Mastering Percentages for GMAT Success

hello everybody and welcome to the next video in our gmat ninja con series my name is harry duffy and today we're going to be taking a look at the next video in our word problem section we're looking at percentages we're going to be starting right at the beginning looking at the basics of percentages how to use them how to convert them between fractions and decimals i'm going to be working right through to the compound interest questions towards the end now before we get into this i'm going to give you an idea about the difficulty rating in this video we probably put this somewhere in the middle so if you're in the 30s or the 40s and you've struggled with percentage questions then we're probably gonna have a few things for you here now while the start the video starts out very foundational we're going to be looking at the basics of everything the second half the video the difficulty level is going to ramp up and there's going to be a couple of questions towards the end that'll really challenge you unless you're in the high 40s already just as a warning for those of you that are in the 48 49 50 region already and are looking for questions the very difficult questions are going to push you onto a 51 we probably don't have much for you today what we're looking for the people that will really benefit from this video are the ones where if you're looking at a percentages question you're just confused by the phrasing of it if you're not really sure where to begin then we're definitely going to have a lot for you and stick around certainly for the beginning of the video the next group if you've yeah if you've memorized all the formulas now if you know that compound interest formula but you're still missing a frustrating number of questions then we're going to be looking at the uh breaking those formulas now breaking the processes down and having a look at the why behind these questions and so yeah if you fit into this box again we're definitely going to have some stuff for you here and all of this can falls into these two bits if you're not if you don't really understand percentage questions but you're just relying on some tricks to get you through them then as i said before we're going to break some of these questions down tell you what to look for and how to look for it and we'll certainly have a few things to help you and this all comes under the category of if if you know everything but you're still struggling with percentage questions and you're not really sure why if you fall into any of these four categories then this video is definitely for you and please stick around we're gonna see what we can do to help you out now i've got nine questions for you today and the way we'll do this is i will put a question up on the screen and give you a couple of minutes to take a look at it then i'll come back we'll take the question away and we'll work through it together on the board and i'll try and explain it in a way that helps you better understand what it is you should be looking for and what it is you should be doing in the question within those questions we're going to cover well we're going to cover as part of them you know translating word problems into math speak this is a feature of our word problems series that we're doing within the overall quant series that if you can be as literal as possible with some of these questions it makes it easier to translate the text into algebra and we'll have a look at what i mean by that later we'll get into just percentage changes single or multiple percentage changes finding a percentage of something or increasing or decreasing something by a percentage but this is the big thing for this video and this is one that i'm going to be really pushing throughout the entire next hour hour and a quarter however long it is how to make fractions your friends if you're not comfortable with fractions then percentages are going to be very difficult so yeah if that sounds like you then i'd take this as a bit of a warning that maybe in the next few days you're going to want to do a lot of work on fractions and that will definitely help you understand percentages better and then we've got this bit looming at the end the compound interest formula it's not it's not the most important thing but it certainly makes a lot of people nervous and we're going to go dig into the details of that all right so without any further ado i will get the first question up on the screen i'll give you a couple of minutes to take a look at it and we'll see you back here soon so we can go through it together okay i hope you've had enough time to take a look at this one if you feel you need a little more time if you've not quite got your answer yet then feel free to pause the video now we'll still be here when you're done but for now i'm going to take the question away so we can take a look at this one together now as we set up at the start of the video a lot of this the challenge in some of these questions is translating the text literally into algebra now we've done a full video on this already and if you want to go take a look at our algebra translations videos after you've watched this one then that'll give you many many more examples about putting this process into action but we'll take a look at how it's put in place in this question and we'll use that technique again and again and again as we go through the rest of this video now the big clue to get this question going is this final sentence the fox 8 what percentage of the hence that's the question and the challenge is to translate into that into algebra so i've rewritten this up at the top left and we're going to go just go word by word converting this from text into algebra so the first section the fox 8 well from the question we were told the fox 816 hence so the fox is going to eat 16 of these hens now the next bit what percentage so oh i've got an unknown percentage here that that's basically the question so let's just come up with some sort of letter and i'm going to call that p percent and the percentage so p percent there now the word of when you're converting text into algebra the word of you could just substitute for a multiplication and then the hens we were told that there were 800 000 hens in this hen house and now we end up with yet the word 16p percent times 800 000. we say well 16 that is equal to p percent of 800 000 because that's what we're defining the question to be we want to know what percentage of the hence the fox 8 we know the total was 16 and so we can put an equal sign in between there 16 is equal to p percent of 800 000 and the question here is what is p and so now that you've translated this literally we've got the beginnings of our equation there we just need to tidy this up into a full algebraic expression so the 16 doesn't change here the only real change is this bit p percent whenever you see something like p percent x percent r percent ten percent five percent whatever it is then that percent means you're going to put something over a hundred and that's very important this is one of the one of the areas that lots of our students make those small unnecessary mistakes everything else they do is absolutely spot-on the algebra is fine the understanding of the concepts fine they just forget to put something over a hundred and that ruins the solution so whenever you see let's say p percent that is another way of saying p divided by 100 when you're converting that into algebra then we've got our times 800 000 on the end and now i've got an equation we can begin to solve this first step i'm going to cancel out a couple of zeros here so we get 16 is 8 000 p or dividing both sides by 8 000 i'm going to have p is 16 over 8 000. now as we've said before in some of these videos at this stage you don't need to be able to do 16 divided by 8 000 straight off the bat you could just begin to cancel down top and bottom what's a number that divides both 16 and 8 000 well yeah there's a variety of them i'm actually going to go with eight so 16 divided by eight will give me two eight thousand divided by eight will give me a thousand i could have done it i could have divided top and bottom by two by four by eight by sixty it's just a case of don't think you have to get to an answer in one step just start cancelling work your way through be calm be measured but um don't take your don't waste time and you'll get to an answer in the end it will probably be much more accurate than you just trying to jump there in one step so now 2 divided by a thousand i get i could divide top and bottom by 2 here but just looking over i've got all the answers in decimals so why don't i take that 2 and divide it by a thousand and i'll get p is 0.002 so now i've got a value for p and it's the final question is do i need to do anything more with it another place in a question like this that some of our students fall apart is that well they say that p i found p but to make a percentage i've got to times it by a hundred i'm going to say no what we did at the beginning was define this as p percent so whatever p is that is the percentage i'm looking for we already put it over a hundred we've already baked in the percent part into this problem and so i can just take this number and go and find the answer choice that matches it in this case d and that's going to be my answer all right so the challenge in this one was converting the text straight from the great big wall of writing that the question gave you into this algebraic expression that we could go and solve now as you can see once you get the equation up there the algebra is not that hard and in a lot of these word problem cases that that is the challenge 70 75 something like that of the question is done by the time you get the algebra on written down if that process is still a challenge if you didn't really follow what i was doing there then as i say we've got the algebra translation video that'll include another eight or nine examples of the putting this process into action and at the end of watching this one maybe go and find that one and as i say there'll be more examples for you there in the meantime we're going to keep going with the percentages topic and i will get the next question up on the screen for you i'll see you back here in a couple of minutes time okay now's the time to pause the video if you need a little more time to look at this one for now we're going to take it away so that we can take a closer look at this question together so of the two things we said at the start that we're going to make a big deal of in this video the algebra translations and making fractions your friend we're going to really use fractions in this question again they're going to be used throughout this video so now is a good time to sort of pause and say well okay which are the the big conversions you should be able to do and yeah what what are the what are the percentages and how do they link to fractions now there's very few things that we really really ask our students to memorize when we're getting ready for the gmat we try and stay away from formulas because when it's question 30 and the clock's ticking away the pressure on you it's likely to make you misremember the formulas in some respects but there are a string of percentages that are very helpful to know and if you can convert between percentages and fractions your life will be so much simpler so we've got some of the easy ones like 50 30 that's just a half most people know that in the same way 33 and a third as a percentage that's a third if you convert that to a fraction 25 is a quarter 20 percent is a fifth those are the ones that most people are very comfortable with the next three are ones that fewer people are aware of sixteen and two-thirds percent as a fraction that's a sixth twelve and a half percent that's an eighth and eleven and one-ninth that could be a ninth and then we're back to the the simple ones that most people know ten percent is a tenth and five percent is a twentieth now if if you're aware of these fractions and you can very quickly convert between them then your life will be made easier in percentages but we can go a little bit further these are the bases and in particular for this question i want to take a look at 11 and a ninth as we said that's one nine percent but you can begin to get multiples of this so 22 and two-ninths as a percentage that's two over 9 33 and 3 9 well that's 3 over 9 which is a third which is what we've got here 33 and a third is the same as 33 and 3 9. but the big important one for us right now 44 and 4 9 as a percentage that's going to be 4 over 9. and we're going to make use of that as we get into this question so just as an aside these percentages and the conversions between percentage and fractions very very useful for dealing with percentage questions and if you don't know these i suggest you spend a little bit of time memorizing them and you'll you will see as we go through the rest of this video how helpful they can be now as i said we're going to use that in this question i come to the question now eventually i want to know what was the percentage change in internet users from 2000 to 2003 but for now i'm going to ignore that part of the question just look at this bit at the top so in 2000 the number of people using the internet was 44 and 4 9 so 4 over 9 if we convert that into a fraction of the number in 2003. now i don't have any numbers to put here and here one of the themes of this video is going to be please stay away from just launching numbers in and hoping they'll work out i don't want to say there were 100 internet users in 2003 as you're going to see as we get later on that way of thinking is going to get you in a lot of trouble i'm just going to put some sort of unknown there i don't know how many there were in in 2003 but i know that the number in 2000 was 44 and 4 9 of that and our conversion says that that's four ninths of that number so four and forty four and four ninety percent of the number would be four ninths of the number or four ninths of x so the number in two thousand is four ninths of the number in two thousand and three now i want to reverse that i want to know what the percentage change in internet uses from 2000 to 2003 was so what i'm going to do is i'm going to say that that x there i want to make that the subject of this transformation from 2000 to 2003. so if i have the 4 9 multiplied by the x in order to make this x the subject i'm going to times both of these by 9 so 4x to 9x there and then divide both of them by 4. so x to 9 over 4 x or other words to go in this direction i times the x by 4 over 9 to go in that direction i times it by 9 over 4. so in 2003 we actually had nine fourths of the number we had in 2000. now that that's not us done because i don't want to know what the what percentage of internet users i want to know what the percentage change was so we need to think about this in a slightly different way this nine over four x well i could think about that as the x plus five over four x this is the original this is the change and so what can i make 5 over 4 into a percentage well as we said 1 over 4 was 25 4 over 4 is going to be 100 because that's full so 5 over 4 is going to be 125 of x so the percentage change is going to be 125 percent which is d so there's a few places you can slip up in this question as i said once you get it into fractions it's much nicer this way of thinking about the conversion back and forth steers you away from these two answers at the beginning yeah if this is four ninths you don't just add 5 9 to get back to the original this is more about multipliers i want you to always be thinking about multiplying from one way to the other when we're dealing with percentages rather than just saying i'm going to add five nights on that i had before that's not going to work in this scenario and then this bit yeah separating that nine over four into the two values because it said percentage change and be very careful with the wording in these questions they're not trying to trick you but if you're not paying attention you can just slip and go down the wrong path all right so as i said the big thing about this one is translating literally and using fractions and we're going to get that into more of the questions as we go through i'll get the next question up on the screen and i'll see you back here in a couple of minutes time okay i hope you've had a little more time to look at this one if you need if you need any more time feel free to pause it now but i'm going to take the question away so that we can look at it together um right so we've got a few things going on in this one we're going to have to find two different percentages push them together um and then at the end figure out yeah figure out another percentage change what percentage higher at store c than at store b but before we get to that final bit we've got to look at the difference between store a and store b and there's something in the wording of this question that i want to take a look at as it says here i've rewritten bits the question the price at a is 25 higher than at b now just this freezing and begins to confuse some of our students here the difference between price at a is higher than at b and a is some percentage of b and this is one of those scenarios where i will take a question and always rewrite it so that i'm doing the same process every time in this case the price of a is 25 higher than the price at b well then i could say the price at a is 125 of the price at b if it's 25 higher then we get the price at b plus 25 of that so the original price is 100 plus the 25 so it's 125 of the price at b now reason i do that is is mainly just so that i'm doing the same thing every time and because that word of it just means multiply so the price at a is equal to 125 so 125 over 100 of the price at b or i could simplify that a is five-fourths of b and i've got myself a very simple equation that i can work with there and it's all in the wording of the the text so i can translate directly from the price at a is 125 percent the price of b it's much harder to translate the price at a is 12.5 percent higher than at c directly into algebra yeah which is why i'm rewriting these so for the second one if the price at a is 12.5 percent higher than at c then we get the original hundred percent of c's price and we add on 12.5 so the price at a is a hundred and twelve point five percent of the price at c and again i can convert that more directly into algebra so a is 112.5 over 100 times c now this one might be a little more difficult to convert straight in or to to simplify down yeah if you need a bit more time to do the canceling go for it but if you can see that 12.5 or if you can remember from the last question 12.5 percent was an eighth in this case we've got and the whole of c 8 8 plus one more so this is a is 9 8 of c and so by taking the question rewriting this bit about some price being higher than another bit as to being some prices this percent of the other bit you can much more quickly convert into algebra and that'll give us a couple of nice equations that we can go and use so we know that a is 9 8 of c and we know that a is five fourths of b and i want some relationship so the question now the price of the shirt is what percentage higher at store c than at store b so i'm going to want to get c and b into a single equation with no a in there and then i want to get c equals something and that'll give me something to work with so i've got two equations here i've got a equals something an a equals something so i can push these two bits together and say 9 8 of c is equal to 5 4 of b now as i said i want to make c the subject of this so if i multiply both sides by 8 i'm going to get eventually going to get 10 b and then divide by 9 c is 10 over 9 b area 10 over 9 if i was to convert that into a percentage that's going to be more than 100 which is none of my answers thankfully this question is about what's percentage higher so i could rewrite that 10 over 9b as the original plus 1 over 9b it is this much higher than the original that's the bit i'm looking for now yeah we've in the last question we said that 1 over 9 was equal to 11 and 1 9 percent so we could say that c is equal to b plus 11 and one-ninth percent of b and so c is 11 and one-ninth percent higher than the price at b which means that the correct answer here is c so what yeah one of the things we we do when we're dealing with word questions like this and this will come it'll definitely come back in number properties this is the other obvious place of doing it is to rewrite the question so the translation from the text into algebra is more literal and if you can get the hang of saying what yeah a is 25 higher than b that means a is 125 of b or the reverse if a is 25 lower than b then a is 75 of b in that literal translation just because the word off means multiply you can create these equations with the fractions that work much more simply than those fiddly ones with lots of percentage signs and it's just an easier way of dealing with them you want to be doing the same thing every time so that when the time pressure is on towards the end of the con section you don't get muddled up between two different processes that's why i always rewrite them this way and hopefully that can help you too all right i'll get the next question up on the screen and i'll see you back here in a couple of minutes time okay now's your chance to pause the video if you need a bit more time to look at this one i'm going to take it away so we can take a closer look okay the the big things in this question are bringing some of the things we've looked at in the last two problems together and so we're we're dealing with percentage changes we're dealing with the change in the tax and we're putting it all together into one equation here what we're looking at is the firm's revenue and we want to know what percentage of it was it this year or the year in question compared to the previous year before the changes before they charged more money and sold fewer hours so the first thing we need to do is we need to come up with some sort of equation for the firm's revenue because that's what we're looking at now in this question all we're told about is the number of hours that the firm sold and the cost per hour so our our revenue formula is going to be let's say n for the number of hours the firm sold multiplied by the cost now we could say that this is the revenue for the previous year again i could put numbers into this but if i go down that path and make that into a habit i'm going to run into a lot of trouble what i want to do here is stick with algebra and i'm going to say that was the previous year and i'm going to make some changes and see what the revenue is like in the new year so i'll i'll still use the revenue but we'll say this is the revenue for the new year now during the sorry this year the firm raised its hourly rate by 40 so we could look at the the cost value the cost value in our new year is going to be 40 higher than it was the previous year or if you remember back to what we're doing the last question 40 higher is the same as 140 of the previous year so i could say that my new cost per hour of consulting is going to be 140 over 100 times by c i could do this very similar thing with the number of hours of consulting we're told here they sold 30 percent fewer hours 30 percent fewer that's the old value was 100 i take 30 off that my new value is going to be 70 of the previous value and so number of hours is going to be 70 over 100 times by n and so this whole expression i can simplify the 7 over 100 sorry 70 over 100 to be 7 over 10 i can simplify the 40 100 sorry 140 over 100 that'll actually cancel down to be seven over five and then i've got the n and the c now we can do the multiplication here 7 times 7 will be 49 10 times 5 will be 50. and just because i'm working with percentages here i could multiply the top and bottom by this fraction of this fraction by 2 and get 98 over 100 times n times c so this new revenue value that we've been working out is 98 over 100 times by nc now the nc that's the old revenue value so the new the new revenue is 98 over a hundred times the old revenue value converting this into a percentage well we've got 98 over 100 as we said way back in question two a percentage is just defined as something over a hundred so we can again literally rewrite this as the new revenue is 98 over 198 percent of the old revenue which means that our answer here is going to be b now full disclosure with this one this is one of those questions where you could have taken the old revenue and picked a number said it was a hundred fine yes it will work in this question but as you're about to see in the next few questions if you were to try and do that elsewhere you're going to get into all kinds of trouble and as we've said before one of the things that we really push is to make sure that you have a technique that will work in every question so that you're never stood there at the start of a question just going is this one where i should use algebra as this one where i should pick 100 and see where it goes it's safer and you'll do less thinking you'll be less stressed if you just do the same thing every time so if i can convince you of one thing in percentage questions using this video is to please not throw numbers at it and just see where it goes stick to algebra stick to general arguments if you can and as you can see with this one it can actually work out very nicely right so on that theme i'm going to get rid of this question we'll get the next one up on the screen and i'll see you back here very soon okay now's the chance to pause the video if you need a little more time to work on this one i'm going to take this away now so that we can take a closer look so these these questions about products being bought at stores and they're given discounts or they're being increased and then they're being sold they're being sold at a markup or they're marked up then discounted that the actual algebra behind them is not usually not particularly difficult the problem with these questions is keeping track of the variables and what you're doing with them so i'm going to be very careful with how i define things and the way we build this question up so that you can see the the simplicity in the question and the the places where you might go wrong so to begin with i'm just going to work through the text here a store brought a product okay so the cost price to the store i'm not giving a value for that i don't know what it is i'll just call it c to begin with we know that it has some recommended retail price of 150 right the next sentence tells us that the store sold this product at a 40 discount now you're gonna get sick of hearing me say this as we go through the video 40 discount means 40 off the original price so 100 take away the 40 gives me 60 of the recommended retail price my sale price is 60 of 150 turning that into algebra 60 over 100 times 150 and you can work that out that the sale price is going to be 90 now we're then told that the store made a profit a 150 profit on the wholesale cost or in other words the profit is 150 of the wholesale cost so my profit that's going to be 150 of c or just 3 over two of c as we're simplifying that down now what i want to get to as we go through this is what was the store's profit as a percentage of the recommended retail price so the two things i need are the stores profit and the recommended retail price well i've got the rrp i've got the pre retail price i need to find the cup that need to find the profit now that may involve finding the cost as we go what i can do with this is to say that my my profit well that's going to be my sale price minus my cost how much money i've got coming in versus how much money i paid for the item to begin with profit we know that's 3 over 2c the sale price i already know that 90 i don't know what the cost is so now i've got an equation just with one variable and i can use this to go and find c so adding c to both sides will give me 5 over 2 c equals 90 and going to calculate this value will give me that c is equal to 36. now i know my profit here was 3 over 2 times c so 3 over 2 times 36 will give me a profit of 54. and now i've got the two values i need i've got the profit i've got the recommended retail price and i want to know what was the store's profit as a percentage of the recommended retail price or in other words i can just say that the value i'm looking for is 54 divided by 150 multiplied by a hundred the profit divided by the recommended retail price times 100 will give me the percentage of the recommended retail price represented by the profit now it's just a case of simplifying this expression down well a hundred over a hundred and fifty as i'm saying this thing's two-thirds divide 54 by 3 i'm going to get 18 times it by 2 will give me a final value of 36 and so my answer here is c so as i said with these questions this is all about being very very careful with how you define the variables and you see i've taken a fair bit of time to write all this out that might feel like it's going to take a long time in the exam but it'll mean that you don't ever get confused between well what did i do with the sale value which is the recommended retail price what value goes where because as you can see by the time we get down into the algebra this is not difficult but the challenge is avoiding getting muddled as you go through and if you take the time to be very clear about what you're doing then this question is not too bad all right i will get the next question up on the screen and we'll see you back here in two minutes okay i hope you've had enough time to take a look at this one if you need some more time now's the time to pause before i take this one away okay this is another question where there's a lot of work to do and we need to be very systematic in how we go about it otherwise we're going to get in a huge model now i've said in previous questions that i would avoid picking a number to start with and using that to judge percentage changes as we go through this is a prime example of where that will get you in serious trouble for example let's say you pick a hundred for your start point okay then the value of this goes up by 20 in the first year that's not too bad we know we've got 120 dollars then it's going to go down by 25 and it's going to go up 16 and 2 3 and it's going to down 11 and 1 90 you're going to end up dealing with something like 74.8 dollars and then trying to raise that by um 33 and a third percent right the numbers there are going to get horrific very quickly what i would suggest you do with a question like this is we're going to see i'm going to stick to algebra as you can probably expect from what we've done so far i'm going to say that my starting value is just some some unknown x and then we're going to use the multipliers that we have done before and again just be very careful with the algebra as we go through so in year one i'm told the value of the asset increased by 20 so if as we've done before if i increase by 20 then that's the same as having 120 of the original so what i'm doing here is 120 of the original which simplifies down to 6 over 5 times x now the question's asking on which of the following dates was the asset closest to its original value so i can see what's the difference between its value at the end of year one to the original value there and the difference i'll have a little column over on this far side the difference here is a fifth of x again i don't know what x is but i can use this column to compare the different years and see how close they were to that original value so i know at the end of year one the value of the asset was six over five times its original one and then in the second year it's going to decrease by 25 percent so again turning this from a decrease into a percentage of something if i decrease the value of something by 25 i have 75 of its original so year two i would have 75 over a hundred times by six fifths of x what i finished with at the end of year one now that simplifies to three over four times 6 over 5 times x which will end up being 18 over 20 times x or 9 over 10x so the difference now between the original value and my year 2 value is only one tenth of x they are one-tenth apart and so i can see year two is closer to the original value than year one because year one i'm one fifth of x away year two and one tenth of x away i'm closer in year two so i know that a is not going to be the answer and this is the sort of process we're gonna go through for the rest of these but i'm gonna speed things up a bit so you can see where this ends up in the third year the value's going to increase by 16 and two now if you remember back to when we had that list of percentages in question two sixteen and two-thirds is the same as a sixth i'm going to increase the value of this by a sixth now if i increase it by one-sixth that means i have seven-sixths of the original so in year 3 i'm going to have 7 6 times 9 10 of my original now the numbers are going to get pretty nasty here unless i start cancelling some things well i can cancel this a 3 on the 9 and the 6 and that will mean that this will end up being 21 over 20 times x and so the difference between this value and the original is going to be 1 20th of x now i just need to do the same thing for the fourth and the fifth year in the fourth year it's going to decrease by 11 and 1 9 percent or i'm going to decrease this value by a ninth which means i have in year four eight ninths of this 21 over 20 x now again there's things that cancel here i could divide by 3 top and bottom there and i could divide by 4 top and bottom there which would leave me with 14 over 15x and so the difference between year 4 and the original is 1 15 of x and i've got one more to go in the final year i'm going to increase it by a third increasing something by a third means i have four thirds of my original now in this case there's nothing that cancels here so i'm just going to have to multiply and say that's 56 over 45. x which means the difference here is going to be 11 over 45 x which which is much bigger like the difference here if i multiply that up that's 3 over 45 x so year 4 is definitely closer than your 5. so e is definitely not my answer as we said year 2 is closer than you one so a is not my answer but one twentieth is closer than your two and one twentieth is closer than your four so in this one the correct answer there is c now as you can see that yeah there is a lot of work in this question there's no way around it you're going to have to do some fraction multiplication here but this way of having an unknown as your starting point and then treating everything as fractions just using constant multipliers you're setting up a multiplier canceling wherever you can making the fractions as simple as possible this will give you a way of looking at this problem that is much simpler than saying okay if my start point is a hundred what do the values become as we go through so as we've used a lot of times up till now we're constantly trying to use fractions in these questions trying to cancel wherever possible and make the fractions as simple as possible because when we get into the next question that begins looking at compound interest that's where this is going to become very helpful all right so i'm going to get the next question up on the screen i'll see you back here in a couple of minutes time okay now's your chance to pause it if you need a little more time with this question i'm going to give you another second before i take it away and we take a look at this one together all right so we're going to look at interest rates now um which obviously comes under the banner percentages here um first thing i want to take a look at is what the difference is between a simple interest rate and a and a compound rate so we use this question in order to drive this i'm actually going to look at pavel's investment first and then we'll go and look at vladimir's compound interest so pavel invest four thousand dollars at a five percent sim annual sorry simple annual interest rate now simple interest just means that we find in this case it'll be five percent we find five percent of four thousand and that's what pavel will earn every year we'll just add that on and add that on and add that on and add that on each year and the amount we add on won't change so five percent of four thousand which is what we're looking to find so five percent of four thousand as we've done all along five we can convert that into algebra so five percent five over a hundred of means multiply of four thousand well i can just begin cancelling that that down and the 5 times the 40 that we're left with there will mean that it'll be 200. so the amount of interest pavel will learn will be hundred dollars every year and that will be the same every year so you'll start with four thousand dollars at the end of year one he'll have 4 200 at the end of year two if we were to keep going i know this question is only about one year but if we were to keep going pablo would have four thousand four hundred after year two four thousand six hundred after year three four thousand eight hundred after year four and it would just keep going he would get two hundred dollars every year no change so in terms of this question we are looking for how much does a year and after one year well we can say that pavel after one you would have his four thousand dollars original plus the 200 here and in interest so pavel sitting at 4 200. well let's to tuck that away and we'll use that later on now we go looking at compound interest now this is one where the amount of interest you would earn will change with time because we take that interest and put it back on top of the principle and then take the interest rate off that now the thing to look out for is this when is it compounded how many compounding periods do we have so in this case we're looking semi-annually but we're only looking for one year so we will have two compounding periods now there is a formula we there is a huge great compound interest formula that we get with this one where we could say the future value is the present value multiplied by 1 plus the interest rate over 100 times the number of compounding periods to the power of the number of compounding periods times the number of years as we've said before yes there are formulas for these things but do you want to have to remember that formula perfectly when you're on question 30 of the exam and the clock's ticking away particularly in this one where you're you're going to end up having to do 1.025 squared and that sounds horrendous i certainly don't want to have to do that level of horrible arithmetic as the clock's ticking down and if we can get to the stage where we understand the process behind what's going on in these questions yeah you might write a few more lines of algebra but the likelihood of you might making a mistake diminishes rapidly and so the way that we can work with this one is that yeah i can say we're dealing with four thousand at five percent interest rate here so i could say that well as we've seen ups up above five percent of four thousand is two hundred dollars now we could say that vlad if he invested four thousand dollars at five percent interest for one year compounded annual so no compounding periods just that one period five thousand sorry four thousand dollars at five percent for the one year he would earn two hundred dollars but we're going to compound this semi-annually so we're going to compound it after six months so that would be half of the full year so rather than earning two dollars in sorry 200 in half a year vlad's going to earn 100 now then we're going to take that amount 100 and add that to the 4 000. so in the second compounding period in the second half of the year he's actually going to be dealing with 4100 and it's as if we invested that at five percent now again if that was invested for one year then he would earn 205 dollars but we only have the second half of the year to deal with so he's only got that invested for half a year so he's only going to earn half of that 205 or he's going to get 102.50 now i can take that 102.50 and add that on to the 4100 we started with so the total that vlad has at the end of the year is going to be 4 202 50. now this this process of saying well okay what would i earn if i invested it for one year how much of the portion of the year do i have in this case it's semi-annually so i'm using a half so we divide the 200 by two but if this was compounded quarterly i would divide the 200 by four add that on and i would have to go through this process four times to take into account each quarter of the year if it was compounded monthly i would do the 200 divided by 12 and i would have to go through this process 12 times to get through the entire year so it's yes it's going to take more time but it means that the algebra and the arithmetic are more simple i just say in if we were to use the formula in this question i'm doing 1.025 squared which is going to be hard and then i've got to multiply whatever that comes out by by my original 4 000 times 1.025 again the numbers are going to get horrible very quickly whereas when you're working down this way yes it's going to take longer yes you're going to do more writing but you're going to be able to keep track of where you are in the question the numbers involved are a bit more simple and you're much less likely to make a mistake to come back to this question we know that vlad's total was four thousand two hundred and two fifty and i want to know how much more does vlad earn than pavel after one year so the difference between them is going to be the four thousand two hundred and two fifty that vlad earned minus the 4200 that pavel learned which means that the difference between them is 250 and so my answer to this question is b right so yeah i hope that made sense the difference between simple interest and compound interest simple interest you just get one value at the beginning you just add that on each year or each comp however often you're compounding the interest compound interest is a bit more complicated and yes there's a formula but if you understand the process behind it and what's going on these questions are much less complicated the less intricate the algebra the arithmetic is less nasty and you can keep track of going what's going on rather than just blindly following a formula and hoping that it works out and we're going to take that compound interest idea on and play with it a little more in the next couple of questions so i will get rid of this one and i'll see you back here in a couple of minutes time okay this is a very heavy algebra question so if you need a little more time to crack this one feel free to pause the video now but i'm going to take it away so that we can take a closer look at this one all right so as i've said huge amounts of algebra here i hope the thing that you've noticed there is that we're given this value p in the question but then none of the answer choices have any p in it now the way that it's presented means that you're likely to use this p value throughout your solution but at some point we're going to have to eliminate it and that's the clue we're going to use as we go through this question now what the the scenario we've got is a car is losing value over time so i'm going to build up this solution by looking at how things change across time so i'll start on january the 1st 2019 we're told that the value of this car is a dollars and this changes it loses a fixed percentage p of its value every year now the process that we're going to follow here to get to the next year is exactly the same sort of thing that we've been doing throughout this video but up until now i've known what the percentage change i'm applying is for example if i if the car was going to lose 25 of its value then i would say well 25 decrease that's the same as 100 minus 25 so i've got 75 of the original value turn that into 75 over 100 times by a that's what i'd like to do but i can't do that here because i don't know what p is but i'm going to follow through the same sort of process so on january the 1st 2020 i've got my original value a and i would like to multiply that by the percentage of what new percentage of it if yes as i said if i decrease by 25 i have 75 of the original in this case i'm going to decrease by p percent but i don't know what p is so i can't take that off the 100 but i can represent it in the same way one that's the same as a hundred percent 100 over 100 we could just write as one minus p percent now that would be the percentage of the original and as i said if you're decreasing by 25 that would be this would be 75 of the original but i don't know what p is so this is the best i can do but i know the value on january the 1st 2020 is going to be a times 1 minus p over 100. now if that's the price on january the 1st 2020 then the price on january the 1st 2021 would be this value that i've arrived at and then i'm going to reduce that by p percent again so i would multiply that again by 1 minus p over 100 now because i've got the same thing in both these parentheses i could say that's 1 minus p over 100 squared and we begin to build up a pattern here that as we go through time each year i would multiply the value i had by one minus p over a hundred so after one year i'd have a times one minus p over 100 to the power of one after two years i'd have a times one minus p over 100 to the power of two and so that can take me through to the third year and i'd have a times 1 minus p over 100 to the power of 3. but we're told what the car's value here in that year that's worth b dollars so that equals b now up until now i've just been building up this pattern and the question is how much will this card be worth on january the 1st 2023 so i'm actually going to change what i do here here and say well i'm going to take that b value and multiply that by 1 minus p over 100. now that gives me an expression to start with january the 1st 2023 the value of the car is b times 1 minus p over 100 but as we said at the start none of the answer choices have any p-values in them they've all got variations of a's and b's so i've got two equations and i'm going to have to use those to get rid of the 1 minus p over 100 in this final one and see what i end up with so if i take this value from january the 1st 2022 and say that a is 1 minus p over 100 sorry a times 1 minus p over 400 cubed is equal to b if i divide both sides by a and then take the cube root i've made 1 minus p over 100 the subject here and i can take this value and substitute it back into my value for january the 1st 2023 and that'll get rid of anything with a p in it so i've got b times the cube root of b over a now i can rewrite that cube root of b over a as b times b over a all to the power of a third which is b times b to the third all over a to the third or simplifying the exponents there b to the power of 4 over 3 over a to the power of a third which comparing it to the answer choices means that the answer here is d so we stepped up the difficulty level a little bit here this is about turning those um percentage changes that we dealt with before from numerical values into algebraic ones and still being comfortable with multiplying by the percentage the new percentage value that we're dealing with if you can get your head around this then you can deal with many multiple percentage change questions even involving lots of algebra all right i've got one more question for you again we're dealing with multiple percentage changes and interest rates and various things like that so i will get the final question up on the screen and i'll see you back here in a couple of minutes time okay now's the chance to pause the video if you need more time to take a look at this one and but i will take it away so that we can take a closer look all right so yeah lots and lots going on in this question um in the end we're going to be dealing with some compound interest again so we're going to dig deeper into how that works but first job here i'm going to make things more simple um all the numbers here so 9.6 million users and they gain hundred twenty 400 thousand four hundred seventy five thousand things are going to be a bit nicer if i divide everything by a thousand so we've got our users i'm gonna say we're starting with nine thousand six hundred and there are increases in the second third and fourth quarter total if we divide everything by a thousand it'll be 400 425 and 475. the numbers will still work out right now what we're aiming to do here is increase this number by 16 and two-thirds percent i know how much i get in the second third and fourth quarter and eventually when we get to the solution i'm going to want to know what was the annual rate of interest approximately what annual rate of interest would the company have needed in the first quarter to ensure they reached their goal so we're looking at that first quarter i know what happened in the second third and fourth i'm looking at that first quarter if i want to know the percentage rate of increase i would need above and beyond that 9600 i'm going to need to know what the absolute number of new customers i would need in order to work out the percentages so that's our first job now if we're starting with 9600 i want to increase that number by 16 and two-thirds well that means that our overall target our annual target is going to be nine thousand six hundred increased by sixteen and two thirty percent or in other words i'm increasing this by a sixth now a sixth of nine thousand six hundred that's one thousand six hundred so my target for the year is to increase by one thousand six hundred users now i know that in the second third and fourth quarter i get 400 425 and 475 users and if i subtract all those away from the 1600 that'll tell me how many users i need to gain in that first quarter so in this case if you do that subtraction we should end up with 300 users that's how many i need in the first quarter in order to make this target work and what i'm looking for is the annual rate of increase that will get me to that value if this figure is compounded every quarter because i'm only looking for the annual rate of interest the company have needed in the first quarter so this is a compound interest question we've got one year and we've got four compounding periods so this is we're going to work this in a very similar way to the vlad and pavel question except what we know and what we don't know is different so what we would do with this if we're going to use compound interest here i would say there is some unknown percentage increase in that first quarter and that's what we're going to go and find now the way we would work this we say x percent of the 9 600 that would give me some percentage value that would give me some number that would be the annual increase i would get if i was remaining at this percentage but the way this works with the compound interest is i'm only having that interest rate for the well i've got a compounding period in that first quarter of the year so i'm going to take that overall increase whatever it was and i would divide that by four because we're only having one quarter of a year so we would take whatever that value is and divide it by four and that would tell me how many new users i would gain in that first compounding period now the good bit is we know that we know that the number of users we're going to get in that first compounding period is 300. so whatever this value is so now let's just call it v there if we divide it by 4 we get 300. so this original one would be 300 times 4 we get 1200 so at the percentage rate the percentage rate we want if it was one compounding period over a year we would get 1200 new users from our 9600 and so what we need to do now is go and turn this line into an equation and go and find x now thankfully we're bringing all the skills we've done so far because i've written this in such a way that i can convert straight from the text into algebra as you can see here x percent of nine thousand six hundred equals twelve hundred so i could write that as x percent or x over a hundred of so multiply nine thousand six hundred equals twelve hundred and now as we've done before we can cancel wherever we can i'm going to divide by a hundred on both sides just by cancelling out those zeros and we could say that x over a 100 is 12 over 96 and now again it's a case of simplifying as much as you can here we can divide top and bottom by 12 and so that's one over eight now what i've got x over 100 is equal to 1 over 8 or x is 100 over 8. again now it's just a case of cancelling down this as much as we possibly can i could divide top and bottom by 4 so that's 25 over 2 which is 12.5 and that would be the interest rate the company would need the annual interest rate the company would need for that first quarter in order to meet their target which means that our answer to this question is d so yeah as you can see here again we've got a compound interest problem here we've got an annual interest rate but it only applies in the first quarter so we would have four compounding periods in this year yes there's a great big formula that you could use in order to work your way through this question the algebra is going to get quite nasty there are plenty of places to make a mistake but if you can understand how this compound interest process works as we said before it's going to take a little bit longer you're going to have to put more steps in but you're far less likely to make a mistake as you can see throughout this yes it's a tough problem but the algebra itself isn't that complex and so if you can get your head around the concepts here if you can understand exactly what we're aiming to do with this compound interest work then you can make your life easier you can minimize the chance of making an unnecessary mistake and it's that skill that will help you on the gmat missing a difficult question far more damaging to your score than missing a very difficult one and so if you've got the opportunity to minimize an algebra mistake arithmetic mistake i would take it right so yeah i hope this has helped you understand percentages a little better as we said at the top i think the key things you can take away here are from the text translate as literally as possible and make fractions your friend as much as possible stay away from choosing a starting value and using that to work through the problem as you go if you can stick to the algebra turn everything into these fractional multipliers simplify the fractions as you go you're far less likely to make a mistake all right that's all i've got for you today we'll hopefully see you back here again soon for the next in our gmat ninja quant series thanks very much for watching everybody we'll hopefully see you back here soon you

hello everybody and welcome to the next video in our gmat ninja con series my name is harry duffy and today we&#39;re going to be taking a look at the next video in our word problem section we&#39;re looking at percentages we&#39;re going to be starting right at the beginning looking at the basics of percentages how to use them how to convert them between fractions and decimals i&#39;m going to be working right through to the compound interest questions towards the end now before we get into this i&#39;m going to give you an idea about the difficulty rating in this video we probably put this somewhere in the middle so if you&#39;re in the 30s or the 40s and you&#39;ve struggled with percentage questions then we&#39;re probably gonna have a few things for you here now while the start the video starts out very foundational we&#39;re going to be looking at the basics of everything the second half the video the difficulty level is going to ramp up and there&#39;s going to be a couple of questions towards the end that&#39;ll really challenge you unless you&#39;re in the high 40s already just as a warning for those of you that are in the 48 49 50 region already and are looking for questions the very difficult questions are going to push you onto a 51 we probably don&#39;t have much for you today what we&#39;re looking for the people that will really benefit from this video are the ones where if you&#39;re looking at a percentages question you&#39;re just confused by the phrasing of it if you&#39;re not really sure where to begin then we&#39;re definitely going to have a lot for you and stick around certainly for the beginning of the video the next group if you&#39;ve yeah if you&#39;ve memorized all the formulas now if you know that compound interest formula but you&#39;re still missing a frustrating number of questions then we&#39;re going to be looking at the uh breaking those formulas now breaking the processes down and having a look at the why behind these questions and so yeah if you fit into this box again we&#39;re definitely going to have some stuff for you here and all of this can falls into these two bits if you&#39;re not if you don&#39;t really understand percentage questions but you&#39;re just relying on some tricks to get you through them then as i said before we&#39;re going to break some of these questions down tell you what to look for and how to look for it and we&#39;ll certainly have a few things to help you and this all comes under the category of if if you know everything but you&#39;re still struggling with percentage questions and you&#39;re not really sure why if you fall into any of these four categories then this video is definitely for you and please stick around we&#39;re gonna see what we can do to help you out now i&#39;ve got nine questions for you today and the way we&#39;ll do this is i will put a question up on the screen and give you a couple of minutes to take a look at it then i&#39;ll come back we&#39;ll take the question away and we&#39;ll work through it together on the board and i&#39;ll try and explain it in a way that helps you better understand what it is you should be looking for and what it is you should be doing in the question within those questions we&#39;re going to cover well we&#39;re going to cover as part of them you know translating word problems into math speak this is a feature of our word problems series that we&#39;re doing within the overall quant series that if you can be as literal as possible with some of these questions it makes it easier to translate the text into algebra and we&#39;ll have a look at what i mean by that later we&#39;ll get into just percentage changes single or multiple percentage changes finding a percentage of something or increasing or decreasing something by a percentage but this is the big thing for this video and this is one that i&#39;m going to be really pushing throughout the entire next hour hour and a quarter however long it is how to make fractions your friends if you&#39;re not comfortable with fractions then percentages are going to be very difficult so yeah if that sounds like you then i&#39;d take this as a bit of a warning that maybe in the next few days you&#39;re going to want to do a lot of work on fractions and that will definitely help you understand percentages better and then we&#39;ve got this bit looming at the end the compound interest formula it&#39;s not it&#39;s not the most important thing but it certainly makes a lot of people nervous and we&#39;re going to go dig into the details of that all right so without any further ado i will get the first question up on the screen i&#39;ll give you a couple of minutes to take a look at it and we&#39;ll see you back here soon so we can go through it together okay i hope you&#39;ve had enough time to take a look at this one if you feel you need a little more time if you&#39;ve not quite got your answer yet then feel free to pause the video now we&#39;ll still be here when you&#39;re done but for now i&#39;m going to take the question away so we can take a look at this one together now as we set up at the start of the video a lot of this the challenge in some of these questions is translating the text literally into algebra now we&#39;ve done a full video on this already and if you want to go take a look at our algebra translations videos after you&#39;ve watched this one then that&#39;ll give you many many more examples about putting this process into action but we&#39;ll take a look at how it&#39;s put in place in this question and we&#39;ll use that technique again and again and again as we go through the rest of this video now the big clue to get this question going is this final sentence the fox 8 what percentage of the hence that&#39;s the question and the challenge is to translate into that into algebra so i&#39;ve rewritten this up at the top left and we&#39;re going to go just go word by word converting this from text into algebra so the first section the fox 8 well from the question we were told the fox 816 hence so the fox is going to eat 16 of these hens now the next bit what percentage so oh i&#39;ve got an unknown percentage here that that&#39;s basically the question so let&#39;s just come up with some sort of letter and i&#39;m going to call that p percent and the percentage so p percent there now the word of when you&#39;re converting text into algebra the word of you could just substitute for a multiplication and then the hens we were told that there were 800 000 hens in this hen house and now we end up with yet the word 16p percent times 800 000. we say well 16 that is equal to p percent of 800 000 because that&#39;s what we&#39;re defining the question to be we want to know what percentage of the hence the fox 8 we know the total was 16 and so we can put an equal sign in between there 16 is equal to p percent of 800 000 and the question here is what is p and so now that you&#39;ve translated this literally we&#39;ve got the beginnings of our equation there we just need to tidy this up into a full algebraic expression so the 16 doesn&#39;t change here the only real change is this bit p percent whenever you see something like p percent x percent r percent ten percent five percent whatever it is then that percent means you&#39;re going to put something over a hundred and that&#39;s very important this is one of the one of the areas that lots of our students make those small unnecessary mistakes everything else they do is absolutely spot-on the algebra is fine the understanding of the concepts fine they just forget to put something over a hundred and that ruins the solution so whenever you see let&#39;s say p percent that is another way of saying p divided by 100 when you&#39;re converting that into algebra then we&#39;ve got our times 800 000 on the end and now i&#39;ve got an equation we can begin to solve this first step i&#39;m going to cancel out a couple of zeros here so we get 16 is 8 000 p or dividing both sides by 8 000 i&#39;m going to have p is 16 over 8 000. now as we&#39;ve said before in some of these videos at this stage you don&#39;t need to be able to do 16 divided by 8 000 straight off the bat you could just begin to cancel down top and bottom what&#39;s a number that divides both 16 and 8 000 well yeah there&#39;s a variety of them i&#39;m actually going to go with eight so 16 divided by eight will give me two eight thousand divided by eight will give me a thousand i could have done it i could have divided top and bottom by two by four by eight by sixty it&#39;s just a case of don&#39;t think you have to get to an answer in one step just start cancelling work your way through be calm be measured but um don&#39;t take your don&#39;t waste time and you&#39;ll get to an answer in the end it will probably be much more accurate than you just trying to jump there in one step so now 2 divided by a thousand i get i could divide top and bottom by 2 here but just looking over i&#39;ve got all the answers in decimals so why don&#39;t i take that 2 and divide it by a thousand and i&#39;ll get p is 0.002 so now i&#39;ve got a value for p and it&#39;s the final question is do i need to do anything more with it another place in a question like this that some of our students fall apart is that well they say that p i found p but to make a percentage i&#39;ve got to times it by a hundred i&#39;m going to say no what we did at the beginning was define this as p percent so whatever p is that is the percentage i&#39;m looking for we already put it over a hundred we&#39;ve already baked in the percent part into this problem and so i can just take this number and go and find the answer choice that matches it in this case d and that&#39;s going to be my answer all right so the challenge in this one was converting the text straight from the great big wall of writing that the question gave you into this algebraic expression that we could go and solve now as you can see once you get the equation up there the algebra is not that hard and in a lot of these word problem cases that that is the challenge 70 75 something like that of the question is done by the time you get the algebra on written down if that process is still a challenge if you didn&#39;t really follow what i was doing there then as i say we&#39;ve got the algebra translation video that&#39;ll include another eight or nine examples of the putting this process into action and at the end of watching this one maybe go and find that one and as i say there&#39;ll be more examples for you there in the meantime we&#39;re going to keep going with the percentages topic and i will get the next question up on the screen for you i&#39;ll see you back here in a couple of minutes time okay now&#39;s the time to pause the video if you need a little more time to look at this one for now we&#39;re going to take it away so that we can take a closer look at this question together so of the two things we said at the start that we&#39;re going to make a big deal of in this video the algebra translations and making fractions your friend we&#39;re going to really use fractions in this question again they&#39;re going to be used throughout this video so now is a good time to sort of pause and say well okay which are the the big conversions you should be able to do and yeah what what are the what are the percentages and how do they link to fractions now there&#39;s very few things that we really really ask our students to memorize when we&#39;re getting ready for the gmat we try and stay away from formulas because when it&#39;s question 30 and the clock&#39;s ticking away the pressure on you it&#39;s likely to make you misremember the formulas in some respects but there are a string of percentages that are very helpful to know and if you can convert between percentages and fractions your life will be so much simpler so we&#39;ve got some of the easy ones like 50 30 that&#39;s just a half most people know that in the same way 33 and a third as a percentage that&#39;s a third if you convert that to a fraction 25 is a quarter 20 percent is a fifth those are the ones that most people are very comfortable with the next three are ones that fewer people are aware of sixteen and two-thirds percent as a fraction that&#39;s a sixth twelve and a half percent that&#39;s an eighth and eleven and one-ninth that could be a ninth and then we&#39;re back to the the simple ones that most people know ten percent is a tenth and five percent is a twentieth now if if you&#39;re aware of these fractions and you can very quickly convert between them then your life will be made easier in percentages but we can go a little bit further these are the bases and in particular for this question i want to take a look at 11 and a ninth as we said that&#39;s one nine percent but you can begin to get multiples of this so 22 and two-ninths as a percentage that&#39;s two over 9 33 and 3 9 well that&#39;s 3 over 9 which is a third which is what we&#39;ve got here 33 and a third is the same as 33 and 3 9. but the big important one for us right now 44 and 4 9 as a percentage that&#39;s going to be 4 over 9. and we&#39;re going to make use of that as we get into this question so just as an aside these percentages and the conversions between percentage and fractions very very useful for dealing with percentage questions and if you don&#39;t know these i suggest you spend a little bit of time memorizing them and you&#39;ll you will see as we go through the rest of this video how helpful they can be now as i said we&#39;re going to use that in this question i come to the question now eventually i want to know what was the percentage change in internet users from 2000 to 2003 but for now i&#39;m going to ignore that part of the question just look at this bit at the top so in 2000 the number of people using the internet was 44 and 4 9 so 4 over 9 if we convert that into a fraction of the number in 2003. now i don&#39;t have any numbers to put here and here one of the themes of this video is going to be please stay away from just launching numbers in and hoping they&#39;ll work out i don&#39;t want to say there were 100 internet users in 2003 as you&#39;re going to see as we get later on that way of thinking is going to get you in a lot of trouble i&#39;m just going to put some sort of unknown there i don&#39;t know how many there were in in 2003 but i know that the number in 2000 was 44 and 4 9 of that and our conversion says that that&#39;s four ninths of that number so four and forty four and four ninety percent of the number would be four ninths of the number or four ninths of x so the number in two thousand is four ninths of the number in two thousand and three now i want to reverse that i want to know what the percentage change in internet uses from 2000 to 2003 was so what i&#39;m going to do is i&#39;m going to say that that x there i want to make that the subject of this transformation from 2000 to 2003. so if i have the 4 9 multiplied by the x in order to make this x the subject i&#39;m going to times both of these by 9 so 4x to 9x there and then divide both of them by 4. so x to 9 over 4 x or other words to go in this direction i times the x by 4 over 9 to go in that direction i times it by 9 over 4. so in 2003 we actually had nine fourths of the number we had in 2000. now that that&#39;s not us done because i don&#39;t want to know what the what percentage of internet users i want to know what the percentage change was so we need to think about this in a slightly different way this nine over four x well i could think about that as the x plus five over four x this is the original this is the change and so what can i make 5 over 4 into a percentage well as we said 1 over 4 was 25 4 over 4 is going to be 100 because that&#39;s full so 5 over 4 is going to be 125 of x so the percentage change is going to be 125 percent which is d so there&#39;s a few places you can slip up in this question as i said once you get it into fractions it&#39;s much nicer this way of thinking about the conversion back and forth steers you away from these two answers at the beginning yeah if this is four ninths you don&#39;t just add 5 9 to get back to the original this is more about multipliers i want you to always be thinking about multiplying from one way to the other when we&#39;re dealing with percentages rather than just saying i&#39;m going to add five nights on that i had before that&#39;s not going to work in this scenario and then this bit yeah separating that nine over four into the two values because it said percentage change and be very careful with the wording in these questions they&#39;re not trying to trick you but if you&#39;re not paying attention you can just slip and go down the wrong path all right so as i said the big thing about this one is translating literally and using fractions and we&#39;re going to get that into more of the questions as we go through i&#39;ll get the next question up on the screen and i&#39;ll see you back here in a couple of minutes time okay i hope you&#39;ve had a little more time to look at this one if you need if you need any more time feel free to pause it now but i&#39;m going to take the question away so that we can look at it together um right so we&#39;ve got a few things going on in this one we&#39;re going to have to find two different percentages push them together um and then at the end figure out yeah figure out another percentage change what percentage higher at store c than at store b but before we get to that final bit we&#39;ve got to look at the difference between store a and store b and there&#39;s something in the wording of this question that i want to take a look at as it says here i&#39;ve rewritten bits the question the price at a is 25 higher than at b now just this freezing and begins to confuse some of our students here the difference between price at a is higher than at b and a is some percentage of b and this is one of those scenarios where i will take a question and always rewrite it so that i&#39;m doing the same process every time in this case the price of a is 25 higher than the price at b well then i could say the price at a is 125 of the price at b if it&#39;s 25 higher then we get the price at b plus 25 of that so the original price is 100 plus the 25 so it&#39;s 125 of the price at b now reason i do that is is mainly just so that i&#39;m doing the same thing every time and because that word of it just means multiply so the price at a is equal to 125 so 125 over 100 of the price at b or i could simplify that a is five-fourths of b and i&#39;ve got myself a very simple equation that i can work with there and it&#39;s all in the wording of the the text so i can translate directly from the price at a is 125 percent the price of b it&#39;s much harder to translate the price at a is 12.5 percent higher than at c directly into algebra yeah which is why i&#39;m rewriting these so for the second one if the price at a is 12.5 percent higher than at c then we get the original hundred percent of c&#39;s price and we add on 12.5 so the price at a is a hundred and twelve point five percent of the price at c and again i can convert that more directly into algebra so a is 112.5 over 100 times c now this one might be a little more difficult to convert straight in or to to simplify down yeah if you need a bit more time to do the canceling go for it but if you can see that 12.5 or if you can remember from the last question 12.5 percent was an eighth in this case we&#39;ve got and the whole of c 8 8 plus one more so this is a is 9 8 of c and so by taking the question rewriting this bit about some price being higher than another bit as to being some prices this percent of the other bit you can much more quickly convert into algebra and that&#39;ll give us a couple of nice equations that we can go and use so we know that a is 9 8 of c and we know that a is five fourths of b and i want some relationship so the question now the price of the shirt is what percentage higher at store c than at store b so i&#39;m going to want to get c and b into a single equation with no a in there and then i want to get c equals something and that&#39;ll give me something to work with so i&#39;ve got two equations here i&#39;ve got a equals something an a equals something so i can push these two bits together and say 9 8 of c is equal to 5 4 of b now as i said i want to make c the subject of this so if i multiply both sides by 8 i&#39;m going to get eventually going to get 10 b and then divide by 9 c is 10 over 9 b area 10 over 9 if i was to convert that into a percentage that&#39;s going to be more than 100 which is none of my answers thankfully this question is about what&#39;s percentage higher so i could rewrite that 10 over 9b as the original plus 1 over 9b it is this much higher than the original that&#39;s the bit i&#39;m looking for now yeah we&#39;ve in the last question we said that 1 over 9 was equal to 11 and 1 9 percent so we could say that c is equal to b plus 11 and one-ninth percent of b and so c is 11 and one-ninth percent higher than the price at b which means that the correct answer here is c so what yeah one of the things we we do when we&#39;re dealing with word questions like this and this will come it&#39;ll definitely come back in number properties this is the other obvious place of doing it is to rewrite the question so the translation from the text into algebra is more literal and if you can get the hang of saying what yeah a is 25 higher than b that means a is 125 of b or the reverse if a is 25 lower than b then a is 75 of b in that literal translation just because the word off means multiply you can create these equations with the fractions that work much more simply than those fiddly ones with lots of percentage signs and it&#39;s just an easier way of dealing with them you want to be doing the same thing every time so that when the time pressure is on towards the end of the con section you don&#39;t get muddled up between two different processes that&#39;s why i always rewrite them this way and hopefully that can help you too all right i&#39;ll get the next question up on the screen and i&#39;ll see you back here in a couple of minutes time okay now&#39;s your chance to pause the video if you need a bit more time to look at this one i&#39;m going to take it away so we can take a closer look okay the the big things in this question are bringing some of the things we&#39;ve looked at in the last two problems together and so we&#39;re we&#39;re dealing with percentage changes we&#39;re dealing with the change in the tax and we&#39;re putting it all together into one equation here what we&#39;re looking at is the firm&#39;s revenue and we want to know what percentage of it was it this year or the year in question compared to the previous year before the changes before they charged more money and sold fewer hours so the first thing we need to do is we need to come up with some sort of equation for the firm&#39;s revenue because that&#39;s what we&#39;re looking at now in this question all we&#39;re told about is the number of hours that the firm sold and the cost per hour so our our revenue formula is going to be let&#39;s say n for the number of hours the firm sold multiplied by the cost now we could say that this is the revenue for the previous year again i could put numbers into this but if i go down that path and make that into a habit i&#39;m going to run into a lot of trouble what i want to do here is stick with algebra and i&#39;m going to say that was the previous year and i&#39;m going to make some changes and see what the revenue is like in the new year so i&#39;ll i&#39;ll still use the revenue but we&#39;ll say this is the revenue for the new year now during the sorry this year the firm raised its hourly rate by 40 so we could look at the the cost value the cost value in our new year is going to be 40 higher than it was the previous year or if you remember back to what we&#39;re doing the last question 40 higher is the same as 140 of the previous year so i could say that my new cost per hour of consulting is going to be 140 over 100 times by c i could do this very similar thing with the number of hours of consulting we&#39;re told here they sold 30 percent fewer hours 30 percent fewer that&#39;s the old value was 100 i take 30 off that my new value is going to be 70 of the previous value and so number of hours is going to be 70 over 100 times by n and so this whole expression i can simplify the 7 over 100 sorry 70 over 100 to be 7 over 10 i can simplify the 40 100 sorry 140 over 100 that&#39;ll actually cancel down to be seven over five and then i&#39;ve got the n and the c now we can do the multiplication here 7 times 7 will be 49 10 times 5 will be 50. and just because i&#39;m working with percentages here i could multiply the top and bottom by this fraction of this fraction by 2 and get 98 over 100 times n times c so this new revenue value that we&#39;ve been working out is 98 over 100 times by nc now the nc that&#39;s the old revenue value so the new the new revenue is 98 over a hundred times the old revenue value converting this into a percentage well we&#39;ve got 98 over 100 as we said way back in question two a percentage is just defined as something over a hundred so we can again literally rewrite this as the new revenue is 98 over 198 percent of the old revenue which means that our answer here is going to be b now full disclosure with this one this is one of those questions where you could have taken the old revenue and picked a number said it was a hundred fine yes it will work in this question but as you&#39;re about to see in the next few questions if you were to try and do that elsewhere you&#39;re going to get into all kinds of trouble and as we&#39;ve said before one of the things that we really push is to make sure that you have a technique that will work in every question so that you&#39;re never stood there at the start of a question just going is this one where i should use algebra as this one where i should pick 100 and see where it goes it&#39;s safer and you&#39;ll do less thinking you&#39;ll be less stressed if you just do the same thing every time so if i can convince you of one thing in percentage questions using this video is to please not throw numbers at it and just see where it goes stick to algebra stick to general arguments if you can and as you can see with this one it can actually work out very nicely right so on that theme i&#39;m going to get rid of this question we&#39;ll get the next one up on the screen and i&#39;ll see you back here very soon okay now&#39;s the chance to pause the video if you need a little more time to work on this one i&#39;m going to take this away now so that we can take a closer look so these these questions about products being bought at stores and they&#39;re given discounts or they&#39;re being increased and then they&#39;re being sold they&#39;re being sold at a markup or they&#39;re marked up then discounted that the actual algebra behind them is not usually not particularly difficult the problem with these questions is keeping track of the variables and what you&#39;re doing with them so i&#39;m going to be very careful with how i define things and the way we build this question up so that you can see the the simplicity in the question and the the places where you might go wrong so to begin with i&#39;m just going to work through the text here a store brought a product okay so the cost price to the store i&#39;m not giving a value for that i don&#39;t know what it is i&#39;ll just call it c to begin with we know that it has some recommended retail price of 150 right the next sentence tells us that the store sold this product at a 40 discount now you&#39;re gonna get sick of hearing me say this as we go through the video 40 discount means 40 off the original price so 100 take away the 40 gives me 60 of the recommended retail price my sale price is 60 of 150 turning that into algebra 60 over 100 times 150 and you can work that out that the sale price is going to be 90 now we&#39;re then told that the store made a profit a 150 profit on the wholesale cost or in other words the profit is 150 of the wholesale cost so my profit that&#39;s going to be 150 of c or just 3 over two of c as we&#39;re simplifying that down now what i want to get to as we go through this is what was the store&#39;s profit as a percentage of the recommended retail price so the two things i need are the stores profit and the recommended retail price well i&#39;ve got the rrp i&#39;ve got the pre retail price i need to find the cup that need to find the profit now that may involve finding the cost as we go what i can do with this is to say that my my profit well that&#39;s going to be my sale price minus my cost how much money i&#39;ve got coming in versus how much money i paid for the item to begin with profit we know that&#39;s 3 over 2c the sale price i already know that 90 i don&#39;t know what the cost is so now i&#39;ve got an equation just with one variable and i can use this to go and find c so adding c to both sides will give me 5 over 2 c equals 90 and going to calculate this value will give me that c is equal to 36. now i know my profit here was 3 over 2 times c so 3 over 2 times 36 will give me a profit of 54. and now i&#39;ve got the two values i need i&#39;ve got the profit i&#39;ve got the recommended retail price and i want to know what was the store&#39;s profit as a percentage of the recommended retail price or in other words i can just say that the value i&#39;m looking for is 54 divided by 150 multiplied by a hundred the profit divided by the recommended retail price times 100 will give me the percentage of the recommended retail price represented by the profit now it&#39;s just a case of simplifying this expression down well a hundred over a hundred and fifty as i&#39;m saying this thing&#39;s two-thirds divide 54 by 3 i&#39;m going to get 18 times it by 2 will give me a final value of 36 and so my answer here is c so as i said with these questions this is all about being very very careful with how you define the variables and you see i&#39;ve taken a fair bit of time to write all this out that might feel like it&#39;s going to take a long time in the exam but it&#39;ll mean that you don&#39;t ever get confused between well what did i do with the sale value which is the recommended retail price what value goes where because as you can see by the time we get down into the algebra this is not difficult but the challenge is avoiding getting muddled as you go through and if you take the time to be very clear about what you&#39;re doing then this question is not too bad all right i will get the next question up on the screen and we&#39;ll see you back here in two minutes okay i hope you&#39;ve had enough time to take a look at this one if you need some more time now&#39;s the time to pause before i take this one away okay this is another question where there&#39;s a lot of work to do and we need to be very systematic in how we go about it otherwise we&#39;re going to get in a huge model now i&#39;ve said in previous questions that i would avoid picking a number to start with and using that to judge percentage changes as we go through this is a prime example of where that will get you in serious trouble for example let&#39;s say you pick a hundred for your start point okay then the value of this goes up by 20 in the first year that&#39;s not too bad we know we&#39;ve got 120 dollars then it&#39;s going to go down by 25 and it&#39;s going to go up 16 and 2 3 and it&#39;s going to down 11 and 1 90 you&#39;re going to end up dealing with something like 74.8 dollars and then trying to raise that by um 33 and a third percent right the numbers there are going to get horrific very quickly what i would suggest you do with a question like this is we&#39;re going to see i&#39;m going to stick to algebra as you can probably expect from what we&#39;ve done so far i&#39;m going to say that my starting value is just some some unknown x and then we&#39;re going to use the multipliers that we have done before and again just be very careful with the algebra as we go through so in year one i&#39;m told the value of the asset increased by 20 so if as we&#39;ve done before if i increase by 20 then that&#39;s the same as having 120 of the original so what i&#39;m doing here is 120 of the original which simplifies down to 6 over 5 times x now the question&#39;s asking on which of the following dates was the asset closest to its original value so i can see what&#39;s the difference between its value at the end of year one to the original value there and the difference i&#39;ll have a little column over on this far side the difference here is a fifth of x again i don&#39;t know what x is but i can use this column to compare the different years and see how close they were to that original value so i know at the end of year one the value of the asset was six over five times its original one and then in the second year it&#39;s going to decrease by 25 percent so again turning this from a decrease into a percentage of something if i decrease the value of something by 25 i have 75 of its original so year two i would have 75 over a hundred times by six fifths of x what i finished with at the end of year one now that simplifies to three over four times 6 over 5 times x which will end up being 18 over 20 times x or 9 over 10x so the difference now between the original value and my year 2 value is only one tenth of x they are one-tenth apart and so i can see year two is closer to the original value than year one because year one i&#39;m one fifth of x away year two and one tenth of x away i&#39;m closer in year two so i know that a is not going to be the answer and this is the sort of process we&#39;re gonna go through for the rest of these but i&#39;m gonna speed things up a bit so you can see where this ends up in the third year the value&#39;s going to increase by 16 and two now if you remember back to when we had that list of percentages in question two sixteen and two-thirds is the same as a sixth i&#39;m going to increase the value of this by a sixth now if i increase it by one-sixth that means i have seven-sixths of the original so in year 3 i&#39;m going to have 7 6 times 9 10 of my original now the numbers are going to get pretty nasty here unless i start cancelling some things well i can cancel this a 3 on the 9 and the 6 and that will mean that this will end up being 21 over 20 times x and so the difference between this value and the original is going to be 1 20th of x now i just need to do the same thing for the fourth and the fifth year in the fourth year it&#39;s going to decrease by 11 and 1 9 percent or i&#39;m going to decrease this value by a ninth which means i have in year four eight ninths of this 21 over 20 x now again there&#39;s things that cancel here i could divide by 3 top and bottom there and i could divide by 4 top and bottom there which would leave me with 14 over 15x and so the difference between year 4 and the original is 1 15 of x and i&#39;ve got one more to go in the final year i&#39;m going to increase it by a third increasing something by a third means i have four thirds of my original now in this case there&#39;s nothing that cancels here so i&#39;m just going to have to multiply and say that&#39;s 56 over 45. x which means the difference here is going to be 11 over 45 x which which is much bigger like the difference here if i multiply that up that&#39;s 3 over 45 x so year 4 is definitely closer than your 5. so e is definitely not my answer as we said year 2 is closer than you one so a is not my answer but one twentieth is closer than your two and one twentieth is closer than your four so in this one the correct answer there is c now as you can see that yeah there is a lot of work in this question there&#39;s no way around it you&#39;re going to have to do some fraction multiplication here but this way of having an unknown as your starting point and then treating everything as fractions just using constant multipliers you&#39;re setting up a multiplier canceling wherever you can making the fractions as simple as possible this will give you a way of looking at this problem that is much simpler than saying okay if my start point is a hundred what do the values become as we go through so as we&#39;ve used a lot of times up till now we&#39;re constantly trying to use fractions in these questions trying to cancel wherever possible and make the fractions as simple as possible because when we get into the next question that begins looking at compound interest that&#39;s where this is going to become very helpful all right so i&#39;m going to get the next question up on the screen i&#39;ll see you back here in a couple of minutes time okay now&#39;s your chance to pause it if you need a little more time with this question i&#39;m going to give you another second before i take it away and we take a look at this one together all right so we&#39;re going to look at interest rates now um which obviously comes under the banner percentages here um first thing i want to take a look at is what the difference is between a simple interest rate and a and a compound rate so we use this question in order to drive this i&#39;m actually going to look at pavel&#39;s investment first and then we&#39;ll go and look at vladimir&#39;s compound interest so pavel invest four thousand dollars at a five percent sim annual sorry simple annual interest rate now simple interest just means that we find in this case it&#39;ll be five percent we find five percent of four thousand and that&#39;s what pavel will earn every year we&#39;ll just add that on and add that on and add that on and add that on each year and the amount we add on won&#39;t change so five percent of four thousand which is what we&#39;re looking to find so five percent of four thousand as we&#39;ve done all along five we can convert that into algebra so five percent five over a hundred of means multiply of four thousand well i can just begin cancelling that that down and the 5 times the 40 that we&#39;re left with there will mean that it&#39;ll be 200. so the amount of interest pavel will learn will be hundred dollars every year and that will be the same every year so you&#39;ll start with four thousand dollars at the end of year one he&#39;ll have 4 200 at the end of year two if we were to keep going i know this question is only about one year but if we were to keep going pablo would have four thousand four hundred after year two four thousand six hundred after year three four thousand eight hundred after year four and it would just keep going he would get two hundred dollars every year no change so in terms of this question we are looking for how much does a year and after one year well we can say that pavel after one you would have his four thousand dollars original plus the 200 here and in interest so pavel sitting at 4 200. well let&#39;s to tuck that away and we&#39;ll use that later on now we go looking at compound interest now this is one where the amount of interest you would earn will change with time because we take that interest and put it back on top of the principle and then take the interest rate off that now the thing to look out for is this when is it compounded how many compounding periods do we have so in this case we&#39;re looking semi-annually but we&#39;re only looking for one year so we will have two compounding periods now there is a formula we there is a huge great compound interest formula that we get with this one where we could say the future value is the present value multiplied by 1 plus the interest rate over 100 times the number of compounding periods to the power of the number of compounding periods times the number of years as we&#39;ve said before yes there are formulas for these things but do you want to have to remember that formula perfectly when you&#39;re on question 30 of the exam and the clock&#39;s ticking away particularly in this one where you&#39;re you&#39;re going to end up having to do 1.025 squared and that sounds horrendous i certainly don&#39;t want to have to do that level of horrible arithmetic as the clock&#39;s ticking down and if we can get to the stage where we understand the process behind what&#39;s going on in these questions yeah you might write a few more lines of algebra but the likelihood of you might making a mistake diminishes rapidly and so the way that we can work with this one is that yeah i can say we&#39;re dealing with four thousand at five percent interest rate here so i could say that well as we&#39;ve seen ups up above five percent of four thousand is two hundred dollars now we could say that vlad if he invested four thousand dollars at five percent interest for one year compounded annual so no compounding periods just that one period five thousand sorry four thousand dollars at five percent for the one year he would earn two hundred dollars but we&#39;re going to compound this semi-annually so we&#39;re going to compound it after six months so that would be half of the full year so rather than earning two dollars in sorry 200 in half a year vlad&#39;s going to earn 100 now then we&#39;re going to take that amount 100 and add that to the 4 000. so in the second compounding period in the second half of the year he&#39;s actually going to be dealing with 4100 and it&#39;s as if we invested that at five percent now again if that was invested for one year then he would earn 205 dollars but we only have the second half of the year to deal with so he&#39;s only got that invested for half a year so he&#39;s only going to earn half of that 205 or he&#39;s going to get 102.50 now i can take that 102.50 and add that on to the 4100 we started with so the total that vlad has at the end of the year is going to be 4 202 50. now this this process of saying well okay what would i earn if i invested it for one year how much of the portion of the year do i have in this case it&#39;s semi-annually so i&#39;m using a half so we divide the 200 by two but if this was compounded quarterly i would divide the 200 by four add that on and i would have to go through this process four times to take into account each quarter of the year if it was compounded monthly i would do the 200 divided by 12 and i would have to go through this process 12 times to get through the entire year so it&#39;s yes it&#39;s going to take more time but it means that the algebra and the arithmetic are more simple i just say in if we were to use the formula in this question i&#39;m doing 1.025 squared which is going to be hard and then i&#39;ve got to multiply whatever that comes out by by my original 4 000 times 1.025 again the numbers are going to get horrible very quickly whereas when you&#39;re working down this way yes it&#39;s going to take longer yes you&#39;re going to do more writing but you&#39;re going to be able to keep track of where you are in the question the numbers involved are a bit more simple and you&#39;re much less likely to make a mistake to come back to this question we know that vlad&#39;s total was four thousand two hundred and two fifty and i want to know how much more does vlad earn than pavel after one year so the difference between them is going to be the four thousand two hundred and two fifty that vlad earned minus the 4200 that pavel learned which means that the difference between them is 250 and so my answer to this question is b right so yeah i hope that made sense the difference between simple interest and compound interest simple interest you just get one value at the beginning you just add that on each year or each comp however often you&#39;re compounding the interest compound interest is a bit more complicated and yes there&#39;s a formula but if you understand the process behind it and what&#39;s going on these questions are much less complicated the less intricate the algebra the arithmetic is less nasty and you can keep track of going what&#39;s going on rather than just blindly following a formula and hoping that it works out and we&#39;re going to take that compound interest idea on and play with it a little more in the next couple of questions so i will get rid of this one and i&#39;ll see you back here in a couple of minutes time okay this is a very heavy algebra question so if you need a little more time to crack this one feel free to pause the video now but i&#39;m going to take it away so that we can take a closer look at this one all right so as i&#39;ve said huge amounts of algebra here i hope the thing that you&#39;ve noticed there is that we&#39;re given this value p in the question but then none of the answer choices have any p in it now the way that it&#39;s presented means that you&#39;re likely to use this p value throughout your solution but at some point we&#39;re going to have to eliminate it and that&#39;s the clue we&#39;re going to use as we go through this question now what the the scenario we&#39;ve got is a car is losing value over time so i&#39;m going to build up this solution by looking at how things change across time so i&#39;ll start on january the 1st 2019 we&#39;re told that the value of this car is a dollars and this changes it loses a fixed percentage p of its value every year now the process that we&#39;re going to follow here to get to the next year is exactly the same sort of thing that we&#39;ve been doing throughout this video but up until now i&#39;ve known what the percentage change i&#39;m applying is for example if i if the car was going to lose 25 of its value then i would say well 25 decrease that&#39;s the same as 100 minus 25 so i&#39;ve got 75 of the original value turn that into 75 over 100 times by a that&#39;s what i&#39;d like to do but i can&#39;t do that here because i don&#39;t know what p is but i&#39;m going to follow through the same sort of process so on january the 1st 2020 i&#39;ve got my original value a and i would like to multiply that by the percentage of what new percentage of it if yes as i said if i decrease by 25 i have 75 of the original in this case i&#39;m going to decrease by p percent but i don&#39;t know what p is so i can&#39;t take that off the 100 but i can represent it in the same way one that&#39;s the same as a hundred percent 100 over 100 we could just write as one minus p percent now that would be the percentage of the original and as i said if you&#39;re decreasing by 25 that would be this would be 75 of the original but i don&#39;t know what p is so this is the best i can do but i know the value on january the 1st 2020 is going to be a times 1 minus p over 100. now if that&#39;s the price on january the 1st 2020 then the price on january the 1st 2021 would be this value that i&#39;ve arrived at and then i&#39;m going to reduce that by p percent again so i would multiply that again by 1 minus p over 100 now because i&#39;ve got the same thing in both these parentheses i could say that&#39;s 1 minus p over 100 squared and we begin to build up a pattern here that as we go through time each year i would multiply the value i had by one minus p over a hundred so after one year i&#39;d have a times one minus p over 100 to the power of one after two years i&#39;d have a times one minus p over 100 to the power of two and so that can take me through to the third year and i&#39;d have a times 1 minus p over 100 to the power of 3. but we&#39;re told what the car&#39;s value here in that year that&#39;s worth b dollars so that equals b now up until now i&#39;ve just been building up this pattern and the question is how much will this card be worth on january the 1st 2023 so i&#39;m actually going to change what i do here here and say well i&#39;m going to take that b value and multiply that by 1 minus p over 100. now that gives me an expression to start with january the 1st 2023 the value of the car is b times 1 minus p over 100 but as we said at the start none of the answer choices have any p-values in them they&#39;ve all got variations of a&#39;s and b&#39;s so i&#39;ve got two equations and i&#39;m going to have to use those to get rid of the 1 minus p over 100 in this final one and see what i end up with so if i take this value from january the 1st 2022 and say that a is 1 minus p over 100 sorry a times 1 minus p over 400 cubed is equal to b if i divide both sides by a and then take the cube root i&#39;ve made 1 minus p over 100 the subject here and i can take this value and substitute it back into my value for january the 1st 2023 and that&#39;ll get rid of anything with a p in it so i&#39;ve got b times the cube root of b over a now i can rewrite that cube root of b over a as b times b over a all to the power of a third which is b times b to the third all over a to the third or simplifying the exponents there b to the power of 4 over 3 over a to the power of a third which comparing it to the answer choices means that the answer here is d so we stepped up the difficulty level a little bit here this is about turning those um percentage changes that we dealt with before from numerical values into algebraic ones and still being comfortable with multiplying by the percentage the new percentage value that we&#39;re dealing with if you can get your head around this then you can deal with many multiple percentage change questions even involving lots of algebra all right i&#39;ve got one more question for you again we&#39;re dealing with multiple percentage changes and interest rates and various things like that so i will get the final question up on the screen and i&#39;ll see you back here in a couple of minutes time okay now&#39;s the chance to pause the video if you need more time to take a look at this one and but i will take it away so that we can take a closer look all right so yeah lots and lots going on in this question um in the end we&#39;re going to be dealing with some compound interest again so we&#39;re going to dig deeper into how that works but first job here i&#39;m going to make things more simple um all the numbers here so 9.6 million users and they gain hundred twenty 400 thousand four hundred seventy five thousand things are going to be a bit nicer if i divide everything by a thousand so we&#39;ve got our users i&#39;m gonna say we&#39;re starting with nine thousand six hundred and there are increases in the second third and fourth quarter total if we divide everything by a thousand it&#39;ll be 400 425 and 475. the numbers will still work out right now what we&#39;re aiming to do here is increase this number by 16 and two-thirds percent i know how much i get in the second third and fourth quarter and eventually when we get to the solution i&#39;m going to want to know what was the annual rate of interest approximately what annual rate of interest would the company have needed in the first quarter to ensure they reached their goal so we&#39;re looking at that first quarter i know what happened in the second third and fourth i&#39;m looking at that first quarter if i want to know the percentage rate of increase i would need above and beyond that 9600 i&#39;m going to need to know what the absolute number of new customers i would need in order to work out the percentages so that&#39;s our first job now if we&#39;re starting with 9600 i want to increase that number by 16 and two-thirds well that means that our overall target our annual target is going to be nine thousand six hundred increased by sixteen and two thirty percent or in other words i&#39;m increasing this by a sixth now a sixth of nine thousand six hundred that&#39;s one thousand six hundred so my target for the year is to increase by one thousand six hundred users now i know that in the second third and fourth quarter i get 400 425 and 475 users and if i subtract all those away from the 1600 that&#39;ll tell me how many users i need to gain in that first quarter so in this case if you do that subtraction we should end up with 300 users that&#39;s how many i need in the first quarter in order to make this target work and what i&#39;m looking for is the annual rate of increase that will get me to that value if this figure is compounded every quarter because i&#39;m only looking for the annual rate of interest the company have needed in the first quarter so this is a compound interest question we&#39;ve got one year and we&#39;ve got four compounding periods so this is we&#39;re going to work this in a very similar way to the vlad and pavel question except what we know and what we don&#39;t know is different so what we would do with this if we&#39;re going to use compound interest here i would say there is some unknown percentage increase in that first quarter and that&#39;s what we&#39;re going to go and find now the way we would work this we say x percent of the 9 600 that would give me some percentage value that would give me some number that would be the annual increase i would get if i was remaining at this percentage but the way this works with the compound interest is i&#39;m only having that interest rate for the well i&#39;ve got a compounding period in that first quarter of the year so i&#39;m going to take that overall increase whatever it was and i would divide that by four because we&#39;re only having one quarter of a year so we would take whatever that value is and divide it by four and that would tell me how many new users i would gain in that first compounding period now the good bit is we know that we know that the number of users we&#39;re going to get in that first compounding period is 300. so whatever this value is so now let&#39;s just call it v there if we divide it by 4 we get 300. so this original one would be 300 times 4 we get 1200 so at the percentage rate the percentage rate we want if it was one compounding period over a year we would get 1200 new users from our 9600 and so what we need to do now is go and turn this line into an equation and go and find x now thankfully we&#39;re bringing all the skills we&#39;ve done so far because i&#39;ve written this in such a way that i can convert straight from the text into algebra as you can see here x percent of nine thousand six hundred equals twelve hundred so i could write that as x percent or x over a hundred of so multiply nine thousand six hundred equals twelve hundred and now as we&#39;ve done before we can cancel wherever we can i&#39;m going to divide by a hundred on both sides just by cancelling out those zeros and we could say that x over a 100 is 12 over 96 and now again it&#39;s a case of simplifying as much as you can here we can divide top and bottom by 12 and so that&#39;s one over eight now what i&#39;ve got x over 100 is equal to 1 over 8 or x is 100 over 8. again now it&#39;s just a case of cancelling down this as much as we possibly can i could divide top and bottom by 4 so that&#39;s 25 over 2 which is 12.5 and that would be the interest rate the company would need the annual interest rate the company would need for that first quarter in order to meet their target which means that our answer to this question is d so yeah as you can see here again we&#39;ve got a compound interest problem here we&#39;ve got an annual interest rate but it only applies in the first quarter so we would have four compounding periods in this year yes there&#39;s a great big formula that you could use in order to work your way through this question the algebra is going to get quite nasty there are plenty of places to make a mistake but if you can understand how this compound interest process works as we said before it&#39;s going to take a little bit longer you&#39;re going to have to put more steps in but you&#39;re far less likely to make a mistake as you can see throughout this yes it&#39;s a tough problem but the algebra itself isn&#39;t that complex and so if you can get your head around the concepts here if you can understand exactly what we&#39;re aiming to do with this compound interest work then you can make your life easier you can minimize the chance of making an unnecessary mistake and it&#39;s that skill that will help you on the gmat missing a difficult question far more damaging to your score than missing a very difficult one and so if you&#39;ve got the opportunity to minimize an algebra mistake arithmetic mistake i would take it right so yeah i hope this has helped you understand percentages a little better as we said at the top i think the key things you can take away here are from the text translate as literally as possible and make fractions your friend as much as possible stay away from choosing a starting value and using that to work through the problem as you go if you can stick to the algebra turn everything into these fractional multipliers simplify the fractions as you go you&#39;re far less likely to make a mistake all right that&#39;s all i&#39;ve got for you today we&#39;ll hopefully see you back here again soon for the next in our gmat ninja quant series thanks very much for watching everybody we&#39;ll hopefully see you back here soon you

Transcript for:Mastering Percentages for GMAT Success

Transcript for:
Mastering Percentages for GMAT Success