10 because it's 210 we don't necessarily need to write it in its simplest form but there we go different ways that you could write this it's going to help us potentially with a Part B question so so here it says there are two 12 blue cubes in the Box work out the total number of cubes in the Box you could be asked one of two questions here sometimes it can say you know we're going to pick a certain amount of Cubes out or there are a certain amount of cubes in the Box what's what's the probability of picking a blue or how many Blues are in there so but this is actually told us specifically that there are 12 blue cubes so we know and you can either use a decimal percentage or a fraction here but we know that 20% of the cube cubes is equal to 12 because blue is 20% and that is also 12 blue cubes so to work out the total number well the total percentage is 100% so how do we get from 20% to 100% well we Times by five of course you could write that as 0.2 and think how do we get from 0.2 to 1 and that would also be timesing by five but here we go we can actually just times both sides by five that will give us the total 12 * 5 is 60 so our final answer here is 60 now of course that question could be give could be given to us in Reverse it could say there are 60 cubes in the Box how many blue cubes are there and in that circumstance if it had told us that there was 60 we would want to work out what 20% is which is the blue cubes to do that we'd start by working out 10% so 10% is 6 and then we could times that by two and get that 20% is 12 so two different ways that you could do it I do like that percentage method more in that scenario because you could probably get from 0.2 to one relatively nicely but there we go two different options that you could have in a Part B question similar to this so we're looking at probability on a tree is very similar when we are thinking about probability from a table except instead of the table adding up to one we sort of have these pairs of branches that add up to one so we've got one pair of branches there we've got another pair of branches here and another scenario where we have a pair of branches there so read the question carefully it says here we have a round pencil case and a square pencil case and we have the titles up here as well where the round and square pencil cases are says there are four blue pens and three red pens in the round pencil case so for the first one the round pencil case we have four blue and three red so we can write the probability on those branches here where it's asking us to complete the probability Tree by writing as fractions you can write them as decimals as well I wouldn't recommend decimals if you can write them as fractions because fractions is a lot easier just to sort of write them down here rather than trying to convert them into a decimal so for blue we have four blue pens and that's out of 4 + 3 which is equal to 7even so that's going to be four out of seven for the red pens they have to add up to seven anyway but it tells us that there's three so that's going to be three out of seven now that's the round pencil case done because on the second part of the tree these are both to do with the square pencil case we just have two different scenarios so this scenario over here is if we chose a blue pen out of the round pencil case and this scenario here is if we chose a red pen out of that pencil case so when we are drawing the next probabilities on we have got four or sorry we've got three and five pens 3 + 5 is equal to 8 so there's eight pens in that pencil case three of them are blue so three out of eight and five of them are red so five out of eight now it doesn't matter which one we pick out of the round pencil case the square pencil case doesn't change so down below here we have the same two probabilities three out of eight and five out of eight and that is our probability tree constructed we can now calculate probabilities of roots on this so for this one it's asked us work out the probability that the pens sinina takes are both red so taking both red ones would be taking a red one out of the first pencil case and then taking a red one out of the second pencil case so we almost go on a little journey through the tree and all we do is take the probabilities that we've passed through so three out of seven and 5 out of 8 and multiply those together so the calculation we need to do is 3 out of 7 multiplied by 5 out of 8 and multiplying fractions is one of the nicest topics that we can do because all you do is multiply the top 3 * 5 is 15 and 7 * 8 is 56 so our final answer for this probability is 15 over 56 you don't have to worry about simplifying a fraction in these types of questions it doesn't say to write it in a simpl form and actually not simplifying it is more useful because it tells us there that there are 15 different ways of doing this because there are three Reds and five Reds in the other one so there's 15 different combinations out of a total possible combin combination of 56 so here we don't need to simplify and nice and easy there just to multiply those fractions now of course you could get a slightly more complex question it could actually ask you to work out the probability of both blues that would be the same but if it said different colors or the same colors that could potentially be a little trickier because in that scenario you could have multiple branches that you need to work out so for example if it's a different colors you could take out a blue and then a red and likewise you could take out a red and then a blue and in that scenario you would need to work out both of those roots and you would add your answers together but quite nice because they both would have 56 on the bottom in fact if we did that scenario this one we would get 20 over 56 as it's 4 * 5 on the top this one's 3 * 3 on the top so we' get nine out of 56 and all you would do is combine those two and that would be 29 over 56 and that would be the probability of getting different colors and likewise you could do that for the same colors as well if you had two blues and Two Reds you could also do that so there we go that is some probability trees now probability trees can also have decimals on we treat them in exactly the same way but this question is actually asking us something that's gone wrong with this probability tree so it says here when a biased six-sided dice is thrown the probability it will land on a four is 0.65 and the bias dice is thrown twice air draws this probability tree but the diagram is not correct write down two things that are wrong with the probability tree diagram there's only a couple of things that can go wrong as we've already mentioned before these branches here have to add up to one so we can have a look at these and we can see if they add up to one let's look at the first one we have 0.65 and 0.25 you might be out a spot already but if we add those two together let's see what we get 10 6 7 8 9 that only adds up to 0.9 so this probability here is incorrect it tells us in the question that the probability of landing on a four is 0.65 so this here should be 0.35 so that is the first mistake that we can see as well as that the only other thing that can go wrong on a probability tree is that these ones could then be potentially MISD drawn now you can see they've got the 0.35 in both of these but they've been switched around it says in the question the probability to land on a four is 0.65 but here it says the probability of landing on a four is 0.35 and the .65 has been put down here so these have been switched around on that part of the branch they've got it correct on this side of the branch but here they have been switched so we could say 0.65 and 0.35 are switched or in the wrong positions and that would be our second reason there in terms of what's gone wrong here now if you were working out probabilities from this type of tree obviously if we had the correct numbers so 0.35 here instead you could also do this with decimals so it might say what's the probability of it not landing on a four U twice so here we would go well that would be going down this route and down this route that passes through 0.35 on both routes so to work out that probability you do 0.35 multipli by 0.35 most likely on a calculator and obviously giving your answer to that obviously this didn't ask for that but you can actually do the same process that we've just looked at but with decimals okay so looking at a ven diagram where we have some set theory involved we just need to get all of these numbers in the correct place so it says here at the start and we have that symbol there which just refers to the universal set or in other words all the numbers that need to go into our vend diagram it says odd numbers less than 30 now that gives us quite a lot of options we've got 1 three five and I would recommend writing them out but this one is a bit of a pain because we have a lot of numbers to write out but I would recommend it even if there's a lot of numbers there because the worst thing you can do is do all of this and then miss some numbers out so there we go that's all the odd numbers less than 30 it tells us that in a we have 3 n 15 21 and 27 and in B we have 5 15 and 25 now the first thing you want to look for is there a number that's in both and you can see here 15 is actually in both of them and five and 25 which are in B they're just in B so in the intersection in the middle of the ven diagram we can put the 15 we can then cross that off from our main list because we're not going to need to use that again we're going to take this similar approach as we go through them so in a we can put the remaining numbers so 3 9 21 and 27 we can cross them off from our main list three nine 21 and 27 then put the numbers that were in only B so five and 25 again Crossing those off the main list five and 25 and then all of those remaining numbers need to go on the Outside Inside the Box but outside of the circles so we have 1 7 11 13 17 19 23 and 29 and then go through and cross them off make sure you've not missed any so 1 7 11 13 are all in the diagram 17 19 23 29 they're all crossed off quick double check and they're all done for Part B here it says a number is chosen at random from the universal set so from all of these numbers in the vend diagram what's the probability that the number is in set a u now we've got two things that we need to really know when we are looking at set theory so you got the U which refers to the union now the Union in terms of a VIN diagram I won't draw the Box on the outside but the union refers to everything here everything in the middle and everything here so we're just looking at the numbers that are inside the circles including what's in the middle you can have the symbol upside down as well and that refers to the intersection and the intersection hopefully that seems quite logical with the language but the intersection is only referring to the numbers in the part that's where it's crossing over so in this in this question here it is looking at the union and the union we have got and if we go through we can count them up 1 2 3 4 five six seven numbers that are in the union so we know that as a probability that would be seven out of and then we can count how many numbers there are in total either from our list or we can count from the diagram 8 9 10 11 12 13 14 15 so that is going to be 7even out of 15 again if it did simplify we wouldn't need to but that would be the probability that it is in the union if it had asked us the probability of A and B in terms of the intersection well there's only one number in the intersection and that would be one out of 15 instead as there's only one number there but there we go that's a bit of diagrams and set theory okay so moving on to some compound interest now here it says that KT invests 200,000 in a savings account for four years any account pays a compound interest rate of 1.5% calculate the total amount of Interest KY will get at the end of four years now when we are looking at compound interest it means that the interest we get after the first year is going to be added onto the total and we're going to get a new amount of interest in the second year obviously that makes sense in real life we want to gain interest on the money that we've already earned in our bank account so here what we are going to do is use a calculator method for this one now if we think about how we would approach this let's think logically about what we have so at the moment in the bank account we've got2 200,000 and let's think about that as being a 100% of the money that we have after the first year we are going to add on an additional extra amount and that's going to be the extra 1.5% now in terms of finding a number that we can multiply 200,000 by to increase by 1.5% we call that a multiplier and to find it we just need to think logically about what this means now if we gained 0% interest so nothing ever changed we just had the same amount of money you could say that we multiplied it by one and that's what this 100% represents or in other words if we do 200,000 and multiply it by one we still have 200,000 we just get the same amount and that would be equivalent to a 0% interest rate in other words it doesn't increase by anything now if we want to increase that multiplier we can add extra bits onto the one so for this one we want to increase by 1.5% so we need to think about what decimal we would add after the one to increase by just 1.5% and we can do that by turning this percentage into a decimal now it can help when doing this to think about some some sort of easier decimal maybe something like 1% 1% as a decimal remember you can just divide it by 100 on the calculator to convert it to a decimal but you will get 0.01 then the pattern for a 1.5% percentage becomes quite apparent because if we have 1.5% it would be. not one and then a five at the end so we go right into that third decimal place for this percent percentage so here if we are adding on an additional 1.5% we are adding on an additional 0.015 and if we add those two together our multiplier becomes 1.015 and that's the number that you can multiply by to actually increase straight away by 1.5% now what you can do you can obviously just add these together as a percentage so that would be 10.5% that you have in the second year and then you can convert that into a decimal by dividing by a thousand so you can always convert a percentage to a decimal Sorry by dividing by 100 definitely not by a thousand so there if we divide by 100 we get our multiplier anyway or you can think logically in terms of adding the decimal onto one so here if we were to do £200,000 and we were to multiply that by our multiplier here 1.015 that's going to add on 1.5% now we have to do this for four years so it you could if you wanted write out the answer repeat the process four times so that you have that answer at the end of four years but if you have four years there is a shortcut that you can take and that is just putting in a power of four which of course just means the Times by 1.015 four times so we'll do it all in one go for us basically on the calculator if we can save a bit of time in the exam then we definitely want to stick to that method so I'm going to type it into the calculator 200,000 * 1.015 use my power button and put a four in there and we get a final answer here of 212,00 272 and then it goes 7101 now our monetary system only goes up to two decimal places that's to the nearest penny so we can either round to two decimal places or we can do something we looked at earlier which which is truncate it to two decimal places essentially just delete anything that comes after that second decimal it would be fair to use either method this one here it doesn't actually matter because following our rules of rounding it it' be 71 P anyway but if it was a five after the line there it wouldn't be incorrect for us to just round it down even following those rules of rounding thinking logically why we might do that and that's because we haven't actually hit 72 P of Interest so we wouldn't expect the bank to give us money that we haven't actually earned so that's the total amount of money in the bank at the end of the four years of course this question though did say calculate the total amount of interest and because we are working out the total amount of interest that just means that we need to subtract the amount that we started with just to figure out what that extra amount of money that was that we actually got so if we do that we can do it on the calculator take away 200,000 and it gives us our final answer here which is 12,272 and of course that that 71 P left over so there we go that is compound interest if you had simple interest simple interest is a lot nicer than doing compound interest because you can just work out the 1.5% and then that's just what we get every year so for example we could work out 1.5% of 200,000 which we can do by using the multiplier here as well 0.015 or you can work out 1% half a percent and add it together but if we do two 200,000 * 0.015 that tells us that we would get £3,000 interest in the first year now if it was simple interest of course this is compound interest but if it was simple interest we would then just multiply that by four and we would get £12,000 so you can see with simple interest we would obviously get less money there only 272 and 71 P less um but of course we are getting less there with simple IND interest so there we go there's a bit of compound interest and a little thought about if it did say simple interest now you can do compound interest without a calculator so we'll have a think about how you would do this if you were to do it without a calculator so it says here Toby paid £7,500 or invested £7,500 for two years in a savings account he was paid 4% per anom compound interest how much money did Toby have in his savings account at the end of two years well if we're going to do this I like to split the page up so here we've got two years so it's not too bad to do a calculator so we'll first work out year one and then we'll work out year two and almost lay this out like a bit of a bank account so we're going to start off with £7,500 and we are getting each year 4% interest so we can start by working out some percentages now you can either work out 10% and then 1% or you could just go straight to 10% I'm going to work up sorry straight to 1% I'm going to go for 10% which is £750 and divide it by 10 again which is £75 from there we can get 4% really nicely because we just need to multiply this by four to get from 1% to 4% and that is going to be300 so in the first year 4% is £300 and if we add that on that gives us a total of £ 7,800 now because this is compound interest that now means that at the start of year two we now have £ 7,800 I'm going to earn the interest on that new amount of money so again I would take the same process either working out 10% or 1% so 10% is 780 1% is going to be 78 and this time again getting 4% obviously it's going to be slightly higher than it was before so we need to do 78 * 4 we can obviously do that to the side if we don't have a calculator 4 * 8 is 32 7 * 4 is 28 8 plus the 3 is 31 and then we have £312 that we are earning in interest in the second year finishing that off we just need to add it on to 7,800 you might know what that is but we can obviously just do that to the side adding £312 onto that and that gives us two 1 11 and8 so after two years Toby is going to have in his bank account 8,1 £12 and of course if we had to go for a a third year here we would start to get decimals involved but we would just follow the exact same process getting 1% timesing by four and then adding it on so there you go compound interest can be done as well without a calculator now when it comes to depreciation this is going to be approached in pretty much the same way that we approach compound interest except with depreciation it's when it's falling in value so rather ra than going up in value it's actually going down in value and again we're going to use a multiplier method for this so here it says Natalia pays 13,99 and then we have Lauren as well but we have the rate of depreciation for Natalia being 12% and Lauren being 133% says whose car will have the greater value at the end of three years so if we have a look at this now if we think about depreciation if our car is currently at its 100% of its value and we're going to have a look at the Natalia's car to start with which is 12% depreciation well if it's losing 12% or in other words we're taking away 12% we need to think about what percentage is actually going to be left over so 100% take away 12 would leave us with only 88% of its value now both of these percentages can be written as decimals this one's 0.12 and this one is .88 so whatever whichever one we want to use we can actually just multiply Again by that number like we did with compound interest except we multiplied by one point something here because it's going down in value we're taking it away from one and multiplying by this not Point number now if we were to multiply by 0.12 that would tell us what 12% of the car's value was but we don't necessarily want to know what 12% is we want to know what it's worth after 12% has been taken away so if we multiply by 0 888 that will tell us straight away what the value of the car is after it's lost 12% so we would take the value of her car 13,99 and multiply that by 0.88 now that would tell us the value after one year but this question of course it says the value after three years so just like with compound interest we can add a power onto our multiplier in this case a power of three and that will multiply it three times for us so 0.88 to the power of three and that gives us a value of Natalia's car which comes out as 9,537 and 20 if we round it to the nearest two decimal places so that's that person's car done we now need to have a look at Lauren's car now Lauren's car cost 14,495 and her car is losing 133% of its value so 100% take away 133% leaves us with 87% so very close to the other one so we would Times by 0.87 again we are looking at three years we would still use a power of three we just need to type that into our calculator and then we're going to compare these values so 0.87 to the power of three and Lauren's car is worth 9,545 and it's actually 0 P at the end so looking at both these numbers now it says which car or whose car will have the greater value at the end of the three years well we just need to write a conclusion here you can see that although it's very close we can see that Lauren's car has the greater value so we would make the statement Lauren you could say Lauren's car has the greater value you can write a little sentence to go with it if you like but we've got all the working out there that backs it up and we can see very clearly that Lauren's car has the greater value now Part B here says the rate of depreciation assumed for Natalia's car was actually too low how does this affect the value of her car at the end of three years well with this sort of question here if it's too low or it's assumed that it the car was the depreciation rate was too low well let's just imagine well what if it was too low what if it was actually 20% you can pick any number of course now if we actually worked that out and it was going to lose more value then the value of her car at the end of three years would of course be lower of course you can actually work that out on the calculator if you want if we do that we'd do 13 995 we can work it out down here if you want 13 995 times taking away 20% would leave you 80% so 0.8 to the^ of three and if we typ that into the calculator if that was the case her car's value would be £ 7,165 44 so you can see of course course the actual value of the car has gone down quite significantly so in terms of how does this affect the value of her car at the end of three years well the value of her car would be lower so that's all we'd need to say you don't really need to give an example there but the important part is the value of her car would be lower and there we go that would be a fine answer for that particular question now when it comes to a reverse percentage I tend to use what I called or what what we tend to call the bar model because I like to think about this visually now it says here it doesn't say anywhere that it is a reverse percentage it says Jules buys a washing machine 20% V8 is added onto the price and then Jules has to pay 600 what the price of the washing machine with no vat added now obviously the price that we've been given is the current price of the washing machine after some money or after that 20% was added on now where it asks us what's the price of the washing machine with no vat added that's essentially asking us to go back into the past and if we ever have to find a past price after a percentage change has already taken place that's what we tend to call a reverse percentage and drawing this as a bar model can kind of how to visualize that so if we think about the price of the washing machine which was originally 100% of its price and then somebody came along and added 20% on well this new price of the washing machine is now going to be in terms of the original price that's now 120% of its value now in a question it's not well this type of question it's not going to give us the 100% because obviously that was the price with no vat added on but the question might have given us this part it might have said what the vat was or it might tell us here the new price and in this question here it's told us that the new price is £600 so to find this out would run a write down that 120% is now equal to £600 or £600 is equal to 120% now the whole aim of this is to try and get this percentage back to 100% it's currently at 120 so if you have a calculator then you can always divide by the percentage that would take you down to 1% so you could divide by 120 and then you can very quickly get that to 100 by timesing by 100 and you would just do that to the £600 on the right and that would get us our price but this could be a non-calculator question so instead we can also try and think about what we would do if we didn't have a calculator well it's always usually quite nice down uh to divide down to 10% so just think to yourself what do you have to times 10 by to get to 120 hopefully you can see that that's timesing by 12 so we would have to divide by 12 to get down to 10% it might be that we have to use some bus stop to do that so 600 ided by 12 12 goes into 65 times and then zero so 10% is 50 from there we can turn 10% back into 100% really nicely because we just have to multiply by 10 and multiplying by 10 really nice with a calculator because we just add the zero on and the price therefore must have been £500 so our answer here is500 now with reverse percentages there's quite a lot of different scenarios that you could have obviously we've mentioned two here where money is added on and you're given the amount that's added on or you're given the new price which is the total after it's added on but with a reverse percentage you can also have a sale so it might be that there is something like a 20% sale and if there is a 20% sale that's quite similar to when we were doing depreciation because that's where money is getting taken off and in a 20% sale you could be given the 80% that's left or the 20% that was taken off so for example this here if they said the sale price was then that would be our sale price of course if it's to do with a sale and this would be almost what we would call the discount or the money that was actually taken off the original price together of course they make 100% just like our previous one together made 120 so there we go that would be how we would approach a reverse percentage if it was going up or if it was going down and this little process on the left here where we are looking at how to break it down to a smaller percentage which can then turn into 100% And of course if you have a calculator you can always divide by the percentage to get 1% and then just Times by 100 now when it comes to density questions where we can see the word density involved there are a couple of different ways that you can actually approach these but as soon as we see that word density we most likely want to write down our formula now the formula is density is equal to mass / by volume and you may also use this in a formula triangle so you might write this formula out as density is equal to mass divided by volume whichever one you use it's absolutely fine of course if you use the formula you need to know how to rearrange that to find mass or volume as well but let's read this question and see what where we need to go with our formula so it says a gold bar has a mass of 12 .5 kg the density of gold is 19.3 G per cm cubed work out the volume of the gold bar and give your answer correct to three significant figures now that three significant figures there does give us a hint that we may have some decimals involved but we'll wait and see how the question progresses now the first thing that stands out to me in this question this is what you have to watch out for quite a lot with these density questions is that the units are given to us in kilograms when it talks about the gold bar and grams when it talks about the density we want these units to be the same and it is much easier for us to turn the weight or the mass of this SK bar into grams than it is to try and change our density so let's turn that into grams there are a th000 grams in a kilogram so we would want to times this by a th and if we times that by a th we get 125 0 0 12,500 G now we can actually use our formula just in one go because if we know that we now have the units all the same well to work out the volume depending on which way method you use you can either cover up the volume on your formula triangle which means that you know it is mass divided by density or you can rearrange your formula to get volume equals and you will see you get mass divided by density as well so here the only working out that we need to do is take the mass which is 12,500 divide it by the density which is 19 to three we've confirmed that it's all in the same units so if we type this in on the calculator divided by 19.3 we get an answer which comes out as 647 668 a few more decimals here but it did want my answer to three significant figures so here we've got the six the four the seven is the third significant figure so we'll cut it there so it's also going to be to the nearest whole number after the line is a six so it's going to round it up that would give us a six 648 now this question was talking about volume so we need to know what units to give here and the units are always given to you in the density so here you are given that it's in grams and it's in cenm cubed so here our volume units would be cm cubed and there we go that would be our final answer 648 cm cubed now if you get one of the harder density questions you can get something that looks like this where things are getting mixed together so here it says the density of apple juice is 1.05 we have the density of fruit syrup the density of water and then it tells us some volumes there that are mixed together so it says we have 25 cm cubed of apple juice 15 cm cubed of the fruit syrup and 280 cm cubed of the water this is a particularly horrible one because we have three things that are getting mixed together to make a drink and it says work out the density of the drink giving your answer to two decimal places now when we have quite a large question like this the first thing is it asks us to work out a density so straight away no matter which method you use whether it's a formula triangle or the formula itself we need to write those down so that we don't forget what we do have to use a formula so density is equal to mass over volume and we've got a lot of information that we need to sort of figure out here now you could go line by line trying to match everything up and figure it out but when we have a mixture like this I like to put all the information into a two-way table so I'm going to give myself enough space to have the apple juice the fruit syrup the carbonated water and the actual drink mixture at the end and then along the side I'm just going to list the densities the masses and the volumes so I'm going to put here density mass and volume you can of course write those in any order here I'm going to put the apple juice here I'm going to have the fruit here I'm going to have the water and then then at the end it's called a drink there but you can put whatever you can put the total or the mixture whatever it is that you prefer to write now if I go through I can actually tick these off and start putting them in my table so the apple juice the density is 1.05 the fruit syrup is 1.4 the water is 0.99 now the only issue is here we can't just add the densities together all of these are getting mixed up and it's going to form a new density that's going to be somewhere in between all three of them so density is going to tell us how dense that drink is and each one is different so depending on how much the volume of each that is mixed in is going to depend what the actual density comes out as so we'll leave that for the moment and we'll put in some either the masses or the volumes now the next bit of information is given to us is volumes we have 25 cm cubed of apple juice so that's going to be 25 just here we've got 15 cm cubed of fruit syrup and 280 cm cubed of the water now of course if we pour a load of volumes into a cup or into a jug or something they will all mix together and we will have a larger volume it does tell us that in the question here as well it does say we've got 320 cm cubed so how would we go about working out the density well we are missing three items here now of course we know that to get density you take the mass divided by the volume well we have the total volume right here the only thing that we are missing to work out the density is this total mass so we can work out the mass for each of these three items here because we have the volume and the density and to work out the mass if we use our formula triangle we can cover up the mass and it's just density multiplied by volume so for the three boxes there we have three things to work out we have the Apple which is going to be 1.05 multiplied by the volume 25 we have the fruit which is going to be 1.4 multipli by 15 and we have the water which is 0.99 multiplied by 280 so if we work all of these out and this is a calculator question so 1.05 * 25 and that comes out as 26.25 the 1.4 multiplied by 15 that comes out as 21 and 0.99 multip by 280 and that comes out as 2772 now of course masses can all be added together you add these three things onto a scale all of those masses will combine so if we add them all together plus 21 + 26.25 we get a total mass of 32445 so we're almost done all we actually need to work out now is the overall density we have the overall Mass we have the overall volume and we know of course from our formula or from our formula triangle that density is equal to mass divided volume so here we want the density so we would take the overall Mass 32445 divide it by the volume 320 and we'll type that into our calculator and it gives us an answer here and I'm going to have to type that one back in again so 32445 / 320 and I get an answer that comes out as 1.01 39 some more decimals but again I'm not going to write everything down because it does only ask for my answer to two decimal places so we're going to chop there after the one and my final answer for this density would be 1.01 check on the dotted line to see if it's given you the units already this one hasn't but check the question it does say in each of these these were in grams per centimeter cubed so we can write that as G per cm cubed and that would be our final answer there for the density of the drink now when it comes to speed distance and time we can take a similar approach we can use tables as well to sort of keep all this information together but you don't always need to now here just assess the question it says olle drove 56 km from Liverpool to Manchester he then drove 61 km from Manchester to Sheffield and then it gives us a speed a time and says to work out the average speed for the total drive from Liverpool to Sheffield now for this particular type of question as it is a combination or a mixture of Journeys we can actually draw this all in a table as well so we can put a table in we can put the journeys down and we need to think about our formula when we are looking at speed so speed is equal to distance divided by time so distance divided by time again that can be put into a formula triangle and we can have speed distance and time there we go and we can start putting the information into our table so speed distance and time down the side we have Liverpool to Manchester for our first journey we have Manchester to Sheffield field for the second journey and we have the overall Journey or the total at the end so we've got 56 km from Liverpool to Manchester that is a distance so we put 56 in here 61 km from Manchester to Sheffield another distance and you can add those two together straight away because if we've gone 56 kmers and another 61 well in total we have gone 117 km for the overall Journey it says O's average speed speed from Liverpool to Manchester was 70 so for the speed in the first journey we have 70 and for the second one we have 75 minutes now typically when we are working out these speeds you can see we have given them or it's been given to us in colmet per hour but this here has been given to us in minutes so it is worth converting this straight into hours that if you were to write it out is 1 hour and it's 15 minutes above 60 so that's 1 hour and 15 minutes and we need to write that as a decimal as well so that we can actually type it into our calculator and use it now 15 minutes is 15 out of 60 Minutes in terms of an hour and if we type that into our calculator 15 divided 60 that comes out as one4 or 0.25 as a decimal 0.25 is the decimal for one quar so not to be confused with 0.15 because it's 15 minutes now obviously minutes are not out of 100 they are out of 60 as there are 60 minutes in an hour so it's 0.25 so here it's 1 hour and 15 minutes so for that Journey it is 1.25 for the time now we're trying to work out the average speed for the overall Journey now the average speed is just up here and we can get the average speed as long as we have the distance and the time we have the distance just below we've got 117 but we are missing the overall time so problem here is actually to find that now we can find the time in this box just here because we have the distance and the speed and if we look at our formula triangle to find time we cover up time and it's distance divided by speed so the working out for this box would be 56 ided by the speed which is 70 so we can type that into our calculator 56 ided by 70 and that comes out as 0.8 so overall we have 0.8 hours 1.25 hours and in total if we add those two together that gives us 2.05 hours we now have all the information that we will need in order to get the overall speed we know that speed is equal to dist distance divided by time from our formula so here if we are going to do distance divided by time we have 117 divided by 2.05 and if we take that into our calculator 117 divid 2.05 gives us it's not the nicest number here we get 57. 07317 make sure you look at this very carefully because this is actually a recurring pattern of 037 717 so that would be our overall speed 57. 07317 with a recurring decimal above those numbers now at no point in the question does it say to give your answer to one decimal place or to any certain amount of significant figures so if it doesn't we won't round the answer we'll write the answer as it is 57 07317 and not forgetting our units which may or may not be given to us on the dotted line here but that is in kilometers per hour and there we go we would leave it like that even though it looks a little bit strange obviously check through you're working but don't doubt yourself if you do get a strange looking number as long as you're confident with the methods that you've used a and you've double checked your answer we should be absolutely fine with that obviously take note of the question here we didn't actually need the speed for this for this box here so just because we've drawn a table doesn't necessarily need mean that we need every box to be filled in because for this one we only needed the overall distance and the overall time there we go that's looking at a speed distance time mixture now if we are looking at speed distance time without a calculator we can do that as well so this one here says Gary drove from London to Sheffield it took him three hours at an average speed of 80 kilm per hour Lynn drove from London to Sheffield and she took 5 hours assuming that Lynn drove along the same roads as Gary and did not take a break work out Lynn's average speed from London to Sheffield now here we can obviously think about this in terms of a use of a formula or we can just think about how these are proportional to one another so we can kind of do this in a bit of a table we can have the time and we can have the distance and we can put our pieces into here to figure out anything missing that we don't already have so here we are given a speed which is 80 km in 1 hour obviously see we it took him 3 hours but we know that in 1 hour he is going to go 80 km so whenever we see a speed we can always input it into a table like this and it's going to help us actually to find any distance in a particular amount of time so here it tells us that it t tells us that it takes him 3 hours so to get to three hours we would just multiply each side by three and that's going to tell us how far he goes in 3 hours so that comes out as 240 km that's just 80 multiplied by 3 so here we know that Gary has done 240 km and of course if you did use a formula triangle for this you would do the same bit of working out speed distance time and we are working out the distance so we would cover this up we would do speed times time and the speed is 80 the time is three so it's 80 * 3 which is exactly what we've done the next part it says Lynn drove from London to Sheffield she took 5 hours assuming that she drove along the same roads and did not take a break work out her speed well that means she's done the same distance but it took her five hours so again if we were to draw a table and of course you can use the formula triangle as well but this is just something a little bit different for you we can put these both into a table we know that she did 240 km and it took her 5 hours to work out a speed we want this down here to be 1 hour our speed is given as a kilometer per hour to get there we just have to divide by five we would do the same to the other side dividing by five and of course we don't have a calculator for this so we just need to do a bit of bus stop division so we've got 240 divided by five five goes into 24 four times up to 20 remainder of four and then it goes into 40 eight times so her speed she's going 48 km in 1 hour and that is a speed of 48 kilm per hour and there we go that would be our final answer for that one 48 km in 1 hour which is a speed of 48 km an hour now of course you don't have to write those tables out hopefully that's just a nice visual way for you to see what we are doing but of course you can stick to using a formula triangle and in that second step there we had a time we had a distance so we're just doing distance divided by time which was why we did 240 divided by five for Part B it says if Lynn did not drive along the same rounds as Gary explain how this could affect your answer to part A well if she didn't drive along the same roads as Gary and we won't write this out onto the screen would just talk about this one if she didn't drive along the same roads as Gary or she could have driven a higher distance she could have driven a lower distance so in terms of our answer to part a bearing in mind that our answer to part A was an average speed well we could say that Lynn could have driven a greater or a shorter distance and therefore her speed could actually have been higher or lower and the important part there is to say that it could actually be either of course if she actually drove um along more roads or more distance than car then that would have meant that her speed was definitely not the 48 km/ hour that we've worked out as the same as if she took a shorter journey and of course it ended up taking her 5 hours rather than three then again her speed would not have been the 48 km an hour so it can change dependent either way it could be higher it could be lower okay so when doing a perpendicular bis sector you do need to use a ruler and a compass for these when we're drawing them we are going to leave our construction lines in but typically a question will say You must show all your construction lines now this is quite a unique perpendicular bis sector because it's asking us to draw one that also passes through the point P typically if we just have a line we can do a perpendicular bis sector and we don't need to worry about it going through a particular point because we're looking at halfway but if we have a point P such as this we just need to make sure that that point is halfway through the line so to do this we would get the point of our compass and put it right on that dot there so we're going to put that there and then we are going to extend our Compass so that our pencil is touching the edge of the line if we now spin our Compass through that up to the point of the line there Point p is now perfectly in the middle so if we now construct a perpendicular bis sector of our new point so our new point now is looking at the line from there there to there so what we can do is we can now do a perpendicular bis sector as normal so the way we do a perpendicular bis sector we are going to put our compass point on one of the ends I'm going to put it on the letter d and I'm going to extend my uh Compass Beyond halfway if I go less than halfway my arcs are not going to cross over so I want to go past the point P somewhere here and then we're going to do a nice semicircle with both points of the compass hoping it doesn't go off the paper there mine has so I'm going to go a little bit shorter with mine so we'll get rid of that one we'll go a little bit shorter so it's still passed halfway but hopefully we get a slightly smaller Arc there we go and just a slightly smaller Arc now I don't want to change the length of the compass so I'm going to keep the compass exactly as it is I'm going to pull it around the other point and then I'm going to continue into a semicircle the other way and what you'll see is we get these two crossover points so here I've got one crossover point just here I've got another one just above that and then I'm going to get my ruler I'm going to draw a nice straight line going through those two points and that will go perfectly through the point at the top obviously we need to use a ruler and a pencil to do this it's a little bit more difficult for me to draw a perfectly straight line I'm going to do it as best as I can and that line should go perfectly up and through Point P so there we go we have constructed a perpendicular line that goes perfectly up through the line CD and into Point P of course if you only had to draw a perpendicular bis sector then obviously we'd only do the second steps there we wouldn't have had to have worried about this curve here which was cutting the line down so that P was in the middle now when we are looking at drawing an angle bis sector we are obviously going to be given an angle and the angle may be on a slightly different drawing it may be on sort of like like a sometimes quite a common one is like a drawing of a garden and you have to split the garden sort of via an angle so here we've just got an angle it just says to construct uh a line that bcts angle a and again it shows you must show all your construction lines now this is probably the nicest construction to do we just put our compass on the point we're going to extend our Compass we're going to draw two curves we're going to sort of draw one here and we're going to draw a slightly bigger one as well so bear that in mind we'll try and draw a small one to start with so we'll spin the compass round we get a nice Arc we are then going to extend the compass slightly further obviously I don't want it to go off the line so just double check that it's not going to be too long there and we'll draw another Arc we're now done with the compass so we can pull the compass out the way and then we're going to get our ruler and we're going to join up what I call like a crisscross so the top one there with the bottom one here you're going to do that using a ruler and a pencil I'm going to just join it up and then we are going to join the lower one on the left with with the upper one on the right and we're going to join those up We join those up again obviously you're going to use a roller and a pencil for that I've not drawn a particularly straight line so we'll try and make it a little better there we go and where those lines cross over so they cross over just here that is going to be where we're going to join up to point a so I'm going to get my roller I'm going to join those two together draw a nice straight line and that has now split the angle in half so if that full angle for example was an 80° angle we would know that on this side it would be 40° and on this side it would be 40 degrees as well now we've not been given an angle here we don't have to measure it and get it perfect we just need to do as best as we can when we are bisecting an angle and that's the best and easiest process I think for constructing this particular bis sector okay so when we are constructing triangles typically we will be given one length of that triangle this one here here says to accurately draw an isoceles triangle with side lengths 8 7 and seven it says one side of the triangle has been drawn for you so here the triangle has been drawn to 8 cm or the base of the triangle has been drawn to 8 cm which we can obviously check now sometimes depending on when you print these off especially depending on the size of the paper that it's been printed off on the ones that you get might not be actually perfectly drawn to the 8 cm just depending on the scale that it's been cut down to but for here when we are practicing we will go by what the question says and it's asked us to draw two of side lengths which are 7 cm if we need to do this we need to get a ruler we need to get a compass again we need to put it on the ruler and measure out our Compass perfectly to 7 cm so you can see there the pencil is pointing right at the seven we can then pull the compass down to one side so I'm going to put it right on the left there of the triangle and I'm going to spin an arc all the way up to the top drawing almost a quarter Circle I'm then going to pull my compass without changing the length because I want it to stay as 7 cm to the other side of the line and I'm going to draw another Arc coming down again you don't have to go all the way down to the line there but you can if you like so this point here where they have joined up at the top and I'm going to just draw a DOT on that line that is the point there that we need to join this up to so we're going to join the bottom left right up to that point of course using a ruler and a pencil drawing a nice line there and then of course I'm going to do the same on the other side so using a roller and a pencil again I'm going to draw a nice straight line coming up to the top of the triangle to finish this off I am just going to label those sides so this one here is 7 cm and this one over here is also 7 cm and that there has drawn an isoceles triangle of course if we had a different type of triangle let's imagine instead it said it wanted this one to be 9 cm then all you would have done is changed the compass length to 9 cm when you were drawing one of the arcs and then it would have given you one where it was seven and nine as the other two side lengths there we go that's how we go about constructing a triangle okay so when we are looking at plans and elevations we are essentially drawing some of the faces or the views that we are able to see from particular directions now this is quite a complicated one as it does involve some scale drawing as well but you can see that we have a 3D shape that's been drawn for us it says on the centimeter grid draw the front and side elevation of the prism and use a scale of 2 cm to 1 meter now we'll worry about the scale in just a second but for starters it says to draw the front elevation and the side elevation and you can see that the front elevation has been given to us here and also the side elevation has been given to us there in fact we'll keep these color coded so if we are going to draw the front elevation it's essentially say if we were standing perfectly on the side of this shape what would we actually see well we wouldn't see any of the sides or the top we would definitely wouldn't see the back or the base we would only see this face here which is in the shape of a trapesium so we need to figure out what that's going to actually look like in terms of on the CM grid now you can probably see to the side it's going to be this sort of shape we just need to make sure that it's the right size when we actually go about drawing it so here if we have been told that 2 cm is going to be equal to 1 M and we have a two or a 1 cm grid to draw on then the 2 m there is going to be 4 cm every meter is 2 cm so this 2 m height is going to be 4 CM so we can start just by drawing that line so 1 2 3 four squares and there we go down at the base that's also 2 m so each meter is 2 cm so that as well is going to be 4 cm so we can go 1 two 3 four along and then it's hidden behind the lines I've drawn now but this little height here is only half of a meter every 2 cm is 1 meter well this is only half of a meter so that is just going to be 1 cm so we're going to go 1 cm up and that's that height there and then that diagonal line we are given the length of that but actually we know that that is now just connecting those two together so we can finish it off without worrying so much about the actual distance there now that is our front elevation so we can label that the front elevation we don't necessarily need to it doesn't say to label them but it does help I think just to label them so we know which one we're looking at now from the side we we will see this rectangle here and we will also see if we are standing over here of course the height isn't going to just disappear we will see some of this we won't see the full length of it because we are not standing above and looking down on it but we will see some of it some of it above the height of the rectangle so if we start by drawing the height of the rectangle again it's 0.5 M High which is only going to be the 1 cm just like before so we can draw this centimeter in it's a meter long so that's going to be 2 cm which perfectly fits our scale 1 meter is 2 cm so that's going to be 2 cm along and then we can join that up and form our rectangle now the rest of the shape is going up to the same height okay so obviously the height of our shape is going up to here so this shape on the side is going to be the same height reaching that 4 cm height and there we go that's what it would look like obviously taking note of the fact and we'll hold on we'll label this side first that we do have this line drawn here now that line indicates where there is a change of sort of direction of the shape or depth of the shape so because the shape starts sort of curving back it goes up in a straight line and then it curves backwards sort of starts slanting backwards we have a change of height there and that's what that solid line actually represents so there we go that would be a front elevation and a side elevation and that was quite a tricky one because we did have some scale drawing involved with that as well and taking note of the fact that we had to convert between the meters on the shape to a different amount of centimeters on the grid it would have been nice there if the scale had been 1 cm to 1 meter then that would have been really nice for us to do because we know that the 2 m then would have just been 2 cm and that would have been a lot nicer for us to draw but for this particular question we did have quite a nasty conversion there but there we go that is plans and elevations now when it comes to bearings there are a few rules that we need to know so a bearing is always measured from North we always measure clockwise and we always give our answer as a three digigit so here if something like 50° although that's not a bearing there if we were to give a bearing of 50° we would write that as 050 that puts it into three and that's what makes it a bearing of course the bearing is measured from North though that's just an angle within the diagram so that's not going to be our answer but this question here says the bearing of B from a is 070 so if we were to draw that as an angle well to get to B from a we are facing north we're standing at a and we would have to turn here 70° so if we label that on the diagram that would just be 70° the question says that angle ABC is 50 that's given to us and it says that A to B is equal to C to b or in other words this line here A to B is the same as this line here C to B which means that that triangle there is an isoceles triangle so this has got a few bits of maths in here particularly that part where we have an isoceles triangle involved it says to work out the bearing of C from a so again C from a would be if we were standing over here at a it's asking us what degree we would have to turn to go all the way down to here so we should be able to find that out because we can use the idea of angles in a triangle so with an isoceles triangle if our top angle is 50° which is what's given to us in this question and we know that in an isoceles triangle base angles are equal what we can do 180 degrees take away 50 which leaves us with 130 and if both the base angles are the same same we can divide that answer by two and that would give us 65° for each of those base angles now in terms of the diagram that means this angle here is 65 and the total of this angle here ignoring that North Line would also be 65° so for this question it's asked us to find the bearing of C from a which just means if you're standing at a how do you get to C well facing north and going clockwise we would go 70 to get to that first line plus an additional 65 and that would give us a total angle of 135° and that be our final answer for this question it's already a three-digit number so we don't need to worry about making it a three-digit number it's just 135° of course if the answer did come out as a two-digit number something like the 65 Dees you would write your answer as 065 so when you are labeling on the diagram it doesn't actually matter if you write it as a three-digit bearing because we are just looking at Angles initially but then when we write the bearing we do make sure we convert it to a three-digit bearing if we need to by adding that zero at the start unless of course it's already a three-digit bearing so there we go there are bearings just remember you measure from North you go clockwise and we always give three digit bearings as answers okay so when we are looking at loky questions we can be asked any of the construction questions we've already looked at so here it says Point T is 250 M from point A It also says that point T is equidistant from points B and point C on the map show one of the possible positions for Point T and it says 1 cm represents 100 m well to start with we'll deal with that first point so if Point T is 250 M from point A well we need to find all of the points that are going to be 250 M from a the scale says that 1 cimeter is 100 Metter so 250 m is going to be 2.5 cm so I'm going to pull the end of my compass to the end of the ruler there and we're going to measure this out to 2.5 CMS which is only going to make a small little jump there on the compass so we have 2.5 cm I can move the roller out the way and we are going to draw a circle around a I'm going to draw a circle for this because there's any point on that Circle would be 250 M away so there we go any of those points there are 250 M away from point A I now also have the problem here that I need to do the point that's equidistant from B and C equidistant just means an equal distance so if I want something which is an equal distance from B and C that's going to be one of my bis sectors now I'm not going to need to draw an angle bis sector because I don't have an angle involved here but if we imagine that we we did connect the lines B and C the equidistant point is going to be somewhere in the middle where we can draw a sort of line that goes through that but we need to find exactly where the middle is so we're going to do a draw a perpendicular bis sector for this one now a perpendicular bis sector is nice for us to do we can put our compass on the end extend past halfway and then we're going to draw a nice semicircular Arc without changing the length of the compass I'm going to go to the other side of the line and I'm going to swing that Arc all the way back round so that it joins those both together we now know that if we connect this point over here and let's put a dot on there instead and this point over here that's going to draw a line which is then equidistant from the line b c so you want to use a ruler and a pencil to do this but we don't just want to join those up we're going to also want to extend the line so that it cross crosses through this circle there we go so those points on the circle are both equidistant from B and C and they are 250 M from point A so you can see we have two points we have this one here and we also have this one here now it says on the map show one of the possible points so obviously we would only need to label one I'm going to label this one and we'll just say that's the point there it's equidistant from B and C and it is 250 M from point A okay so when expanding double brackets there are a couple of methods that you can use I tend to just connect them up using connecting lines now essentially what we actually do have here is two lots of a single bracket you have to first do 5x multipli by the 2x- 3 and we also have the two in the first bracket which is also getting multiplied by the 2x - 3 now you could write it out like this and then add them together that's just technically what we're doing but it's much easier to leave it as it is and we'll just expand the first bracket by the 5x so 5x multiplied 2x would give us 10 and we get x s as we have an X being multiplied by another X then we have 5x multiplied by the three not forgetting to include or incorporate the symbol there that is a negative3 so we can put the negative in and then just do 5 x * 3 which is 15x we can then move on to the two doing our second bracket here so 2 * 2x would give us + 4x and 2 * the 3 would give us 6 and again we have a negative there a positive time a negative makes a negative and there we go we have expanded the double brackets we now have this concept of simplifying as well so where we simplify this double bracket or this Now new quadratic we need to look for the like terms now the 10 x^2 is the only piece there that has a squared with it so that stays as it is but these two middle pieces here are both an X piece so they can be combined the number at the end is on its own so that's going to stay as a negative -6 as well so you could always write that in if you want but we need to simplify and combine these middle two so -5 add 4 would give you1 X and that would be your final answer and that is your expanded and simplified double bracket so our final answer 10 x^2 - 11 x - 6 and there we go that's how we expand a double bracket now the reverse of expanding a double bracket is factorizing a quadratic now we can take a very similar approach but we're obviously going to do this in Reverse now if we're going to factorize we know it's going to go into a double bracket now this one here we know that it's going to start with an x as we have x's in our quadratic there's no coefficient of x squ so there's no number in front of it so we know in both our brackets it's just going to be a single X in both because x * X gives us X squ we can then look at the factor pairs of this number here so three only has one factor pair that's one and three and if we had multiple factor pairs we would obviously have to consider other factor pairs as well but this is quite nice one because we only have one and three then the next step is to look at this number in the middle and think how would we make plus four and that's by combining these numbers because we know the middle two are going to get added together when we expanded the double bracket in the first place so to make +4 we would want + one and + three as 1 + 3 is equal to four so we put those into our brackets so + one and + three now of course when you are factorizing this you could write that the other way around we could have x + 3 and x + 1 that both mean the same thing so you can have either option there doesn't matter what AUD you write them in the important part is that the correct symbol is with the correct number and here both of the symbols are positive so it's quite a nice one now of course you can have other variations you can have something called a difference of two squares which looks like something like this so x^2 - 16 is good example so here you don't have anything in the middle now it does factorize into a double bracket but when there's no X piece in the middle we just need to obviously bear that in mind when looking at our factor pairs factor pairs of 16 you can have one and 16 you could have 2 * 8 and you could have four * 4 but because there's nothing in the middle this time we're looking how we can make zero now the only way that you can make zero is if the two factor pairs are the same number because essentially they cancel each other out so you could have plus 4 and minus 4 and together they would equal zero which is why there's no X piece in the middle so that's a more unique scenario there we could have x + 4 and xus 4 and of course they could be written in either order as long as the positive and negative is with the right number but in the case of this one both numbers are the same so as long as there's a positive and negative with the same number that would be a difference of two squares obviously lots of different variations of this you can have positives and negatives in all the different positions but the important part is to look at them very carefully when you're looking at your factor pair and just think how do we make this number in the middle and that should help you then get towards the symbols that you need in front of those two numbers so here's a good example of a quadratic where there is also a positive and a negative now here we're actually asked to solve this quadratic and when we are solving a quadratic as long as it's equal to zero we can actually go about solving it the first step is to factorize the quadratic so for this particular step we're going to kind of ignore the equal Z for the moment and we're just going to factorize the quadratic on the left we need to look at the factor pairs of 24 so for 24 there's quite a few factor pairs we can have 1 * 24 we can have 2 multiplied by 12 we could have 3 multipli by 8 and we could have four multili by 6 now we're looking for a factor pair that can combine in order to make negative or positive for this one positive 5 and -4 when they multiply so a little bit of logic here can help because we know that if we are multiplying to make -4 that one of the numbers has to be positive and one of the numbers has to be negative otherwise when you times them together in that final multiplication here you would get a positive otherwise so we have to have a positive and a negative negative so using that how are we going to get + 5 well there's no way of doing that with 1 and 24 there's no way of making five with 2 and 12 but we can make five with three and 8 you can make five by having Min -3 + 8 I'm being very careful there not to do it the other way around because if we had positive three takeway eight that would actually give us neg five so here we could have -3 and positive 8 of course these brackets can also be written the other way around as long as we keep the plus with the eight and the minus with the three now when we are solving one of these it's really nice and easy at this point for us to solve now this is equal to zero now essentially when we are solving this we are actually finding and you don't always have to go into this much depth but you are finding the points where a quadratic crosses through the x- axis so this is what we're looking for and that's what equals z means where this x axis is going through that's where Y is equal to zero we know that this point just here is zero on the origin so it's where the x-axis crosses through zero so we have two coordinates and essentially we can take them in two steps now for the first one we say okay well when does X - 3 when does that equal zero that gives us a nice little equation to solve you can add three to both sides and that's when X is equal to 3 and you might notice there a bit of a pattern here it was -3 and here our solution is positive three and this one here should help emphasize that pattern because here we get plus eight and if we look at where x + 8 is equal to Zer we have to take away 8 and we get X is equal to8 so here where the bracket had a plus eight our solution was ne8 so you can see there both the numbers in the brackets our Solutions are basically just the flipped sign so it is quite quick to find them you don't have to write out that equation there as long as you know that that is what's happening it's okay for you to just flip the signs and go right where there's a negative3 in the bracket so our solution for that is positive3 and there's a positive 8 in the other bracket so our solution for that is8 so there we go that's how we would go about factorizing a slightly harder quadratic and also solving it but obviously B pay attention to the words you would not need to do that second step there if it just said to factorize now when it comes to draw dra in a quadratic graph we have a couple of methods that we can use in this one we're going to look at a non-calculator method which is going to mean that we can solve this or answer this no matter what kind of paper it appears on so here it tells us to complete the table for this equation and the equation there says Y is equal to x^2 - x - 6 so here we are also given a list of the X values it tells us here we're going to be using X from -3 to 3 and we want to find these y values and two of them have actually been given to us we've been given the six and the -6 now when I'm solving these I always like to start with the Positive value so I'm going to start over here so that I only have to substitute a positive number into my equation so it says that Y which is what we're trying to work out is equal to x^2 take away X take away 6 so if x is 3 that would be 3^ SAR take away 3 and then take away six so you need to take your time to work that out you can always do it underneath 3 squ is 9 so it's 9 take away 3 takeway six 9 takeway three is six takeway six is zero so this here is going to be zero we can then move on to the next number when we move on to the next number this working out we'll have to start again but we're only going to replace those threes so here we are looking at two so we would have two squar take away two take away six 2^ s is four four takeway 2 is two takeway six is ne4 and again we can just keep repeating this process to find our next numbers we'll get rid of the twos we are now looking at the ones so we have 1 squar take away one take away six 1 squar is 1 one takeway 1 is zero takeway 6 is6 you'll notice in this kind of table here you can have repeated values so don't worry about seeing something like that it'll make sense when we actually go about drawing it now we're moving on to the negative numbers when you move on to the negatives you need to be extra careful if you are using a calculator here then you need to make sure negative numbers are put into brackets so here I'm going to put the negative one now in a bracket that's just going to ensure that my calculator if I am using a calculator doesn't Square the one and then do a takeaway because your calculator will follow the order of operations if you put min-1 squ like this in a calculator it will give you the answer ne1 and actually negative - 1^ SAR is not negative 1 it is positive one which we should know because a negative times a negative will make a positive answer but you can test that out in your calculator and you can see you will get different values so need to be very careful with that part so here - 1^2 is positive1 then we are taking away -1 which is going to add one and then take away six 1 + 1 is 2 take away six is4 and you might actually start to see a pattern you can see here both of these were -6 both of these were -4 and that's going to follow the same pattern here with the zero so we know that actually that's going to be zero over here again you can sub it all in though make sure you're happy with that but you might be able to start to see a pattern as you're going which can save you a little bit of time but just be very careful with spotting patterns so now we can go about actually plotting this it does say on the grid draw this graph so we'll plot each point be very careful when you plot them remember your X and Y coordinates you've got the X labeled on the graph over here and your y up here and then you've got your X and your y to plot so I'm going to start on the left we'll start with -3 and 6 which is the coordinate -3 6 so go along to -3 up to 6 we get a point just here make sure you mark it nice and clearly then we have -2 and 0 so -2 and Z is just here -1 and -4 is going to be here then you have 0 and -6 we have 1 and -6 we have 2 and-4 and then we have the three which is with the zero now when you are drawing one of these a quadratic graph or any graph that has a power in We join these points up with a nice smooth curve meaning we're not going to get our roller and join each one up dot to dot we're going to try and make it a nice smooth curve you can go from one dot to one dot you don't have to do it all in one go it's really important that you hit all of the points and go through as neatly as you can so I'm going to do one point to each one I'm going to try and make sure that when I do it the line does have a little bit of a curve in it and there we go going to try and join it up really nice and neatly and there we go that's a quadratic graph and how we go about drawing them on the graph and plotting the table and being careful with negative numbers so there we go that is drawing a quadratic graph now when we are looking at area of a circle it doesn't necessarily only have to appear on a calculator paper so here is a really good non-calculator version of where we are working on the area of a circle now we have here a guard Garden that's in the shape of a circle with a radius of 10 m now as soon as it says the word Circle and we think we might be looking at area we want to write down our area formula so area is equal to pi multiplied the radius squared or pi r squ says here we're going to cover the garden with grass seed to make a lawn and grass seed is sold in boxes each box of grass seed will cover 46 M squared Bina wants to cover all of the garden with grass seed work out an estimate estimate being a really key word here for the number of boxes of grass seed biler needs and you must show you're working now as soon as we see the word estimate we want to round any numbers that we can see to one significant figure so one significant figure for the 46 M would be 50 m squared and one significant figure for the 10 well that's already written to one significant figure so now we have got an numbers that we can actually estimate with the only other number in this question which isn't written as a number but is written as the symbol pi that is also a number that we're going to want to round to one significant figure now pi to two decimal places is equal to 3.14 so if we were to write that to one significant figure an estimated version we would use Pi is equal to 3 and that's going to mean that we can actually work out area of circles or estimated area of circles using pi is3 so if we work out the area of this circle which we know is p pi r s well we can do three multiply by the radius squared and the radius is 10 so 10 squar that means we're going to do three * and 10 * 10 is 100 so 3 * 100 would give us an area of 300 this is in meters so me squared so that would be the area of the garden we know from the question that we are each box of seed covers 46 M squar which we've rounded to 50 so if we want to figure out how many of those boxes are needed to fit into 300 we can do 300 divided by 50 now when you are dividing large numbers like this when you are estimating if we imagine it's written as a fraction as well two ways that you could write it but we can cancel off a zero from the top and the bottom just to make that division a little easier so 30 divided by five would give us an answer of six so we would need six boxes of grass seed so that'll be our final answer for how many boxes we need and we have shown all our working along the way now it says in Part B is your estimate for part A an underestimate or an overestimate and you must give a reason for your answer now the rounding that we did in terms of Pi hasn't overly affected the answer for the area if we did one or 3.14 * 100 we would have got 314 M squared but the rounding that we have done in terms of the division there is quite a large one so we could actually go about figuring this out you could sort of think to yourself here well if we did 314 M squar and we divided it by the original 46 what would be our answer now typically the the reason we've estimated it is probably because that was relatively complicated to work out so here I mean you can think about this in terms of would it be more or less if we'd had done the real one and you can already see that even if we' had done the 314 M squared we would have had to have got an additional box we' had to have got seven boxes and that's without even thinking about this 46 here which could push it up to another amount of boxes but the important part is we can see very clearly that our estimate is under what we'd actually need so really we'd have needed more than six boxes and therefore ours is an underestimate and we could just say that we rounded down the number that we divided by so because we divided by 46 uh sorry because we divided by 50 and we should have divided by 46 actually our answer would have been bigger than six and that's the reason that we would give to do with that division so there we go it's an underestimate we would give a reason along with that you could phrase it however you wanted so I'm not going to put specific words there but you could phrase that as because when we actually divided we divided by a larger number and therefore we got a smaller answer if we had a divided by 46 we would have got a number bigger than six and therefore ours is an underestimate now we can also have compound shapes where we are looking at area of a circle so here it says a garden is in the shape of a rectangle and a semicircle and a d is the diameter of the semicircle it says here Carol is going to cover the garden with fertilizer and a box of fertilizer cost £4.99 Carol has been told that one box of fertilizer will cover 12 M squar of the garden work out the cost of buying enough fertilizer to cover the garden completely well to do this we need to get the area of both of these shapes and we are going to be using a calculator for this question so to start with we can get the area of the rectangle because that's just going to be the length times the width or 8.4 multiplied by 5.6 if we work that out on the calculator we get 8.4 * 5.6 which comes out as 47.4 this is in meters so me squar for the semicircle we're going to want to first know what the radius is as we know that the formula for the area of circle is equal to piun R 2 so here here for this area we will do PI multiply by and then we need to find the radius now the full length across is 8.4 given to us on the bottom of that rectangle so if we divide that by two we get 4.2 M as the radius and we can put that into our formula pi * 4.2 2 if we type that in on the calculator * 4.2 to the^ of two we get an answer of 55417 now there are some more decimals there but because we are going to figure out how many of these 12 M squar boxes fit into our answer I'm not going to worry too much about going into four decimal places the important part here is that we add both of these together so 55 417 there is a six after that technically it does Round Up to an eight but again it's not going to affect our answer in this case so I'm just going to add that to the 47.4 so Plus 47.0 04 and we get a total here of 102 point and it goes 4 five and some more decimals but again I'm not going to worry too much about writing down all those decimals because I want to know how many boxes actually fit into that so here when I divide by 12 which is our amount of me squar that each box covers well I'll divide by 12 and I get 8 538 again more but I get over eight boxes now we can't go in and ask for 8.53 eight boxes we have to round this up we'll go in we'll buy nine boxes of grass seed we'll have enough to cover the garden we'll have a little bit left over but we will have enough so obviously it does say to cover the garden completely not to just undercover it so here we're going to have to buy nine boxes now it's asked here to work out the cost each box of fertilizer costs £44.99 so we need to do 9 multipli by the £4.99 it's going to cost me for each box so 9 * £4.99 is equal to 44 and 91 p and there we go that would be our final answer now it says here that Carol finds out that one box of fertilizer will actually cover more than 12 M squ of garden explain how this might affect the number of boxes she needs to buy well if she finds out it only covers 12.1 M squ it doesn't tell us how much more there's a chance it might not actually change it at all but it does say explain how it might affect the answer well here if we find out that it covers more than 12 M squ for example let's imagine it covers 15 M squ well if I do the 102 divided by 15 I actually get 6.8 and in that case I'd only need to buy seven boxes so in terms of writing down an answer here and again I'm not going to write down some words for this we'll just talk about it because you can phrase this in different ways but something along the lines of if the grass seed covered more than the 12 M squared Carol may need to buy less boxes we're not going to State any numbers because we don't know the exact amounts but the important part there is it might mean that she needs to buy less boxes and actually that would mean that she'd have to pay less now we obviously don't need to say that it doesn't need and ask us about the actual cost but thinking about real life here it would just mean she would need to buy less and she might not have to spend as much on the grass seed so there we go there is a functional question looking at area of a circle now when it comes to looking at cones you are given the formulas for these in your exam so here we are told the volume of a cone for this one is 1/3 Pi r^ 2 H we're given a little diagram that shows the radius down the bottom of the cone and it shows us the height and the height being the perpendicular height from the base to the top of the cone now this one here it says the radius of the base of the cone is 5.7 and its slant height is 12.6 now the slant height is along the edge of the cone that's not the perpendicular height that we need for our formula so here where it says calculate the volume of the cone and give you answer to three significant figures we have a little bit more to work out than just plugging these numbers into the formula so here you can see and it has highlighted it for you quite nicely on this diagram that it makes this particular shape which you can hopefully see is a right angle triangle so here is another topic that is going to involve Pythagoras Theorem if we have a missing length on a right angle triangle straight away we can use Pythagoras and Cones is a really nice topic that they like to include Pythagoras with so if we were to draw that triangle to the side you can see that we are given the length of the hypotenuse 12.6 and we are given this base length here or one of the shorter sides 5.7 Pythagoras Theorem is a^ s + b^2 = c^2 and if we are working out one of the shorter lengths we have to do a subtraction essentially doing something along the lines of c^ 2 - A 2 is equal to b^ 2 you can mix around the A and the B there the important part is the shorter side that we have we are going to call a as our formula there has the a in it and our longest side is always the C so to work out the Shor side we will do 12.6 squar take away the 5.7 squar and that's going to give us our our value of B squared so 12.6 squar take away 5.7 squar gives us an answer of 126.2 7 to find just the length we have to square root our answer and if we square root that on the calculator we get quite a long number but it's 11.23 6 99248 now because the question here wants that answer to three significant figures we could probably get away with writing our working out down to five significant figures but if I want to be super careful I make sure I don't miss any numbers or get a slight error in my rounding particularly when it's only that many decimal places and it's not going on forever on the calculator I'd be absolutely happy just writing all of those down just to make sure I don't make a mistake so we know here the height is 11236 and the rest of those decimals I won't write them down there but I will use them in my working so for the volume of a cone it just tell us the formula up here 1/3 Pi r^ 2 H so we want to type that into our calculator so 1/3 multiplied by pi multiplied by the radius squared which is 5.7 and then multiplied by the height which is 11236 99248 now most calculators have the answer button that you can use on the calculator so my last answer was the 11236 number so I'm just going to type in 1/3 multiplied by pi multiplied by 5.7 s multiply by and then I'm going to click the answer button which at least saves me a little bit of time typing in that big long number so when we type that in we get an answer which comes out as 382 321 some more decimals but again it does say give your answer to three significant figures I've already written up to six significant figures there so I'm going to chop this after the third significant figure and you can see there that's not going to round up so our final answer will be 382 two this is a volume and our question is in centim so that would be cm cubed and there we go that is our final answer there for the volume of a cone and really it's just using a formula but this question was a little bit more complex because we did have some Pythagoras Theorem involved to find the height of the cone now when it comes to spheres again there are two formulas that you can have you can have the volume and the surface area and you are given those formulas you can see they're given there on the right hand side so it says here the diagram shows a solid Hemisphere and that has a radius of eight work out the total surface area of the hemisphere now here working out the volume would have been a little bit easier because we can plug that straight into the formula just divide the answer by two but when it comes to the surface area we have two surfaces that we're going to have to work out we have the circular surface which is up here on the top and we have the curved surface area that's going round the outside of the sphere around here so we're going to start by working out the circle so for the circle the circle there has a radius of eight the formula for the area of a circle is one that you are expected to remember so area is equal to piun R 2 so we can work out that area by just doing Pi multip by 8^ S now it's completely up to you you can leave this in terms of pi as you are going through so that does out as 64 Pi which can make it a little bit nicer when you have to add it to another number or you can write out the decimal the decimal version comes out as 201.06193 N8 either way I now have to work out the curved surface area as well so we are looking at the formula here where it says the surface area of a sphere so here the formula is 4 pi r 2 now this here is a hemisphere so once we've got our answer for that we are going to want to have the answer so I'm going to work out the full curved surface area to start with so I would do 4 * pi * the radius squared which is 8^ SAR I can work that all out or I could straight away divide it by two if you do divide it by two you could actually simplify this a little further you could just work out 2 piun r^ 2 or 2 * < * 8 purely because this four here can divide by the two so it can simplify down just write 2 piun * 8^ 2 completely up to you you can either work it all out and divide your answer by two or you can simplify it a little bit and do it in one go so I'll work this out 2 * pi * 8^ 2 and that comes out as either 128 Pi if you want to write it in terms of Pi or you can write out the full answer again which is 42.1 2 3 8 5 9 7 now to finish this off we are working out the total surface area so to work out the total surface area we just need to add both of those together we do need to give it to three significant figures so if we did leave it in terms of Pi to start with we're going to have to write the decimal this time but I can do 128 pi and I'm going to write this working out down here so 128 Pi we're going to add to that the 64 Pi there you go so plus 64 pi and that equals 100 and 92 Pi which I'm going to write as a decimal that decimal comes out as 6385 I'm not going to carry on because again in this question it says to write it to three significant figures so chopping it after the three gives us a final answer of 603 and and this is a surface area it was in cenm so that be cm squared so there we go that is our final answer for the surface area of the hemisphere now when we are looking at a volume of a cylinder unfortunately we not given any formulas for this but the volume of any prism is very nice to work out you work out the area of the cross-section which in this case is a circle and you times it by the height or the distance that it goes through the shape and in this case the distance that goes through is 15 cm so the numbers are given to us in the question they are also on the diagram but it says to calculate the volume of the cylinder and give your answer to three significant figures so here first things first we're going to write down the formula for the area of a circle which is pi r squared and then work out the area of this circle here on the top the radius is given to us remember you could be given the diameter in which case you'd need to divide that by two but this question has given us the radius so we would do PI multiply by 4^ 2 you can write that in terms of Pi if you want that comes out as 16 Pi or you can write it as a decimal I do tend to like to leave it in terms of Pi until I get to the final step just to reduce the amount of writing that I do so that's the area of the cross-section now we need to multiply it by the length that it goes through the shape in this case the height of the cylinder so that' be 16 Pi the area of the circle multiplied by 15 which is the height and 16 * 15 is 240 so that's 240 Pi now of course the question might say to give your answer in terms of Pi in which case we could leave it as it is but this one does say to give it to three significant figures and that comes out as 753 982 and a few more decimals but it only wants it to three significant figures so I'm going to chop it after the third significant figure there and that is 754 as we have nine after the line so we have 754 this was a volume it's in centim so cm cubed and there we go that's how we would work out the volume of a cylinder now working out the surface area of a cylinder is a little bit more complicated than working out the volume so when it comes to the surface area of a cylinder yes we are going to need area is equal to piun r s because with the surface area we do have two circles that we need to work out the area of but then we also have this curved surface area that's going around the outside so we'll look at that bit afterwards but we'll start with these two circles now this question has given us the diameter of the circle so we want to divide that by two the radius is going to be 6 cm so we can start by working this out we've got pi multiplied by 6^ SAR which is 36 Pi now we do have two circles so we would want to multiply this by two that's going to give us the surface area of both those two circles which comes out as 72 Pi now again it's up to you if you want to label that as uh in a in a as as a decimal but you can leave it in terms of Pi again I've just like sort of written there that's the two circles you could write that in in words but we've got the two circles done and the area for that is 72 Pi for the outside or the curved surface area if we unraveled that from the circles it would make a rectangle now it's quite nice we know the height of that rectangle which is 18 this is in cenm so 18 cm you can see that on the right here but the top of that rectangle there and this is the complicated part of this if I highlight that on the shape you can see the top of that rectangle wraps all the way around here and that is the circumference of that Circle so you do need to know circumference of a circle anyway the form for for circumference is circumference is equal to Pi multipli the diameter or you might write it as piun * 2 R as two radiuses is the same as the diameter but pi times diameter is quite a quick one nice one to remember we know the diameter of the circle is 12 it's down the bottom of the cylinder so for the top there the working out we would need to do is pi * 12 we could do that on the calculator we can write it as 12 Pi if you like that might say you a little bit of time justes come out as 26 Pi again you can write out the decimal if you like but it does save a lot of time just writing these down in terms of Pi the total surface area then we've got the curved surface area 216 Pi we're going to add to that the two circles which was 72 Pi if we add those two together on the calculator we get a total of 200 and 88 Pi the question does say to give you answer to three significant figures though so we are going to have to write this as a decimal now we've saved a bit of time along the way but that comes out as 94. 778 a few more decimals but it does say three significant figures so if we round this to three significant figures we will chop it after the four there after that decimal place it's a seven after the line so that would Round Up to 95 again this is an area so cm squared and there we go that is the surface area of of a cylinder now when we are looking at fractions obviously there's a couple of different fraction calculations that we can have we can have to add subtract multiply and divide so these are two relatively nice ones when we are adding fractions together we need to have a common denominator at the moment the dominant denominators are seven and five I do have a look to see if one of them can turn into the other in this question here it's not easy to turn a five into a seven and vice versa so we will have a look for the lowest common multiple of seven and five hopefully you can spot that that's 35 if you're not too sure you can always times them both together for a multiple W always give you the lowest common multiple but in this case seven and five the lowest common multiple is 35 now to get 35 on this left fraction you would have to times 7 by 5 and whatever you do to the bottom you just do to the top so 2 * 5 would give you 10 for the right fraction we have to multiply the five by the seven and again you do the same to the top so that would give us seven we can now add these two fractions together and that would give us 17 over 35 just combining those numerators the only thing that would change in this question is if it was a subtraction instead of an add so if this symbol here was instead a subtract if it was then we would do 10 minus 7 on the top and we would get 3 over 35 of course this question is an ADD so we were absolutely fine to add them together now it doesn't say anywhere in the question to give your answer in its simplest form so you don't even have to think about simplifying it if it did then you would obviously have to look to simplify as it turns out this fraction doesn't simplify anyway but there we go even if it did um simplify and it didn't say to simp give it its simplest form we don't actually need to worry about it so for Part B here we have a division the problem is one of these is a mixed number now when we are doing any sort of frac fraction calculations we do need to convert these two improper fractions first so to turn 1 and 2/3 into an improper fraction we do the big number at the start multiply by the bottom and then add the top so 1 * 3 is 3 + 2 and that's going to give us five as our new numerator over three so here we're going to have 5 over3 divided by 3 over 4 now when you dividing fractions all you have to do is multiply by the reciprocal so here we've got 5 over3 we multiply by the reciprocal is where you just flip the numbers over or flip the fraction over so we multiply by four over three multiplying fractions is one of the nicest topics all you do is times the top 5 * 4 is 20 and times the bottom 3 * 3 is 9 so for this one here we get 20 over 9 it seems a little strange to leave an answer like that but it doesn't say in this question to give your answer as a mixed number doesn't say to give it in its simplest form so it is fine to leave it as 20 over9 it would also be fine to convert this which you could do you wouldn't be wrong to do so but just for this particular question you wouldn't actually gain any extra marks for it nine fits into 20 twice up to 18 that leaves you with a remainder of two so you would have two 9ths left over so you could write your answer as two and two 9ths as well obviously if you were given a multiplication question it would be the same as this step here but of course you wouldn't need to do the reciprocal you just times the top times the bottom and give your answer however you've been asked but there we go that is some fraction questions now you can have some slightly more complicated fraction questions as well here we have in the first question two mixed numbers that we're going to have to convert and we are adding there so we're going to get some quite large numbers and in Part B it does ask us to give our answer as a mixed number and it's simplest form so just have a little bit more to think about on these type of fraction questions now we're going to start by taking each of these fractions and writing them as an improper version so 2 * 7 is 14 plus the 1 is 15 over 7 for the second fraction here we have 1 * 4 which is four add the one which is 5 over 4 and we're going to add these fractions together now straight away you've got seven and four we need to make a common denominator the common denominator or the lowest common multiple of 7 and four is 7 * 4 so 28 which means we're going to have to multiply this left fraction by four and this right fraction by seven so straight away you can see we're get some quite large numbers so 4 * 7 is 28 so both of these fractions will be over 28 the left we're multiplying by 4 15 * 4 is 60 the rights we're multiplying by 7 5 * 7 is 35 and now you can just add those two together so 60 add 35 is 95 and on the bottom is 28 so there we go we get 955 over 28 for the first one thankfully there that question doesn't ask us to simplify it or write it as a mixed number but you could write that as a mixed number as well you could say well 28 goes into 95 it goes in three times and there's a remainder of 11 obviously you can take your time to work that out but as we don't need to for this question we won't go through draw the steps for that one that would be quite a horrible one though you'd have to write 28 and some multiples of 28 down to the side particularly as this is more likely to be a non-calculator question for Part B we have a division again but we have this additional step of writing our answer as a mixed number in its simplest form so we'll take the same First Steps 1 * 5 plus the 1 is 6 over 5 you could write this out up here first before we do the reciprocal so divided by 3/4 but of course we are going to Times by the reciprocal 4 over 3 and work that out so 6 * 4 is 24 5 * 3 is 15 now you've got two steps that you can take and it doesn't matter which order you do it in you can either convert it into a mixed number or you can write it in its simplest form so it doesn't really matter either way I'm going to start by converting it into a mixed number so 15 fits into 24 once and the remainder from 15 to 24 is 9 so you get 9 over 15 now this step here it can sometimes look like the answer is finished but 9 over 15 does simplify so it's going to be one and something but you just need to spot what 9 and 15 both divide by well they both divide by three so we can simplify the fraction 9 divid 3 gives us 3 and 15 / 3 gives us five so our final answer there is 1 and 35ths so you can see that one there had a few more steps of course you could have simplified at this point they both divide by three that would have given you 8 over five and then you could have converted it into a mixed number five goes into eight once with a remainder of three so 1 and 3 fifths so you can do either step you can do it in any either order it's completely up to which one you prefer but there we go that's how we would tackle some of these fraction calculations when looking at standard form conversions it's obviously important to know some of the rules so the number at the start when we are looking at standard form it has to be between 1 and 10 which means if we are converting a number into standard form we need to essentially move the decimal to a position where it's between 1 and 10 and looking at the number that we have in this question you can see that we're going to want the decimal to move between the five and the six that's going to mean that the number would read as 5.62 but of course when we're writing something in standard form we're just writing it in a different way so here we need to know how many times we are multiplying it by 10 now in the case of a large number something like 562 and let's imagine it's 0 0 well in that particular circumstance there we would actually be timesing it by 10 to get to that number one two three four times but this is actually a not Point number so although we count the jumps 1 2 3 it's actually going to be a power of -3 but goes with our number so 10 the^ of -3 is just a decimal .1 or or 0. one okay lots of different variations depending on the power there but all that means is that we are timesing by a .01 number or type of number that sounds like that um instead of actually the whole number 10 multiples of times so there we go if you have something like this this would be written with a positive power 5.62 and that that particular one there would be 10 the^ of 4 but here where we have a not Point number we just need to remember to put the negative with the power so here when we are converting back again you could have a positive or A negative power here when we are writing it as an ordinary number but whenever we do this I like to write the numbers out that we have and I know I'm I'm going to move this decimal so kind of put it at the top here and for a positive power we are multiplying it by 10 three times so it would go 1 2 3 the decimal is going to move here to the end as it's at the end of the number we don't even need to write it so the answer there would be 1,452 of course you could also have a negative power involved so let's imagine it was a power of -3 well in that circumstance 1 4 5 2 the decimal would just go one 2 3 in the other direction so the decimal is going to jump to there when when you have that type of question though you do need to make sure you tidy this up and put all your zeros in plus a zero at the start so if it was 1.45 2 * 10 ^ of -3 that's the answer that you would have got instead so there we go it's just being aware that a large number is going to have a positive power and these Noto numbers have negative powers and there we go that is some standard form conversions now you can also have calculations with standard form typic these sorts of questions here where it is a little bit more complicated this would be allowed with a calculator so this is more like a calculator question you can have them without a calculator we'll talk about the process you would do if you didn't have a calculator but when you are multiplying or dividing with these sorts of numbers the 13.8 and the 5.4 are the starting numbers in terms of the actual ordinary numbers that these these would be so all you have to do when you're multiplying them is multiply the starting numbers so 30 13.8 * 5.4 being careful not to confuse this obviously with our method like we do with double brackets where we connect them up here we are just multiplying those starting numbers so 13.8 * 5.4 a good logical way to think about this is if we do something like 30 multipli by let's just make something up 600 you only have to multiply the starting numbers three and 6 is 18 and then you just count the zeros and add them all together so 1 2 three zeros on the end but all we have to do is multiply those starting numbers here it's just a little bit more complicated because it is written in standard form but if you multiply those starting numbers you get 7452 as well as that if we look at these powers of 10 if you remember about index laws when you are multiplying with the same Base number you can add the powers together so here we would have times 10 but the powers we're going to add this is quite a comp complicated one CU we have a negative power so that' be 7 + -12 when you add a negative that's a takeway so that' be 7 - 12 and 7 takeway 12 of course you have a calculator for this so you don't have to think too much but that is5 so times 10 the5 so if you didn't have a calculator that is a process that you could take obviously you wouldn't have numbers as complicated as that it might have been two and three in the brackets and maybe you didn't have to write it as an ordinary number or at this point you could convert it so here if we type this into a calculator I'm just going to type in the 74.5 2 * 10us 5 and see what we get so for that my calculator actually shows it me in standard form so it doesn't really help it does give me the answer which is good for this because we can balance it out the standard form so here if we want the decimal just there we make the number smaller 7.45 to and that means you make the power bigger one bigger than5 is4 so you can balance it out as well if it did say to write your answer in standard form but this does say give your answer as an ordinary number unfortunately my calculator shows it in standard form so we are going to have to convert it so I'm going to write the numbers 7452 it's a negative power which means it's a not Point number which means I'm going to have to hop the decimal four places to the left being really careful with this so it's going to go just here and underneath each loop I need to tidy that up with zeros so there we go my final answer for that is. 745 2 and there is a more complicated standard form calculation when it comes to congruent triangles congruent is just a word for exactly the same or mathematically exactly the same so it says here write down the letters of the two triangles that are congruent now if we look the majority of these triangles have given us some angles some have given us two so some of of them we can actually get some more information for for the first triangle triangle a you can see that those two angles 55 and 45 add up to 100 so our missing angle just here would be 80 as angles in a triangle have to add up to 180 for the triangle B we can't fill anything in same for Triangle C but we can for triangle D you see that kind of matches triangle a we've got an 80 and a 45 so this missing angle here has to be 55 now with a congruent triangle the same length has to be opposite the same angle so what I mean by that is if we look opposite this 80 we have 10 over here 10 is opposite the 45 over here 10 is opposite the 55 so so far all of those have to be different because the 10 cm is opposite different angles so this is going to be our deciding triangle and over here 10 is opposite the 80 so if 10 is opposite the 80 in this one it's also opposite the 80 in this one so our congruent triangles would have to be a and d so there you go you can see in terms of looking at congruent triangles you can only really look at the Angles and the sides obviously the angles have to be all the same in that triangle so if there was a couple of different triangles where 10 was opposite 80 we might also have to look at the angles as well and make sure the other two angles were the same so for example if this one wasn't wasn't the only one and maybe this angle was 35 and this angle here was 65 then they wouldn't be congruent triangles because the angles would be different even though the 10 cm is opposite the 80 but this was quite a nice one you had the 80° opposite the 10 for both of those with all the same angles so they are the congruent triangles now similar shapes can be tricky and it can be relatively nice this particular one here the triangles that we've been given are overlapping one another but when it says that shapes are similar which this question doesn't we'll just discuss how we know it's similar but if if you are told that shapes are similar it just means that one is an enlargement of the other and when it comes to enlargements you are looking at the scale factor between the two so here we know that these shapes are similar because they share this angle here and this line on both of the triangles is parallel you it's going in the same direction so they both share the same direction here they both share the same direction here as their straight lines and then those parallel lines are also the same direction as well so these shapes have to be similar it's just the larger triangle there is an enlargement of the other now when you are given these overlapping shapes you can actually redraw them so I could redraw these triangles I'll try and draw them slightly to scale as well so you can see which one's bigger than the other now on the big triangle this is a this is e and this is C and on the small triangle this is B this is C and this is D so it's given us some lengths here that we can actually label onto the diagram it tells us e to C is 8.1 it tells us D to C is 5.4 and it tells us D to B is 2.6 and it says work out the length of a to e which is this length just here so in order to find the scale factor between the two you need to match up the similar sides so here on both triangles we've been given the base of both to find the scale factor or in other words what we've multiplied by to get from 5.4 to 8.1 you just do the bigger number 8.1 divided by the smaller number 5.4 we'll do that on the calculator so 8.1 divided 5.4 gives us 1.5 and that there is our scale factor so that number can now be used to move between the triangles so here where we are going from the small triangle up to the larger triangle's length we just need to multiply by 1.5 so 2.6 on the calculator time 1.5 gives us 3.9 so that one there is 3.9 and our first answer here would be 3.9 CM this also has a Part B where it tells us that a to c is 6.15 a to c is this length just here here on this larger triangle it wants to know the length of a to B now A to B is quite an interesting length to ask because here that is the gap between these two triangles or in other word the difference between the larger length and the smaller length well we can find the smaller length here on our diagrams that we've drawn because we know the scale factor is 1.5 we can just divide by 1.5 to go down to the smaller length and if I do 6.15 ID 1.5 I get 4.1 so the Gap that's in between them must be the difference between 6.1 6.15 and 4.1 so I can work this out we can do 6.15 the length of the larger triangle subtract 4.1 the length of the smaller triangle and that gives us 2.05 cm and that would be the Gap there which they've called AB so there we go but that's how we approach similar shapes you just need to find the scale factor between the shapes by doing the larger divided by the smaller and then you can use that scale factor to multiply or divide between the shapes okay so for this question we're looking at some column vectors it says shape a is translated by the vector 47 and shape B is then translated by the vector3 and -2 and that makes shape c describe the single transformation that map shape a onto shape c now if we had a grid this would be quite nice to draw because the vector just tells us a movement and if we imagine a starting point the first Vector means four to the right and then down by seven now this would obviously be nice and easy to do if we had a proper grid but the next Vector tells us to go left by three that's what the -3 means and then down by two now if we had a grid it'd be really easy for us to count what this final Vector actually did how far have we gone left and right but because we don't have a scaled Grid it's much easier just to add these together and thankfully vectors are really nice to add together so we can just add them up and see what we get you just need to be very careful because we do have negative numbers involved but when you are adding together vectors you literally just add the top number with the other top number so we have four add -3 if we add a negative that is a sub subtract so it's four take away three and that gives us one on the top we do the same for the bottom we have -7 add -2 again we're adding a negative which is a subtraction so -7 take away two would give us9 and you can kind of think about this in terms of the diagram I'm going to write my answer here 1 and -9 but if you even look at the diagram you can see if we have gone four to the right and then three to the left technically we've only actually gone one to the right overall which is why we have one on the top if we also look at this downwards line it's going down by seven and then we go down by another two and overall we've gone down by nine so you can kind of relay the vector back to a diagram and think about what it looks like but it's much easier just to apply this method up here where we are adding the vectors together now you may have some more complicated column vectors where we have questions like this we are told the vector Vector a is 1 and four Vector B is 3 and two and it says write down as a column vector and for part well the first part here we just have a plus b so a is 1 and four we just want to write that down we have a plus symbol in between and B is three and two that's a nice one to add together 1 + 3 is 4 4 + 2 is 6 and there is our final column Vector but just looking at how that's been written and how it's given us a up here and B here for the second part of this question though we have 2 A's and 3 B's now when you have 2 a it just means that this Vector a that we are given at the top essentially we just want two of those so you could write 2 a as two lots of one and four I prefer just to times them straight by two so that would mean 2 a would be two and8 just doubling both those numbers you can also write the 3 B as three lots of three and two but again I just prefer to times them both by three and write 9 and six this is asking us to add those vectors together so a nice one again 2 + 9 is 11 and 8 + 6 is 14 again you just need to be very careful when it tells you to subtract and also being careful if you have negatives involved but this particular column Vector question was quite a nice one as all of the numbers in those vectors were all positive and we were only adding these vectors together so we didn't have to worry too much about actually worrying about negative numbers and that's where these questions can get a little bit trickier when you have negatives involved okay so here we have plotting a cubic graph now cubic graphs there are lots of different types of equations that we can have for this this one here we have y is equal x cubed + x^2 - 2x + 1 now if you can plot this cubic graph you can pretty much plot any of these graphs but we have to fill in the table now similar to when we are plotting a quadratic graphs we're going to substitute these X values into the equation to find the corresponding y value so for this first one we're going to have to do to start with 2 cubed now because we also have some negative numbers that we're going to need to sub in I'm going to make sure that I put every number in a bracket so to start with I'm going to do 2 cubed then we have X2 so 2 s take away 2 x which means 2 * X so 2 * in Brackets 2 and then + one now if we type that all into a calculator this will typically be a calculator question we could work this out without a calculator as well so just to think about if we didn't have a calculator 2 cubed is 8 2^ SAR is 4 2 * 2 is four we're taking that away so take away four and then plus one 8+ 4 is 12 takeway four is eight add one is nine but you will see if you type that into your calculator you do get nine anyway for each number we now just need to change the two into whichever X number we're substituting in so here we're substituting in zero so if we get rid of these numbers and just change those to zeros you will see for that we get one we can then do the same for the -3 the fact that we've got it in Brackets is going to mean that those negatives are not going to cause any issues so when we sub this in these zeros now are going to change into ne3s and we just need to take some time to actually type that in make sure we type it in really carefully if you've already got it on your calculator screen you can do what I've just done on the paper there which is to just delete the twos change them into a-3 and press equals and you'll see that we get Negative 11 we can now actually plot these on the graph remembering these are just X and Y coordinates so -31 we'll go across to -3 down to1 we're going to have to be really careful here because the graph does have a little bit of a strange scale you can see that down here in between that would be -10 and-1 would be in the middle of that so we need to trace along really carefully ideally I would use a ruler just to line it up just to make sure I don't miss any of these -2 goes to -1 which is just here again just labeling on any extra numbers that you need to that's -2 just there so it's between 0 and -2 -1 goes up to three again being really careful that would be two so three is in the middle so three is just here 0o goes to one which is between 0 and two one goes up to one as well so that would be just here and two goes up to nine this is 10 in between just here so it's in between those two so this would be nine now with this cubic graph cubic graphs when they are positive have this sort of shape here if they are negative they do slope down the other way so they can slope down this way as well this is a positive X cubed so when we join this up we are going to join it up with a nice smooth curve now it's okay to join up until you get about to this point here but now when you come down through this x what you don't want to do is now draw a straight line across you can see that it has this almost flick in the curve you so when we're coming down we can go below that and then back up again and then back up to our final point now we don't need to stress too much about whether it goes through the x-axis or not we're just plotting it as best as we can so you could have as you did this little part here you might have come all the way back down and back up and that's absolutely fine as well the important part is that it looks similar to how an X cubed graph should look and we're really accurate with going through those coordinates so there we go there is our X cubed graph now you can also have to plot reciprocal graphs a reciprocal graph is where we have a fraction involved like this up here where we are dividing by X now they're quite nice to draw because a lot of the time the divisions that we have to do aren't overly complex if you look at the table that we have though you can see it goes up by different amounts it starts by going up by 0.5 and then it continues to go up by one so it doesn't really matter which side we start here is we only tend to plot the positive region so to start with for this one here if we start on the left we've got three divided by 0.5 when you divide by 0.5 it just doubles the answer so 3 divided by a half is six a few have been done for us here we have 3 / 3 3 divid 3 is just 1 here we have 3 / 5 3 ID 5 you could type into your calculator but it is a decimal or you could convert it over 10 just to help you write it as a decimal that would be 6 over 10 6 over 10 is 0.6 and then for the last one there we have 3 divided by 6 that is the fraction 1 12 and that can be written as 0.5 of course that last those last two there aren't the nice but most more than likely you would have a calculator for one of these but if you don't there's a method that you could take we can now plot these points on the graph being careful because 0.5 is down here between 0er and one so that's 0.5 the rest are on whole numbers but that one there we just need to be careful that goes up to six then one goes up to three two goes up to 1.5 so being careful we're in between one and two there three goes up to one four now goes up to 0.75 which is going to be very difficult for us to draw you could just to help you label the halfway point over here so 0.5 you know now it's pretty much in the middle obviously you can only be as accurate as you can see it's difficult to see it's difficult to get it any more accurate than that might help to draw the last one first we can see where halfway is and then the one on the five is just in between those previous two so somewhere around there again we're only going to do our best to be as accurate as we possibly can but with numbers like that it's very difficult to get them absolutely perfect now again a reciprocal graph similar to that of a quadratic and a cubic We join it together using a nice smooth curve so I'm going to come down doing one point at a time trying my best to make it a nice curve if you do make a mistake obviously you've done it in pencil so you can rub it out you can go again the really important point is that it's a nice curve and it goes through each of these points perfectly so there we go there would be our nice curve that is a reciprocal graph we don't tend to draw the negative part of a reciprocal graph because so a reciprocal graph does look like this in the positive region in the negative region it goes down to here so a full reciprocal graph can be really difficult to actually plot so it tends to be that we only tend to look at this particular region here where we are plotting the positive part of that reciprocal graph now when it comes to simultaneous equations the method that I use is I look at the coefficients of Y so when we have simultaneous equations we need one of the coefficients to be the same but the method that I use I look at the coefficients of Y you may have learned to do these where you look at the coefficients of x if you do do that that's absolutely fine but I just think it's a lot nicer to look at the coefficients of Y now I always start by labeling these equations so I'm going to call this equation one and equation two I want to make the coefficients of Y the same so for for four and five I could make them both into 20 which means I need to times the top equation by five and the bottom equation by four once we have a coefficient which is the same we can eliminate that coefficient by either adding or subtracting the equations which will'll obviously talk about so for the top one if I times it by five 2x becomes 10 x the 4 y becomes minus 20 y so we've got Min - 20 Y and the 19 becomes 95 5 * 10 is 50 5 * 9 is 45 so 95 in total for the second equation 3 * 4 would give us 12 x the 5 y we already know is going to become 20 y that was the whole point and one becomes four now when you have these where we have have now a similar coefficient on both I look at the symbols with that coefficient so if the symbols are the same we do same signs subtract and if the symbols are different we do different signs add so in this particular case here the signs are different so we will add these two equations together now typically if you look at these two coefficients at the start more often than not those coefficients are always going to be the same so you will always subtract the equations even if the symbols are different in the Y coordinates or in the Y coefficients there so here I do have different signs so I'm going to add these equations together 10 + 12 x will make 22x - 20 add 20 y will become zero so they disappear that's the whole point of what we are doing and then adding the four onto the 95 gets us 99 so here we can now solve for x we now have a nice equation which does give us um a not so nice answer here because 99 doesn't divide perfectly by 22 now typically questions like this you would normally have a calculator if the answer does come out as a decimal but that doesn't mean that you have to have this using a calculator so if you don't you will have to write it as a fraction 99 over 22 or 99 divided by 22 now if you do have a fraction you just need to look and see can you simplify it because we're going to need to substitute this back in to find y now 99 and 22 both divide by 11 so a good way to simplify this would be to divide them both by 11 and we get 9 over two and that's a much nicer fraction that we might be able to write as a decimal 9 ID 2 is 4.5 so we could write that as 4.5 and that's our value for x now with simultaneous equations we also need to find the value of y we found the value of x so we could put that onto our answer line we've got X is equal to 4.5 we now just need to know what Y is equal to so pick one of the equations typically if I have an option I will always pick the equation that has the positive symbol in it just to avoid using any negatives but you can substitute into either equation I tend to write what I'm doing so I'm going to write sub X = 4.5 into and I have chosen equation 2 which I have labeled so that makes it really clear what I'm doing now that means I need to do 3 * 4.5 as it's 3x and then we have + 5 Y and that is equal to 1 this isn't going to be very nice here but I can work out 3 * 4.5 4.5 add another 4.5 is nine as we already know because that was half of nine that we got and then adding another 4.5 would get us to 13.5 obviously you could add those three together but we get to 133.5 5 so we have 13.5 + 5 Y is equal to 1 here we have a number at the start so we need to get rid of this number from both sides just solving an equation like we normally would so minus 13.5 and we get here that 5 Y is equal to -12.5 now straight away as soon as I see that this does stand out as a very clear calculator style question so this would be very difficult it would certainly be a not very nice question at all to appear on a non-calculator paper so typically this sort of question is more a calculator style question because now we need to do -12.5 and divide that by five and that wouldn't be very nice to do without the calculator so I'm going to type it into the calculator we do get Y is equal to 2.5 if you didn't have a calculator this would be really horrible to do to be fair you could leave your answer at this point as -2.5 over5 but you would probably want to simplify that um so we could think well the top and bottom do both divide by 2.5 that's not very nice at all but 2.5 goes into 12.5 five times obviously it's negative on the top there so it's neg five and it goes into five twice so5 over two five does divide by two it becomes 2.5 so we do get Negative is sorry Y is equal to -2.5 so a very difficult version of a simultaneous equation there but if you can solve that one or at least be happy with the steps there even if you are at a point where you know steps like this here that's really really horrible to come up in a simultaneous equation you've got the steps like this where it simplifies down to 4.5 if those are the bits that you're not liking but the process you're actually okay with then you can probably solve any simultaneous equation because it doesn't get any harder than that so looking at another simultaneous equation this one here again we have different signs but you can see here that these coefficients that the start are actually the same so you could just go about subtracting these equations straight away but I do tend to still focus on those y coefficients so here I'm going to label them one and two and in this scenario the top equation there can just be multiplied by four which makes it have a 4 Y which is going to match our negative4 y below so if we do that we would have 12x + 4 Y and this time we have -4 at the end so we'd get -16 now the bottom equation this time we don't actually have to change it we can leave it as it is because it's got that 4 y so we have two equations with different signs in front of the same coefficients so we will add these equations together when we add them together we get 15x is equal to -16 add 6 is -10 so again we've got quite a horrible sort of solution going on with this question as well because 15 doesn't fit into 10 perfectly so if we divide by 15 we get10 over 15 and notice as well how I've dropped the negative symbol off the fraction there it doesn't have to be on the top or the bottom you can put it out of the fraction this one's not too bad we can divide the top and bottom by five which does give us at least a fraction which doesn't look awful to substitute in five goes into 10 twice and it goes into 15 three times so our value of x is -2/3 so X is equal to -2 over3 again now we have an x value it's not the nicest to substitute in but we are going to sub it into one of the equations so I'm going to sub X is equal to -23 I'm going to pick an equation let's just go for equation one one it doesn't really matter here equation 2 has a takeaway in equation one has A4 at the end either of them none of neither of them are going to be easier than the other so let's just sub it into equation one now we do have three multipli by -23 because we have 3x + 1 Y and that is equal to -4 so we need to work out 3 * 2/3 I might just do that to the side that's three over 1 time -2 over3 so I'll just put a one underneath the three it's just going to help me to times the top times the bottom so 3 * 2 is 6 1 * 3 is 3 and it was negative so -6 / 3 and that's actually -2 6 ID 3 is 2 so that part there just becomes -2 + Y which is equal to4 now again you can either maybe look at that and think well what do you have to add to two to make4 it has to be a negative but or you could just add two to both sides of the equation and you get Y is equal 2 -2 so there we go we got Y is equal to -2 again we've got two solutions there for X and Y neither of them being particularly nice obviously a simultaneous equation that just has all Positive Solutions or positive integers they're the nicest types to solve but it is very common now that you are going to get negative or fractional Solutions and if you do obviously we've gone over two of possibly the worst questions that you could have in order to practice those now when we have shapes where we have algebra on the shapes we are going to be looking at forming and solving an equation this is a relatively unique one here but we are given a rectangle where we are given two opposite sides but with different algebraic pieces and then we also have a y on the top it says here all measurements are in centimet and the area of the rectangle is for 48 CM s show that Y is equal to 3 now that line there where it says to show that y equals 3 is really important if we're going to show that y equals 3 we can't put three up here and then use that as part of our working out because that wouldn't be showing it's three that would just be assuming it's three and then using it to prove that it is three but here what we need to do now is essentially keep that as y we're going to try and figure out what Y is and if we show that it's three then we will have shown it okay so we're not going to use it at all we're going to imagine that the question never even told us this and we're just going to go about working it out so the first thing there are a couple of ways that you could make an equation so either if the area is 48 well we know that length time width is 48 so potentially we could do the length times the width and say that that is equal to 48 the problem is we have two letters involved and we want to avoid that at all costs obviously we can do that if we have simultaneous equations but we are just looking for an easier route on this particular question so if we can avoid that that would be great but here we do have these two lengths which we know are equal but we've been given two different algebraic expressions so what we could say is we know that 2x + 6 is equal to 5x - 9 we now have an equation with X on both sides it only has one unknown in it obviously on both sides but just the One X and we can actually go about solving that now when you have an x on both sides we want to get rid of the smallest X from both sides we'll get rid of 2x that would leave us with 6 is equal to 3x - 9 we want to isolate the X's so we can move the N9 to the other side and we get 15 is equal to 3x leaving us with a nice equation to solve we can divide by three and we get five is equal to X or of course you could write X is equal to 5 we can now use that to find the length of the rectangle so over here if we have 2X + 6 well if x is 5 that's 2 * 5 + 6 2 * 5 is 10 and 10 + 6 gives us 16 so we know that the width of the rectangle or the length of the rectangle is 16 now we can look at this information where it told us that the area is 48 we know that the area of a rectangle is the length times the width so if the area is 48 if we divide that by the length of 16 that would tell us what the width is and 48 ided by 16 is equal to three and therefore y has to be equal to three so there we go that we have shown it we have shown by forming an equation how we have found the value of x we have shown on the left there how we have found the length of the rectangle being 16 and then we've shown how that means the width or the length which however you want to refer to that has to be three of course you could have subbed the xal 5 into this length as well you'd have just got 5 * 5 and then take away 9 and 5 * 5 is 25 takeway 9 is also 16 so it didn't matter there which length we used to actually find that so there we go that is some forming and solving equations forming and solving equations can be one of the hardest topics that we actually look at purely because they are some quite difficult problems here is a good example where we have two rectangles and it says all measurements are in centimeters it says the area of rectangle a is equal to the area of rectangle B and then tells us to work out the per perimeter of rectangle B well here we don't actually know any expressions for the area but because they're rectangles we do know that we can do the width time the length or the length time width to find the area of a rectangle so for rectangle a we can do 4X * 2.5 for rectangle B we can do 2x - 3 and multiply it by 7 and we know that they're then going to be equal to one another so if I do these to the side we've got 4X multiplied by 2.5 4 * 2.5 is 10 so that would be 10 X for the other rectangle we have 2x - 3 * 7 now if you're going to times something like that 2x - 3 by a number typically we would write that using a single bracket so 2x - 3 ultip by 7 7 * 2x is 14 x and 7 * 3 is 21 keeping the symbol the same so that would be our expressions for the area now it says here the area of rectangle a is equal to the area of rectangle B so we can write an equation now where we say okay well the area of one is equal to the area of the other and now we have actually formed an equation we can now just go about solving this so it's up to you what you do first typically I would get rid of the smallest X from both sides if we do that it is a little strange though because when we take away the 10x from both sides we end up with 4x - 21 is equal to 0er that's absolutely fine though because now we can add the 21 onto the zero to isolate the X's we get 4X is equal to 21 and now we actually need to go about solving this and dividing it by four so typically this looks like it would be a calculator question just because that's not a very nice division to have to do but if we divide divide by four at this point we get X is equal to 5.25 so not the nicest of X values to have but we haven't even finished because this question wants us to work out the perimeter of rectangle B so rectangle B is obviously this rectangle here we know now that X is 5.25 now thankfully on this particular rectangle we also know that this length is which means over here must be seven as well so we just need to find the length of 2x - 3 that's going to be the same as the bottom so we just need to work this out by substituting X's 5.25 in so we've got 2 * 5.25 and then take away three two lots of 5.25 is 10.5 minus 3 and minusing three from 10.5 is going to equal 7.5 so the top there is 7.5 and the bottom as well is going to be 7.5 so we now have our four values 7 7 7.5 and another 7.5 so for the perimeter we just need to add them all together so 7 + 7 is 14 7.5 and 7.5 is 15 so 14 + 15 gives us 20 9 it did say all the measurements were in cenm so 29 CM would be our final answer for that one and there we go that is forming and solving an equation and that is a very difficult version of forming and solving equations okay that is the end of the video so we have gone through every question in the ultimate revision guide hopefully that was useful and helpful obviously at any point you can download the guide if you haven't already you can refer back to the questions you you can go into the chapters in the video to access any particular unit or any particular question to go through at your convenience whenever you would like to but hopefully that video was useful and helpful and hopefully I will see you for the next [Music] one [Music] over