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## Compound interest

# Solved example: compound interest

## Video transcript

let's read this question madhu deposits ten thousand rupees at twenty percent per annum compounded quarterly okay I am collecting data from this question ten thousands important twenty percent the interest rate is important and then how often it's compounded compounded quarterly this is like really important and then what's the question okay so what is the total amount and the interest she gets after six months okay so after six months I need to find how much this ten thousand will become now I know that component list is or a component this problem is just a percentage change problem in disguise so let me do this my step one then is just to convert this component is problem into a percentage a percentage change problem convert to percentage change problem convert to percentage change problem that's it that's my first step and that's probably the most important step because after that the problem is not not component rest anymore so I need to first find out how much money should I start with how much am I starting with how much is the amount I start with and that's easy that's ten thousand that's given over here so that's the amount you begin with now what's your next question okay by what percentage should I increase this 10,000 by what percent by what percent are you increasing and the answer to that is not straightforward it's not 20 percent because this is 20 percent per annum compounded quarterly now the moment I read this word quarterly I know in my mind that I want everything in terms of quarters now quarter is my unit that's the time at which I'm gonna find the interest and add it back so I want everything in terms of quarters so I look here this is 20 percent per quarter no it's not it's 20 percent per annum so I need to find what is the interest rate per quarter so I have 20 percent every year but a year has four quarters so if I divide by four I'll get the interest rate per quarter and that's gonna be equal to five so this is my interest rate I have to increase by five percent every quarter okay so I'm done oh no I have one more step how many times I do it because this final uh-huh after how many days I need to find the amount it's just another way of asking how many times you should increase this ten thousand by five percent so let's do that how many times now how many times six months doesn't tell me how many times very directly but if you look here it's compounded quarterly and like I said quarter is now our unit for everything so how many quarters are there in six months that's all I care about so how many times should I increase twice right because there are two quarters in six months so that's it I have my two quarters so I have two times we have succeeded ninety percent of our problem is solved in the sense that we have now converted this component is problem into a very simple question of can you increase ten thousand by five percent two times increase it once and then increase it once again and whatever answer you get will be here Ahmad so step two is to actually do this to actually do this and find that amount so let's look at step number two step two is find this okay so increase ten thousand increase ten thousand by five percent two times by five percent twice percent twice now there are a few ways to do this and depending on what's your favorite way to deal with percentages you can do it so step two let's try we're gonna do two methods let's actually have a line that separates these two methods so the first way the obvious way to do this is find five percent of 10,000 and add it to 10,000 that linkless for 10,000 by five percent once right so what is five percent of ten thousand five percent of 10,000 now the way I like to think about it is I like to ask what is one percent of 10,000 now that gives me what is one percent of 10,000 that's 10,000 divided by 100 if I remove these two zeros I get 100 100 is 1% I want 5 percent so that's 5 times 100 or 500 so that's 5 percent of 10,000 increase 10,000 by that so I just have to add 500 to 10,000 so I have 10,000 10,000 or 4 zeros plus 500 plus 500 that gives me ten thousand five hundred equals ten thousand five hundred now what is this this is my amount after increasing by 5% once right so we're gonna call it once after doing it once I have this but I want to do it twice so now for the second time what should I do I have to find five percent of not ten thousand but 10,500 because I have to increase this number by five percent now so let's do that one zero five zero zero now what is this equal to I again I mean you can do five by 100 into this and if you do that exactly what you'll get is five by 100 this hundred will be cancelled with that but in other words one percent of this number is hundred and five right and then five times that will be hundred and five into five which is 100 times five is five hundred and then five times five is 25 so 500 plus 25 you can verify this but I think that you can also start practicing doing these kinds of questions in your head because it's possible to do them and now you can find the amount after doing it twice because that's just gonna be this number Plus this number so I'm gonna say twice after increasing by 5% twice you'll get this Plus this which is one zero five zero zero plus five 25 and again you can do this I'm gonna try and do this in my head what I will do is because I'm able to see ten thousand five hundred plus five hundred and twenty-five I'll take this 500 part alone so ten thousand five hundred plus five hundred will be eleven thousand I have a 25 remaining so it's 11 thousand twenty-five 11 thousand twenty five seven thousand twenty-five sometimes the eraser doesn't work eleven thousand twenty-five now what is this number this is the amount after increasing 10,000 by 5% twice right that's exactly what we wanted that's what we wanted actually that's the amount that we will have after six months this is the answer to the question and we have it now this is the most intuitive and also simple way to do the problem but it can get tedious if this number is large instead of say six months if he had said two years then two years has eight quarters so do that eight times and this can get really long but you may ask a question you may ask hey I only want to know what the final answer is why am i calculating all these middle answers and that's a great question if you want to not calculate all these middle answers there's a way to do this and let's look at that it's a much quicker way but to do that let's pause for a moment and just take a look at percentage change now we've learned this before but let's take a look quick look at it again if I have any number let's forget 10,000 if I have any number I'm gonna call it X if I want to increase it by five percent what should I do I have to find five percent and add it that's what you did here so what is five percent of this number X that's just 5 by 100 into X right now to increase this by five percent I just have to add these two numbers yeah and this works for any number and no matter what this X is you can take it out common all right you can take it out and write this whole thing this way so there's gonna be 1 over here 1 plus 5 by 100 right and you know that you can write this if you just take this 100 over basically you're taking me entire denominator to be 100 if you do that then you can write this as X in 205 by 101 0 505 by 100 and if this looks like a lot of work we're doing this so that we don't do work later they're doing all this work before so this can be simplified even more if I divide the numerator and denominator by 5 so I get 21 divided by 20 now pause over here and notice where did we start with this number what is this number this is X increased by 5 percent X increased by five percent is entire number just simplify and simplify you get this number this is also X increased by five percent but this is beautiful this is just a product this is just something multiplied by X this is a little bit more complicated than that so what this shows is increasing a number X by five percent is exactly the same as multiplying it by twenty one by twenty no multiplication is much easier than this thing called increasing by some percentage whatever it so this is a much more beautiful and much more useful result so let's keep this let's take this and let's forget these the rest of what we did here and we only care about this result as we go forward so let's actually forget about the rest and let's say that we have this alone with us let's take it away and let's start looking at this once again what does it say multiply a number by 21 by 20 and you and you increased it by five percent how many times though once that's it once what should I do take whatever my number is I don't care that it's ten thousand it could have been anything but take that number take that ten thousand multiplied it by 21 by 20 if you do calculate this we're really suspicious and if you do calculate this this will be equal to ten thousand five hundred but you're right why are we calculating it let it be this way so what should I do for twice I don't care what this X is right so this X can either be ten thousand or this entire number I don't care but if I multiply this entire number by twenty one by twenty again I will get five percent once more I'll increase it by five percent once more that's exactly what I need to do I need to start with this number the entire number so ten thousand into 21 by twenty that's the number I'm starting with and I have to increase this number by five percent so this is my new X increased by five percent just means multiply this entire number by twenty one by twenty now you can see that this scales very very very well so thrice x 21 21 by twenty again four times again and you do this say for example even 20 years writing it will be super easy ten thousand multiplied by 21 by 20 20 times multiplied 20 times or to the power 20 if you want to write it that way but that's it so you can see over here that when the numbers become large this method becomes more useful so amount after after six months you know that this answer and the previous answer has to be same but I would like you to be more suspicious let's actually calculate this so how would I do this I would probably take away these zeros first I'll just divide the numerator and denominator by these these zeros typify these tents are you can't divide by zero and then I would just find the product of these two 21 into 21 that's 441 that's Toni 1 squared yeah you don't have to remember it I just happened to so 441 in 200 so 4 4 1 0 0 so I have 441 times 100 divided by 2 times 2 which is 4 divided by 4 and that is equal to if I take this out it's 1 1 and then I have nothing over here so 0 and then in 10 it goes twice and then again I have 2 with me and then so in 20 it goes 5 times so have 11,000 25 which is exactly what I expected to get 11,000 25 the answer is the same and these are broadly the two methods that you always use in component rest's if the number is really small you'll just be like you'll do this but in fact almost always you'll do this because once you understand why this works this is just a better method just a faster method every single time and you may also notice that if you just take some time you can show that this what we're doing here is exactly the same as what you may have seen otherwise which is P into 1 plus R by hundred whole to the power n you can take some time to show this to yourself this P is basically the amount we start with r is our percentage R by 100 plus 1 is what we did over here to get this 21 by 20 and depending on the number of years you get this power n now we're almost done there's only one more step finding the interest now you find the industry of the amount you have the principle to subtract the principle from the subtractive principle from the amount that's right principle from the amount and then you get the answer so let's do that so step number three step three find interest and that is just going to be equal to one one zero two five eleven thousand twenty-five minus my original principal and that's gonna give me its 1025 and you can do the subtraction to verify this so thousand twenty-five is the interest that I will have and notice that in compound interest finding the amount is easier you first find the amount and then subtract the principle to get the interest you may remember that in simple interest it's actually easier to calculate the interest you first find the interest and then you add it to the principal to find the total amount also notice that we have taken ten thousand here so it may look like the numbers very early simple but no matter what this number was the method remains exactly the same what are the steps convert to a percentage problem let's just go back and summarize so convert the problem to a percentage change problem read this question and do that pay particular attention to the what's the frequency in which it's compounded step number two do the math actually do that person distance problem and then finally find the interest and that's it