AQA GCSE Maths FULL SUMMARY Topic by Topic Part 3: Ratio

hey guys just to let you know this lesson is in collaboration with flip Academy a free online platform for all of your GCSE a-level IB and cbse revision out now on web at www.flip.academy and also as an app for Apple IOS and Android it really is better than all the other GCSE platforms out there it covers all the topics in great depth and has an app that you can download your courses on and use offline hello everyone and welcome to flip Academy today we're going to be doing the ratio proportion and rates of change topic for AQA maths so here again we've just got our spec points so change between the standard units of length area volume and capacity we actually went through this in the numbers section as well so if you want to go back to that you can have a look and you also have to have compound units so for example here the heel of a shoe accepts a pressure of 188 pounds per square inch and we need to convert this into kilograms per square centimeter so one pound is 0.45 kilograms so to go this way we would Times by 0.45 and if you want to go against it we divide by 0.45 okay so we want to go from pounds to kilograms so we're going to times it by 0.45 and again this is normally on a this is going to be on a calculator paper so if we just bring up our calculators we have 198 multiplied by 0.45 that leaves me with 89.1 and that's kg per square inch and the last thing we need to do is convert square inches into square centimeters so once again it's going to be a multiply but we have to be careful here so we're doing this pressure we have pounds and then per square inch the square inch is at the bottom so if we're timesing this by 6.25 what we're doing to the overall thing is actually dividing because the bottom the denominator is getting 6.25 times bigger so divided by 6.25 okay so remember anything that's on the bottom of the fraction so it actually writes out first five six and that is kilograms per square centimeter okay so anything that's on the bottom of a fraction is going to be divide anything that's on the top is going to be times if you're going the correct way about it next thing we have is scale factors and diagrams so a map has a scale of 1 to 75 000 so what that means is for every one unit on the map that's equal to 75 000 units in real life so distance between two points on the map is 12 centimeters so here we have one to 75 000 and then on the back it's 12 so we have to work out what it's two here so all we do is give them one to twelve with times in by 12 so to get from 75 000 to the answer we're going to also Times by 12. once again it's going to be on a calculator paper so 75 123 times 12 gives me 900 000. centimeters now it tells me to go from centimeters into kilometers so centimeters to meters we divide by a thousand uh sorry 100. so that's going to give me a slice of two zeros we have nine one two three so nine thousand meters and then to go to kilometers we divide by further a thousand that gives me nine kilometers nice and easy the other type of scale diagram they can give you is that they give you a accurate scale drawing showing a small box and large bottle now accurate means that you can measure and it'll be correct so using either a ruler and it also works for angles as well so protractors so the Angles and the sides are all the actual length and size that they appear to be on the diagram so it tells me that the small box has a real height of 20 centimeters so we need to find SM for the real height of the large box the problem is we have no scale so we need to work out what the scale is okay now I know that this box the height in real life is 20 centimeters so let's see what that represents on the diagram so if we pull out our ruler and we're going to align it nice and neatly so it should be 90 degrees so you can see here it's one two three four five six so this is actually uh so one centimeter 1.2 four six eight so 1.8 centimeters is equal to 20 centimeters okay so 1.8 centimeters on my diagram is 20 centimeters in real life so if we divide both sides by 1.8 that'll give us what one centimeter equals so 20 divided by 1.8 again these are all calculated questions and we get 100 over nine so 11.1 recurring and again this is an estimate so it's okay if you get a slightly different answer now what we're going to do is measure the height of the large box so we're going to line this up nice and neatly and we have is whoops so we have one centimeter two three four five six seven and then eight point four centimeters so if we want to get to 8.4 centimeters again using our scale we times this side by 8.4 which means we have to times The Other Side by 8.4 and again they do give you a bit of a range so it's not as bad so again answer Times by 8.4 which gives us 93.3 centimeters to three Sig fig and again it doesn't look about four times bigger five times bigger so that is probably correct so we don't need this anymore so the next part for the next part we have to express a quantity as a fraction of another quantity okay so quite simply all we need to do is for a lot of these things it can be quite easy so I picked a slightly trickier example so Lily's age is two years and four months Hugo's age is one year and eight months right Lily's uh age a month as a fraction of Hugo's age so we have we're going to do Lily's age over Hugo's age because it says write Lily's age as a fraction of okay so it's going to do Lily divided by Hugo now two years is 12 months uh is 12 times two months plus four so that's 24 plus 4 is 28 months and Hugo is 12. plus eight so 20 months so what we have here is 28 over 20 it says to give your answer in its simplest form so we can divide the top and bottom by two and that gives us 14 over 5 which is the answer in its simplest form next is using ratio notation so essentially um there aren't really any questions on this because every ratio question will have ratio notation so all you need to realize is that ratio notation looks something like this so it might say number of red pens to Blue is five to four and that's all it means by ratio notation however you should also know that you can simplify ratios so yellow to Green let's say it's six to four so I could always ask myself is there anything I can divide both of these numbers by do they have a common factor and if I look I can divide both sides by two right so I should know that that is completely equivalent to three to two these rate of ratios are completely the same okay so that's all it means in terms of ratio notation so understanding being able to recognize it and then being able to simplify it and that's all that spec Point means okay so it should be too bad now jumping right into dividing a quantity into a certain ratio so again this is the 2021 paper paper three Jeff and Kaz share 270 pounds in the ratio Jeff the cars is 2.6 to 1. Okay so Jeff Getz 2.2 pound 60 for every one pound cares get so normally I'll draw a little bar out and draw the little boxes and everything like that but in this case it's going to be a bit tricky to do that because it's a decimal so we're not going to draw boxes instead we need to work out how much Kaz and Jeff get okay so if we work out how much Kaz gets first the best way to do this is to use fractions okay so again you can use like where you draw out a box and you shade them in and all that fun stuff in this case because we have a decimal there's no point doing that and what we're going to do is we take the number we're interested in so it's Kaz we have we're looking at the number one and that's over 2.6 plus one and then all we need to do is multiply it by the total amount of money so again it's a calculated paper it's paper three so let's do it so we have one over two point six plus one and then multiply by so if we just press equals multiply that by 270 Jeff sorry Kaz gets 75 pounds now again what you could do is do 270 minus 75 to get your answer but in this case I'm actually going to do the exact same thing for Jeff just to illustrate my point the number we're interested in now is 2.6 over 2.6 plus 1 times 270 and again you should get the answer of actually no let's do it the proper way should be 195 but let's do it we have that bracket 2.6 over 2.6 plus one close bracket multiplied by 270 195 okay so of course you can still do you know what might make sense 270 minus 75 that's to illustrating my point now the actual question says how much more than cars does Jeff get so he gets 195 minus 75 120 more than cars okay so in terms of the marks here I'm up for this I'm up for that and then unlock the final point the next bit again is a relatively short topic Express a multiplicative relationship between two quantities as a ratio or a fraction so again November 2021 papers a is half of B right so in other words B is 2 times a okay now if B is 2 times a then that is the same as writing a ratio of B to a as two to one so if you were to write this A to B all you do is we flip it round it's one to two so the answer would be this one okay so if one quantity is twice another it's in a ratio of two to one if it's three times another it's three to one and so on okay the next bit is understanding and using uh proportion as equality of ratios so it's very similar to the previous part right so the idea is that if I say that b is 3 times bigger or more than a hopefully you can see that that's proportion right B is 3 times a then B to a will equal three to one that's what it means by equality of ratios the fact that those two things are equal they mean the same thing okay likewise if you talk about proportion if you're saying that debate uh 10 cookies you need two eggs they can ask you well what if I want to bake 20 cookies but then hopefully you can see that you need twice as many eggs because to go from here to here you times by two so the number of eggs you need should also double and that's all that means okay so this is kind of proportion and then proportional equality of ratios just meaning that you should note that proportion and ratios are the exact same thing okay that's all it means next things we need to relate ratios to fractions and to linear functions so again similar to what we've done before if I have a so let's say I'm splitting money or splitting 24 sweets between Bradley and Stephen and the ratio of Bradley to Stephen is three to one I should recognize that this means that Bradley gets three over three plus one so three quarters of the sweets okay so a lot of these spec points are about notation so being able to understand what we mean by ratios and linking them to fractions and things like that okay so it's not necessarily there's a lot to remember it's just that there's a lot that you have to really process and really understand okay so that's all that means here is that ratios are the same as fractions likewise if I say that in a different uh Universe Stephen gets two-thirds of the sweets you should have to change that into a ratio of Bradley to Stephen well Stephen he gets two parts because he's the number on top and then two plus something gives me three it's a one so that's the ratio they share in that Universe okay she needs to be really really aware of that okay you should be able to go from fractions to ratios in the way that I've just described and from ratios into fractions again in the way I've just described hopefully that all is completely fine the next part is all about percentages okay so there's a lot of stuff you need to remember for percentages um and again we do have a couple of examples here but we need to go through some Basics first so first of all a percentage is just a number of parts per 100 okay so the idea is that 30 is the same as saying 30 parts per 100 this doesn't really come up in questions at all but the idea is that you should be able to say 30 is the same as 30 over 100. okay likewise 21 is the same as 21 over 100. you need to be able to link percentages to fractions of a hundred and the fact that it is just kind of always out of 100 okay you should also be able to interpret percentages so again 21 as 21 over 100 you should be able to also write them as fractions but also decimals okay so 21 divided by 100 hopefully you know if you have 21 divided by a hundred you're just moving the decimal place twice which means it's the same as having 0.21 likewise if you have 30 just add a decimal point in a few zeros if you're dividing by 100 you take the decimal point and you move it to the left twice you get 0.3 okay so you should be able to change between percentages fractions and decimals again it's really easy to go from percentage to decimals you just divide by 100 which is just moving the decimal point twice to the left likewise going from decimals to percentages you just move it twice to the right so 0.21 is just 21. 0.3 is just 30 and so on you should also be able to interpret these multiplicatively so what that means if I say 21 percent of 200 that's the same as doing 21 times 200 so the best way to write that is as a decimal so 0.21 times up 200 and then working out what that is okay so and again you should be able to do that as a fraction or as a decimal right so all of these things are the same you need to interpret fractions decimals and percentages are just multiplying and they'll all give you the same answer Okay so now that's a bit obscure so let's get on to actual some hardcore maths the next thing you need to be able to do is to work with percentages greater than 100 okay this is a really key thing okay in general if something is a three percent increase that is the same as saying it's at 103 percent of the original amount okay that's all that means three percent increase so it's at 103. so all I did to do that to get that is 100 is its original price right then I add on the three percent that it's increased by okay so in this case David is getting paid 34 000 pounds per year he's going to get a three percent increase in the amount of money he's paid so you can either do the thing where you find three percent and then you add it on or you can say thirty four thousand pounds and I'm timesing that by 100 plus three percent so it's the same as just doing 103 percent of 34 000 okay now in terms of different ways you can do this uh one way which I'm quite partial to is changing that to a decimal so remember to change to a decimal you move it to the left twice and you just multiply it nice and easy so again it's all calculated stuff so it's not too bad 1.03 multiplied by 34. 1000 35 000 and 20 pounds and again this is my preferred method of doing it or again you can leave it as a percentage so 103 percent of uh so you can do it where you literally Times by 103 the percentage sign is just shift and the ANS button and that gives you the same thing okay likewise if you wanted to do it as a fraction right you have 103 out of 100 multiplied by 34 000 again you get the same thing so it doesn't matter if you use a decimal or percentage or a fraction it's all the same thing which is exactly what we just said okay uh the next thing that you have to do is also calculate percentage loss and percentage profit right so um I really like percentage of profit and loss because there's a one formula okay it's the final amount minus the original over the original and this works every single time okay so with this final amount it was 213 590 pounds minus the original price of 201 500. we're dividing it by the original so 21 500. they're multiplying it by a hundred so again let's whip out our calculators so we have that and then fraction button we have two one three uh 590 and minus 201 500 over 201 500 multiplied by 100 gives me lots of six so in other words she made a six percent profit that's it now there are a few different ways to do that okay a few different ways so some people teach you just to do final over original times 100 but then you need to minus 100 as well that's why it's too complicated for loss some people teach you to like reverse this to original minus final you can do that but look what happens if you use our exact same formula we take the final minus the original the final was 325 000 600 pounds take away the original over 300 52 000. times 100 okay so look what happens when we do the exact same thing so we're using the same formula okay so some people teach you a different formula there's really no need for it at all this is way way way easier because you're remembering fewer things yeah a negative number so this is where people panic and that's why they want another formula but let's think about it for one second right she's lost money right that's where the negative comes from right the word loss is the negative so this translates to a 7.5 percent loss for those of you that for those of you that use a different uh formula do the exact same question but use your other formula you get the same answer okay so best thing to do is if they ask you how work out percentage profit or loss use the same formula if it's a positive number it's profit because our money is six percent more it's a negative number it's a loss because she's lost 7.5 percent of her money makes sense right there's no point in Remembering multiple formulas in my opinion okay but you need to be able to do both of those whichever way makes you comfortable the um last thing yeah the last thing is you need to be able to apply percentages for more than one year so that sounds really complicated all it means is interest or depreciation so Perry invests 25 000 pounds for three years in a savings account and she gets 2.7 compound interest so what that means is her money increases by 2.7 so again if we use our little thing that's a hundred plus two point seven percent her money is increasing as so that's 102.7 that's how our money is increasing every year okay to write that as a decimal let's move the decimal point two spaces over to the left that's 1.027 each year okay now here's the funny here's an interesting thing right to work out the amount she has at the end okay we take our 25 000 pounds we times it by this little multiplier we have right now that only works for one year so if I want to write this again the next year you'd times it again and for the third year you times it once again so instead of writing that because that's a bit long and tedious get rid of it all you need to do is put brackets and put it to the number of years so if you want a kind of formula the final amount equals the initial amount times your multiplier to the power of the number of years you're investing or the number of years past whatever you want to call it right simple as so again a lot of students find compound interest difficult but once you really understand where it comes from it's really not that bad and that's a common theme with GCSE maths it's difficult until you understand it so you need to accelerate the understanding so 27 000 so let's think for a second there's a lot of decimal points what should I round it to when it's money right so we should only run to two decimal places right you don't have point one six seven zero eight of a pound so 27 000 pounds uh twenty seven thousand and eighty pounds and 17 Pence not too bad unless everything you need to know about percentages okay you need to take percentages of numbers you need to do um percentage loss and and Gain profit and compound interest as well to have some proportionality I'm quite partial to portionality it's quite fun this is part a this is part B so L is directly proportional to D Squared that means as d goes up D Squared goes up L also goes up so work out an equation connecting lnd well what we do is we put the L and directly proportional is this symbol okay so if it said L is equal to D Squared then we would put the equal sign but because this is directly proportional we put the proportional sign which looks like an alpha from science or just like a fish with no with the tail missing going the other way now to get rid of that sign we need to introduce a constant and that's how we get rid of that sign so L is not equal to D Squared it's equal to something times D Squared now some students leave it here this gives you one Mark it's a three mark question it's not going to be one step okay they also give you this information they never give you information you don't need okay so we need to work out what K is okay will be L over D Squared is 85 over 10 squared so that's 85 over 100 so it's a 0.85 uh just in case you don't believe me 0.85 cool so I'm not done yet that's the second mark the actual answer is L is equal to 0.85 D Squared hopefully that's not too bad so for Part B it's nice and easy find out the value of L when D is five well L is equal to 0.85 5 squared shoving the calculator so answer times 25 is 21.25 that's two marks doing about five seconds not too bad I'd say so again that was from paper two uh 2020 paper one also had this now this is a fun one okay it means inversely proportional to not directly proportional so what that looks like is one over whatever is is on this side okay so directly proportional L proportional sign D Squared inversely proportional it's m one over square root of r it then asks us the value of R is multiplied by four so I don't need to do anything else by the way so if I did m equals again you multiply both sides by r by K it says R is multiplied by four so if this became four right so if I multiplied that by 4 what would happen to the value of n well let's have a look the square root of 4 is equal to 2. so now M will equal K over 2. so the value will divide by two hopefully that's not too bad compound units okay so the compound units that you need to know are the main ones are speed and if you remember that's distance over time so you have to use all different distances over all units of time so I'll just I'll show you some common ones but keep in mind they can't ask you about some others you have meters per second you have kilometers per hour you also have miles per hour you need to be able to use those units and convert between them you have um rates of pay which is always just going to be basically pounds per hour or pounds per week or day or whatever you also have density which again is mass over volume so it's usually going to be kilograms per cubic meter or grams per cubic centimeter and lastly we have pressure which is force over area that is nine times out of ten given as per square meter but it kind of given us pascals or as we had before you have PSI which is pounds per square inch those are the main ones you just need to be able to use them okay so what that means is when you do a calculation with density you should be able to say what the units are if you do a calculation with speed you should be able to tell what the units are and so on which I'm hoping you're completely fine with next one is comparing lengths areas and volumes using ratios so again we have the 20 to 21 paper 3 which you've looked at a few times so in general what you want to do is if it's just a length of some kind right what you're going to do is square root area or cube root volume likewise if you want to find the area you're going to just take the length and square it and if you have volume you're going to take the length and Cube it okay so in this question our length is our ratio so we're going from an area to a radius so what we're going to do is just square root it so square root of 2 and then for B we have the square root of 4.5 so our answer is the rate the radius of current a is going B is going to be root 2 to root 4.5 however it's asking me to put it in as one to another number so let's do that we divide by square root two so we need to divide this side by square root 2. and again fairly simple actually let's just quickly do this so we have the square root of 4.5 divided by the square root of 2. 3 over 2. so we don't normally write fractions so let's do one to one point five okay so again it's it's a very a lot of students find it tricky but it's not tricky it's just because they haven't covered it before okay so if you want to get a length which you always do in these questions you either square root whatever the area is or square root the um volume okay and that will give you your answer you can also find the um ratio between them and then square root it but either way that's what you're going to be doing this question um it's really uh a bit obscure but it's not that bad so essentially all it's saying is is that you should know that if you say x is inversely proportional to y that gives you this right so again X right that is the exact same as saying X is directly proportional to 1 over y that's it that's all that matters okay and you need to be able to construct equations from that so if they tell you X is directly proportional to this then you need to do what we did in the previous part up here and create an equation but you should see that um so sometimes they may have multiple choice question say which two are equivalent these two statements here would be equivalent we're going to get onto some again obscure stuff you need to be able to interpret the gradient of a straight line as a rate of change so the idea here is if you have a straight line graph and let's say this is a cost in pounds and this is the amount of time the gradient is going to give you cost per time right and specifically it's going to give you a rate of change so how quickly cost changes with time and the steeper it is so the steeper it is the faster cost changes with time so if you want a generic thing is the gradient is going to give equals y per unit X okay that's all you have to say you say whatever's on the y-axis per unit whatever's on the x-axis in this case it's cost per unit time if this was a volume and time you'd say it's the volume per unit time okay super easy and it will always give you a mark and it will also tell you if they ask you to again that's how you interpret it right it's the y-axis per unit x-axis or how quickly y changes with X okay so in any of these questions if it asks you if it ever asks you interpret the gradient what does the gradient mean always just say it's whatever's on the y-axis per unit whatever's on the x-axis you will always get the mark nice and easy and you'll be able to get that now if you're doing the higher tier this is where things get a bit more interesting okay because what if you have a curve okay so let's say you have something like that well the gradient of let's say a point here is the only difference is that it's the instantaneous rates of change so what it is it's the Y per unit X at that value of x and only that value of x okay and again also one thing you may uh want to remember is if it's a distance time graph what does it mean for it to be distance per unit time that is your speed okay so that's just one other thing to remember okay now the last spec Point says to set up solve and interpret the answers in growth and Decay problems including compound interest and also something they don't mention is depreciation and if it's higher T it's also going to be iterative processes this is this question here is all of those in one okay it's a growth problem which is very similar to compound interest and it's an iterative process okay so assume that the population increases by 12 each day okay so increase by 12 so once again we have a hundred plus twelve percent which is a hundred and twelve percent cool now if you remember I said that final is just equal to initial times your percentage but as a decimal says 1.12 to the power of the number of years or days in this case it's days it really doesn't make a difference this is going to be a very big number I've I'm very very sure so we have 4 200 multiplied by Open Bracket 1.12 to the power of 20. yep uh and I'm not going to write the decimal because it's the number of butterflies and you know I don't know much about butterflies but I'm pretty sure you can't have 0.4 of a butterfly so we'll leave it as that which is indeed greater than 40 000. so what does it mean by iterative process well an iterative process is just where you do it day by day so you'd find out what one day is then you find out two days and so on that's it and it doesn't really seem like a higher topic but it's not too bad anyways that is the end of the ratio topic so slightly shorter than an algebra one so hopefully you found it a bit less taxing on the mind but yeah I hope you enjoyed and I'll see you in the next one