Making Ratio Problem Solving EASY GCSE Maths

hey guys welcome back to another gcsc mats video this time as you can tell from the title and from the screen we're going to be making ratio problem solving questions super super easy now before we get started we're going to start with a very tiny I'm talking like 2 minute at best description of what a ratio actually is so I hopefully no one skips this this is really important I'm want to show you why ratio problems of can be quite easy but before we get started again most people that watch these videos are not subscribed so I would really appreciate it if you found any of this useful or any of my other videos useful if you could just press the Subscribe button it really does help out a lot in the description you'll see a link to my website where you can book tuition classes in case you're interested I've got space in science and maths and of course you'll have my email address where you can email any questions you have to me as well so let's start with what the hell a ratio is so let's say we have Suites and I have one blue suite and I have two red suets okay a lot of people write that as a ratio of blue to red as 1 to two which is true but what a ratio actually tells you is the proportion so this tells me for every one blue Suite I have two red Suites which means even if I had let's say two blue Suites I would have to have four red Suites okay so 2 to four is actually the same ratio and the reason why it's the same is you can simplify both sides by dividing by two and you get one to two so it's actually the exact same thing so the problem that a lot of people face is that some students will think that if I just add these two parts together it would give me the total well as you can see in this example I could add together two and four and get the total but if I add the total of this I get three which is not the total and here's the big problem if they said that the ratio of blue to Red suet was something like four to five and they said work out the number of red Suites that is an impossible question because there are an infinite number of combinations that will give you the exact same ratio an infinite number of solutions so in other words they will never ever just give you a normal ratio and ask for the number of something they can ask for the fraction or the percentage or the proportion they cannot ask you for the exact number without giving you another piece of information and that and the reason for that is if I just have a ratio there are an infinite number of answers for the number of red and blue SES this goes on forever because if I times both sides by the same number I get another combination that gives me the same proportion so let's actually get on to some questions now this is what I'm talking about there are only yellow black and purple buttons in a bag the ratio of the number of yellow buttons to black buttons is two to three so I like to write this out separately and the ratio of black to purple buttons is four to five and it says given that there are less than 100 buttons in the bag what is the greatest number of purple buttons and this is where students get confused what you mean the greatest number isn't there just one answer no because of this reason there's an infinite number of solutions now if I said that there are less than 40 red s uh suets then that means I have 1 2 3 four possible answers but if it's asking for the greatest number all of a sudden there's only one greatest number it's 40 well if it's less than 40 I guess it' be 20 that's it so in this case we're going to work out what the maximum number of purple buttons can be given that the combination of all three of those buttons should add up to less than 100 now the first thing we need to do is we need to actually combine these ratios so as you can tell they give us different proportions yellow to Blue is 2 to three blue to purple is four to 5 and a good way to do this is to make the common number the same so we're going to make three and four the same number some people like to write out like this as like a table so we have two to three and then for this we have four to five if we want to make these two numbers the same we look for the lowest common multiple the lowest common multiple is 12 so what do I times the first one by to get to 12 I Times by four so that' be 8 to 12 or do I times the middle one to get to 12 I times it by three so be 15 so that is our ratio for the number of yellows to Blues to purples so 8 to 12 to 15 now there's less than 100 in the bag well if I quickly add up these parts 8 + 12 is 20 + 15 is 35 okay so we're just going to work out how many times 35 fits into to 100 which is actually quite easy it only fits in twice so we're going to do is we're going to double all of these numbers and again if you add these up 16 + 24 is 40 + 30 is 70 so remember this is the ratio of yellow to Blue to purples that is the greatest number of purple buttons how do I know this well there's two possibilities there's either 15 purple buttons or there's 30 purple buttons they want the greatest well what's bigger 30 or 15 30 that's it how cool is that so they don't always say oh it's less than or whatever or use an inequality sometimes they literally tell you how many there are so here we have the number of houses and the number of flats from the ratio of 5 to3 so let's write H to f as 5 to3 and then Flats to Bungalows are 6 to 5 and they tell me that there's 50 bungalows in the village so again we want to combine the ratio first so some people again like to use a little table HF B so 5 3 65 the lowest common multiple here is six what do we times the top one so 3 * what gives me six we times it by two that's 10 to 6 and we times this one by 1 if you'd like so makes it five so the houses to Flat the Bungalows is simply 10 to 6 to 1 to five oops cool now it says that there are 50 bungalows in the village so I know that this number is actually meant to be 50 now because ratios represent proportion and that proportion won't change what do I times 5 by to get to 50 I times it by 10 which means I times the other two numbers by 10 as well so 10 * 10 is 100 6 * 10 is 60 so how many houses are there there are 100 houses in the village again the proportion should never change which means whatever we do to one number we do to the others this is the simplest way to do it I have got some other funky questions so do bear with me but that's it for now I love this question the perimeter of a right angle triangle is 84 cm I like to write I like to draw out the triangle they're telling me that the perimeter so all of the lengths added up are is 84 cm and the lengths of the side are in the ratio 3 to 4 to 5 what's the area now if you remember to work out the area we just need this side times this side now the first question I got asked by some students is how do I know which sides three four and five and I said it doesn't matter well there's only one side that matters but it doesn't matter which way around we label these two sides it does matter which one's the hypotenuse though the hypotenuse is just the biggest one that's all it is and then three and four it doesn't matter which way around why not because when we times numbers it doesn't matter if we do 3 * 4 or 4 * 3 we get the same answer so all we need to do is if we add up three five and 4 that gives me 12 but we want a perimeter of 84 currently I have a perimeter of 12 so if I want to make this equal to 84 what do I Times by I Times by 7 so I'm going to multiply all of these sides by seven so draw out another triangle 3 * 7 is 21 4 * 7 is 28 5 * 7 is 35 which means the area is just going to be 21 * 28 over 2 and I'm not going to do that in my head because I can't actually could I maybe 294 cm squared and for four marks I actually think that's a very good question because it combines together a few different topics So It's Tricky look how short it is and all we're doing is using the same logic the idea of portion if one side doubles all of the other sides double as well using that fact over and over and over again this is very similar so Rosie Matilda and Ibrahim collect stickers and they give us the ratio so I'm going to rewrite this in a format which doesn't make my eyes bleed we get 4 to 7 to 15 and it says Ibraham has 24 stickers more than Matilda okay now if Ibrahim has 24 more stickers than uh Matilda and the difference in these two parts is eight right 15 takeway 7 is eight that means those eight parts should equal 24 right because he got eight more parts than Matilda and he got 24 more stickers those two things are the same and it says Ibrahim has more stickers than Rosie well first of all let's work out how much one part is that's three stickers and that means we can work out how much each person gets gets so Rosie she has four 4 * 3 is 12 stickers Matilda I write Matt which is probably a bit confusing but Matt uh gets seven time three which is 21 stickers and Ibrahim which technically by the way since we know IAM gets 24 more than Matilda we could do 21 + 24 or we could do 15 * 7 um sorry 15 * 3 not * 7 which is 45 stickers which is again the same as 21 + 24 and it says Abraham has more stickers than Rosie how many more well he has 45 12 which is 33 more stickers than Rosie that's it okay so these ratio questions every year are answered quite poorly but we have to do is break down the proportion work through it take your time these are three or four Mark questions as you've seen so far that means you get three or four minutes at minimum to do them here's a question that probably came up in a lot of your mocks because this question was really badly answered and so quite a few schools that I'm aware of use this question in their marck to see see who really understands ratios and who doesn't so we have four types of card in a game if any of you have played with normal playing cards it's fairly similar to that but each card has either a black or white circle a black or white triangle so any one of those four shapes and colors Express the total number of cards with a black shape as a fraction of the total number of cards with a triangle well the best way we could work this out is if we had the number of black shapes over the number of triangles that's probably the best case scenario right because then we can just put them as a fraction job done and there are so many ways to do this the way that they do it in the mark sche the way I've seen a lot of teachers do it the way I've seen some students do it is using a lot of fractions and in my opinion that's a bit uggers it's a bit ugly this is a four mark question but you can do it very quickly instead I like to use logic so here they've broken up the colors of the shapes into eight parts and the shapes of the Cards into nine parts now logically these number of Parts should be the same because there's the same number of cards in a deck so in a normal deck of playing cards you have 52 cards now if I say one to three of them are you know Spades to not Spades and let's say one to three of them are Aces and not Aces uh maybe we should use a different value but the idea is because they're playing cards they should have the same total because they have the same number of total cards so in other words if we change this into their lowest common multiple which is 72 I should be able to get the exact proportion of each of these of each of these cards because we can make that into one big ratio so to do that I would have to times this one by nine so if we do black to y * by 9 you have 3 * 9 which is 27 5 * 9 45 then we also have Circle to Triangle we're timesing it by8 cuz 9 * 8 is 72 and we get 2 * 8 is 16 7 * 8 which will give you and again remember why this works so my overall ratio is this now I know some of you are going to say could we also do 144 or 288 or anything else yes but remember we're looking for a fraction fractions are proportion so for example if I have 10 SES and you have 20 SES I have half as many as you but if I have 20 sweets and you have 40 sweets I have half as many as you if I have 30 sweets and you have 60 sweets I have half as many as you and so on so even if you made this out of 288 you will get the same answer in the end if you simplify your fraction so again just remember the logic both of these ratios should be out of the same total because the total number of cards is the same for both of them we know all of the cards are black or white and all of the cards are Circle or triangle so in total the number of circles and triangles should be the same as a total of the number of black and white shapes should be the same then all it says is black shape triangle so black is 27 triangle is 56 and that's your answer that's it pretty pretty good and again I'm just using the fact of ratios right I'm just using the proportion fact it's all I'm doing the last the last thing I see very commonly in problem solving questions with ratios is when you have algebra and the only way they can do this is if they tell you what it's equal to so sometimes they'll say 2x - 1 x - 4 = 2 to 1 or something something like that but sometimes they give you another set of algebra now here is the deal I'm going to give you two equivalent ratios quickly hopefully we can all agree that these ratios are equivalent look what happens when I divide the right number by the left number for both of them I get the same value look what happens if I divide the left by the right I get the same value now this should make sense to you because remember it's proportion right if they have the same proportion then that means their fraction should be the exact same as well so we're going to use that fact in this question now here's the only caveat if I did something like divided the left and the right to get 1 over four but divided the right by the left to get four I do not get the same answer so you need to make sure you're dividing in the same order and you can choose sometimes it's easier to do one than the other for example if one side's one you probably want that being the denominator what I'm going to do is I'm going to take the left side divide it by the right side and that should be the same as the left side divided by the right side again remember it has to be the same order and then all you're doing is you're solving this algebraic fraction which of course is uh really easy right no not really first thing you do is You' cross multiply is that 16x or is it 6X no is 16x cool so I get 2x - 1^ 2 in fact I'll write it out like this let's write out properly and then all you do is you expand move to one side yada yada yada job done so 4x^2 - 4x + 1 we have 16 x^2 that would be - 64x + x so - 63x then - 4 let's move everything over to the right hand side so we get 0 = 16 x^2 - 4x^2 so that's 12x2 - 63x + 4x is - 59 x then - 4 -1 give me - 5 uh can we simplify this not that I'm aware of so what we're going to do is use the quadratic formula to solve this um some calculators like my amazing one actually does the quadratic formula for me but again hopefully the quadratic formula isn't too bad I get five b anticlimactic I know or -12 so the question is which one is it well with ratios we can't have negatives right I can't say that I have -2 times the amount of whatever than you there are Niche cases where you could like with money so if I'm £100 in debt and you have £50 in the bank I guess I could say I have -2 times the amount of money but again we don't normally say that so the answer is x equals 5 that basically takes us through every single possible type of ratio question they can ask you now I will say that situations like these they may change them of course right like that that's just we have to expect with Ed EXL so it's not about memorizing oh if they talk about houses and flats this is what I do if they talk about triangles this is what I do they may give you triangles they may give you houses and flats but they may also change it so make sure you understand why I did what I did and why it works more importantly that's the key key learning from this