For some this is the most “interest”ing concept on the Revised GRE (at least for those with a fondness for bad puns). For most, compound vs simple section can be a nuisance. Many think, what is the difference between the two, and/or how did that formula go again?

But remember, this concept involves money, and for many that means its practical (especially if you invest money yourself). But enough rambling…

## Simple Interest

Principal: The Amount of Money initially invested

Interest Rate: The amount return on an investment expressed as a percent of the principal.

Time: The length of time in which a principal is invested

## Sample Problems

1. John invests 100 dollars in account that yields 8% simple interest annually. How much money will John have in his account after one year?

(A) $4

(B) $8

(C) $104

(D) $108

(E) $110

2. Bob invests 100 dollars in a fund that yields 15% simple interest monthly. If Bob invests the principal in the middle of January, which is the first month will he have more than $200 total?

(A) June

(B) July

(C) August

(D) February

(E) March

3. In 2001, John invests x dollars in a special account that yields y% simple interest annually. If he has $250 in his account in 2006 and in 2008 he has $270 in his account, what is x + y?

(A) 5

(B) 25

(C) 200

(D) 205

(E) 210

Answers:

1. D

2. C

3. D

## Compound Interest

Okay, that was the easy part. Now for compound interest. In compound interest things become complicated. We no longer have a nice, clean linear increase. To illustrate:

If Mike invests $100 at 10% simple annual interest, he will have $110. After two years he will have $120. That is his money grows by $10 every year. After 10 years, Mike will have doubled his money.

Now, let’s say Mike’s friend Thomas invests $100 at a 10% rate that is compounded annually. After one year, Thomas will have made the same amount as Mike. But then things start diverging. Remember how Mike always gets 10% of the original 100 (the original 100 is called the principal)? Well, Thomas – because things are getting compounded annually – gets 10% of whatever the value of the account is at the end of the year. Let’s see how this plays out over time.

1^{st} year: 10% of 100 = 110

2^{nd} year: 10% of 110 = 121

3^{rd} year: 10% of 121 = 133.10

4^{th} year: 10% of 133.00 (rounding down) = 146.30…

After 10 years, Thomas will have made $260, which is $60 more than Mike.

Okay, that may all seem like chump change, but the same percent increase applies to numbers with a few more zeroes thrown in. How would $260,000 vs. 160,000 sound?

Of course the point of this lesson is to understand the conceptual difference between the two forms of interest—and not to have you running to the nearest ban, since the numbers above are very generous.

Now for the fun part: Notice how, in the case of Thomas, I seemed to be doing mathematical wizardry. After all, how did I know that 10% compounded annually at 10 years is going to yield 160% of the principal? Well, let’s meet the formula:

V = Total Value

P = Principal

r = annual interest rate

n = number of times per year invested

t = number of years

Pretty unpleasant, no? Well, let’s try to put the formula to the test. And you may want to get your calculators out (this is the Revised GRE after all!)

If $10,000 is invested at 10%, compounded semi-annually, how much will the investment be worth after 18 months?

(A) 11,500

(B) 11,505

(C) 11,576.25

(D) 11,625.30

(E) 12,000.50

Now don’t worry about the semi-annual bit—it just means twice a year. And remember the *n* from the scary little formula above: the number of times per year invested. And that 18 months? That corresponds to t, the number of years, which translates to 1.5.

.

That was easy—once you know where to put everything (and provided you remember the formula)!

Sorry, I’m having trouble with the last compound interest question. When I solve V = 10,000(1+ 10/200)^(1.5)(2) I get 12,155.0625. Am I doing some order of operations wrong?

Thanks for your help!

Hi Liz,

This may be the problem, so let me try to walk through the calculation and maybe you can see where you went wrong!

=10,000 * (1 + 10/200)^(1.5)(2) [Calculate “10/200”]

=10,000 * (1 + 0.05)^(1.5)(2) [Add “1” and “0.05”]

=10,000 * (1.05)^(1.5)(2) [Calculate “1.5 * 2”]

=10,000 * (1.05)^3 [Calculate “(1.05)^3”]

=10,000 * 1.157625 [Calculate terms]

=11,576.25

Hi Chris, I am not so clear why r should be divided by 100. I mean, r is already the annual interest rate like 5%. Or does it alternatively mean in GRE test that annual interest rate merely stand for the number of a percent (like the 5 of 5%)?

Thanks a lot for helping me with this! 🙂

Hi Susan 🙂

In the equation on this post, r refers to annual interest rate expressed as a percent. For example, if the question said that the annual interest rate were 5%, then r = 5. So, in the equation, we need to divide by 100 in order to rewrite the interest rate as a decimal:

interest rate = 5%

r = 5

r/100 = 0.05

That said, since the GRE doesn’t test the equation directly, if you wanted, you could define r as the interest rate as a decimal. In that case, you would not divide by 100 in the equation:

V = P(1+r/n)^(nt)

r = annual interest rate expressed as a decimal

So, if the annual interest rate were 5%, r = 0.05.

Either way works when using this equation. You just have to make sure that in the end, you are adding the interest rate expressed as a decimal to 1 in the parenthesis!

Hope this clears up your doubts 🙂

It is the number which divided by 100 not really. Because the meaning of word percent is per or one of cent or hundred. So if you want to find 8 percent of 200 than

Think that there are 200 apples which should be distributed between 100 persons. Than 8 persons will get how much apples?

One person out of hundred get 200/100

Than 8 persons out of hundred get (200/100)*8

Can you please provide the solution to this problem.

In 2001, John invests x dollars in a special account that yields y% simple interest annually. If he has $250 in his account in 2006 and in 2008 he has $270 in his account, what is x + y?

(A) 5

(B) 25

(C) 200

(D) 205

(E) 210

thanks..:)

Hi Akash,

I’ll just quote Chris from another comment here in the thread: “Sure, I saw that the difference between ’06 and ’08 was $20. Thus each year is a $10 increase since are dealing with simple interest. If the accounted started in 01′ that means in 5 years (from ’01 to ’06), the account went up $50 (ten each year). So in 2001 account was $200 = x. If account increases $10 that is 5% of 200. So y =5. And x + y = 205.” I hope that helps. 🙂

thanks for the quick reply. 🙂

You’re very welcome! Happy studying 😀

I’m still unclear on how to solve this problem I got y=4 and x=208.333. This was based on my using the SI formula twice. Once for a 2-year period and the second time for a 5-year period. Can u please explain?!?

Hi,

Will the GRE calculator enable us to perform the last equation (comp. interest). I’m not sure how I will compute ^3….

thanks,

Hi Yasmine,

The GRE calculator is very limited, and if you want to calculate powers, you have to type it manually. So 4^3 would be 4*4*4= rather than a calculation of ^3 directly. This can get very messy with powers that are larger. Because of this challenge, we like to emphasize estimation skills and mathematical pattern thinking!

I hope that helps. 🙂

Hey! Can any body prove why less compounding period results in more interest?

Hi Tian Tuo,

Good question! 🙂

The interest earned

per periodis greater with fewer compounding periods, but the final value of the interest earned is greater with more compounding periods. I hope that clarifies! 🙂Hi there, thanks for all the information first of all.

I have a question about the last example. I tried to solve it without the formula simply multiplying by 1.1 each time so 10,000 x 1.1 = 1,100 , and then 1,100 x 1.1 = 12,100, and then 12,100 x 1.1 = 12,310… which is incorrect but I don’t understand why… Am i not simply compounding the interest each time? 18 months / 6 months = 3 so three compound interests should equal the correct answer?

Any help would be appreciated!

Hi Holger,

You have almost got it! 🙂

You have to divide the interest (10%) by the number of compounding periods per year. Notice how we have a 200 on the bottom of the fraction? (10/200 inside the parentheses.) This is because we are doing two things at once–dividing 10/100 to get the decimal version of 10% and also dividing that by 2 because there are 2 semi annual periods in the year.

The way you are doing the math means that the investment is actually earning 20% interest! (Which would be awesome. :)) I hope that helps.

THANK YOU. I was having trouble understanding the video lesson.

Hi Chris,

For Question 3, Though i got 205 as my final answer but with another approach i was trying to calculate y (rate of interest) considering P= $250, R= y% , T= 2 years

So, S.I = (P*R*T)/100

S.I = (250*y*2)/100

S.I = 5y

We got $270 after 2 years

So, Amount = P + S.I

270= 250 + 5y

Therefore, y = 4

However, the correct value for y is 5% but with this approach i got 4%. Please let me know where i am wrong.

Simple Interest is always calculated on the original Principal (‘x’ in this case). But u mistook it for 250/- (which is actually, P + S.I after 5 years).

By your approach,

270 = 250 + (S.I for 2 years)

where, S.I for 2 years = P.T.R/100 = x*y*2/100.

=> 270 = 250 + x*y*2/100

=> x*y = 1000.

As we know John is having $250 after 5 years,

P + S.I for 5 years = $250

=> x + (x*y*5/100) = 250

=> x = 250 – (x*y/20)

(substituting x*y with 1000 from earlier equation)

we get, x = 250 – (1000/20) = 250 – 50 = $200

(substituting x with 200 in x*y = 1000)

we get, y = 5%.

Therefore, x + y = 200 + 5 = 205 which is (D)

I set it up like this:

At some year T, the total balance will be P + PRT.

So for 2006, (@ T = 5), x + x*(y/100)*5 = 250, where y/100 = R.

For 2008, x + x (y/100)*7 = 270. {P + PRT = balance at time T = 7}

After subtracting the two equations, I get x*(y/100) = 10. So principal times rate for one year is $10. I originally started down your path, and ended up with a principal of 208.33, but with 4% interest, you will not get $270 at year 7.

I think the big takeaway is P + PRT = balance at some later time T.

This is a good , subtle problem. Hope this helps – John.

Hi,

Could you explain how you solved the problem you setup above?

Thanks,

Jess

Hi,

I am confused about how to calculate such large exponents while calculating compound interest quickly, as gre calculator doesnot allow such calculations.

But remember, this concept involves money, and for many that means ***it’s*** practical (especially if you invest money yourself). But enough rambling…

Don’t persecute me : ) I know grammar, but I don’t know math!

🙂

how many compound interest and simple interest problems on gre?

You might get one. But that’s probably about it.

Hope that helps!

100* (1+ 15/100 * X) = 200

100 + 15X = 200 => X= 6.6 ( it means after July)

my way is a bit complicated!

do you know another way to solve Q2?

Thanks,

R

Hi Ram,

You def. got it! But a more straightforward/intuitive way is the following: each month $15 is added. 7 x 15 = 105, so he will have to wait 7 months after Jan., which is August.

Hope that helps!

How many of these type of questions are actually on the GRE? I am using the Official Guide to study, and there are some scary looking sample problems regarding compound interest and such. I am nervous because I only learned this stuff in high school using calculator programs!

It does look pretty scary. These question types aren’t that common. My rough estimate is 40% of the tests will have such a question. And typically these questions aren’t as scary looking as some they have in the Official Guide.

Hope that helps 🙂

How do I solve for the exponents without taking the log (in the second simple interest problem for example)? I noticed that the GRE calculator does not have a log button?

Hi Jill,

For the second problem you don’t have to use logs at all, since it is dealing with simple, not compound, interest. Does that help?

the formula is not the same in the video clip

Final = P (1+ r/c)nc

which one should stick to it

Rakan,

Just make sure that the “nc” is in exponent form:

P(1 + r/c)^nc

That should do the trick!

Hi, I am completely aware of the difference between simple and compounded interest. However, will the type of interest be explicit on the test? I took a practice version which did not specify, and there was no way to tell which method was desired.

Hmm…that’s a good question. It typically won’t be explicit but will give you an indication via the question. There shouldn’t be any ambiguities in the problem, though it may seem initially tough to figure out which method is required. Remember to determine whether the sum paid out per month/week/year is the same or if that sum is increasing.

Hope that helps!

Hi Chris,

concerning the first sample problem the answer should be 100$ as the interest is collected annually, i.e. the interest is only added to the whole year and not fractions of it by definition of simple interest, otherwise you will need to define the compounding rate.

thank you.

Oh yes, that’s right! Thanks for catching that :). I’ll make the necessary changes.

Chris you are going to kill me 🙂

what I said was wrong to some extend, I confused the mathematical definition of the word simple interest and compound interest and the collecting rate , i.e it’s ok to collect the simple interest per any period of time (a year or a fraction) as long as he didn’t mention in the question a certain collecting rate ( a constrain that a bank can add or something) I’ll use your example to explain my point

e.g. John invests 100 dollars in account that yields 8% simple interest annually, collected annually, How much money will John have in his account after 26 month?

the answer is 16$

on the other hand if it was 8% compound interest, with a monthly compounding rate, with an annual collecting rate then the answer is

100 x ((1+(0.08/12))^(12 x 2) = 117.288$

I’m sorry for that slip, I wasn’t accurate at all in what I wrote the first time, and I assumed that the interest is collected annually out of the blues I guess 🙂

Hi Amr,

Not a problem. Now I understand what you mean :). Sure, you can have all sorts of different non-yearly intervals with simple interest. It shouldn’t affect anything – the way it does with compound interest.

For the first compound interest problem, is there a faster way to calculate the exponent rather than multiplying it by itself multiple times? It seems rather time consuming

Hi Ben,

Actually, there is not a way that I know of. The good news is that with the calculator it is not that time-consuming. Anyhow, explanations of a concept can often time make the actual solution seem longer than it really is when you are simply punching the numbers into the calculator. One alternative though, if you don’t use a calculator, is to look at the answer choices. Usually two are three of them will not match with the numbers given in the question (e.g., round numbers when the answer clearly calls for a decimal).

Hope that helps :).

Thanq very much chris giving good stuff and i suggest u to give more questions like this…

Shiva,

Thanks for the kudos :). An advanced interest post is definitely called for in the not too distant future.

Chris,

Can you explain the solution of Q2.

The question is basically saying that John earns $15 per month (on top of his 100). Thus it will take him 6.67 months to earn 100. Since he start investing at the beginning of January, he will earn 100 by the end of July.

Hope that helps 🙂

Wait, are you saying that when it says “monthly,” we still assume his account balance increases daily/hourly/etc? I understand that it’s 6.67 months, so technically it’s the end of July. But isn’t that assuming that the bank deposits his interest earnings every day or every hour, as opposed to the beginning of each month?

Thanks.

I had the same doubt.

As the account yields 15% interest monthly, shouldn’t the interest be reflected when a new month starts?

Hi Manan,

I responded to Zuhaib’s comment. Again, sorry for any confusion :).

Hmm, that is a good point. I was assuming that it increases every minute, but technically ‘monthly’ would mean only at the beginning of each month. So not the best question – my assumption would be far more valid if the question dealt with a constant growth rate, such as a population of bacteria.

Sorry for any confusion :).

Hi Chris,

So if we say that interest is added at the beginning of the month then the answer would be August, right? As he still has $190 at the beginning of July.

Also, this blog seems to have a few typos.

1. Q2 first it’s John then you say Mike!

Intro to compound interest: you state, after 10 yrs, mike will have 260, which is 60 more than mike

Wouldn’t that be bob has 260, which is 60 more than mike.

Thanks.

Btw, thanks for these blogs and comments. they’re very helpful!

Oh wow, you’re right! That is very confusing :(. Note to self: do not use really generic names :).

Thanks for catching that Shamila!

Dear Chris,

Thanks for the discussion however, I am confused with your response regarding question# 2. According to the question, the investment took place in mid-January, NOT beginning of January, therefore the first $15 supposed to be earned by mid-February…am I right? While it takes 6.67 months to earn $100 with the given simple interest rate therefore the value $200 never be reached until late first week of August.

Hi Milon,

Sorry for the confusion — I changed the wording from beginning of Jan to mid-Jan. last time I edited the post. Therefore, it kind of invalidates much of what I (and the other users) said in the comment post.

Yes, you are right. The first amount will be earned in mid-Feb. and the 200 won’t be reached till the first week of Aug. There is also some ambiguity about whether the simple interest is paid only at the beginning (or now that I’ve changed the question) the middle of the month. In the latter case, the answer will still be Aug. That’s why I changed the verbiage.

Again, sorry for any confusion :).

Hi Chris!

I’m confused with the second question still!!! Is the growth rate constant? As in..15% on 100 is 115 during feb mid! So the next will be calculated as 15% on 115 right? Or it is constantly increasing? moreover, how can the answer be Aug? Because 200 would be collected by mid of July! I’m seriously confused!! 🙁

Sorry for any confusion :).

The amount is invested at simple interest, meaning that $15 dollars will be added to the account at the middle of each month. So it will take 7 months (7×15). Therefore, seven months after the middle of January is the middle of Aug.

Hope that helps!

Hey Chris..I got that part..but what about the growth rate? Is it constant? As in..15% on 100 is 115 during feb mid! So the next will be calculated as 15% on 115 right? Or it is constantly increasing?

Also, in such type of questions, will they specifically mention if the rate is constant?

Ankita,

Sorry that was confusing :). No, the growth rate is not constant but is only calculated on the 15th.

Hope that helps!

Chris:

Do you mind showing the steps you used to solve it? Thanks

Sure, I saw that the difference between ’06 and ’08 was $20. Thus each year is a $10 increase since are dealing with simple interest. If the accounted started in 01′ that means in 5 years (from ’01 to ’06), the account went up $50 (ten each year). So in 2001 account was $200 = x. If account increases $10 that is 5% of 200. So y =5. And x + y = 205.

Hi Chris,

Thank you for the explanation. How come y isn’t equal to 0.05? I thought the 5% would have to be added in decimal form, with x + y equaling 200.05.

Thanks!

Hi Helena,

That’s a tricky one! See the variable is presented as “x percent”. In this case, the number for x is simply 5, as in 5 percent. As an actual number, 5% is represented as .05, but if ‘x’ were to equal .05, we would have .05 percent, which is .0005.

Hope that clears things up :)!

Hi Chris:

How did you come out with option E (210) on question 3?

Oops, that should be (D) 205

x = 200 and y = 5

Thanks for noticing that :).

Hi Chris, this is again regarding Q3.

Please clarify how did you calculate y=5. kindly let me know how you feel about the solution below. I feel the answer should be “E”

The balance in the account in 2006 is $250 which increases to $270 in 2008. Since this questions deals with Annual Simple Interest, we can calculate that interest earned per year is $10.

Thus in 2005 a/c balance = $250 – $10 = $240 ….

So in 2001 a/c balance should be $200.

So x+y should fetch 200+10 = $210.

Hi Abhijeet,

Notice the questions says ‘y’ percent. If ‘y’ were 10, then 10% of 200 is 20, which would lead to an increase of $20 per year, not $10 (or 5%), as the question states. Therefore, y must equal 5.

Hope that helps :)!

Hi Chris,

Sorry, my mistake !!

I mistook ‘y’ to be the annual interest that accrues every year on the principal of $200. $10 is definitely is 5% of $200.

Thanks,

Abhijeet

No problem :)!

Hi Chris,

I still feel the answer should be $210(200+10) or the question needs to be rephrased to “y % simple interest annually”. If we are saying “y simple interest annually”, one expects to calculate the interest rather than the rate. Correct me if I am wrong.

Thanks,

Aditya

Yes, that should be y%. Now I see why this one confusing so many people.

Thanks for point that out :)!

hello

I am very confused about the sample question 3. I find it very hard. Can you please explain me.

I am very weak in math. would appreciate your help.

Hi Sukaina,

Just a note that I’ve sent your question to our remote tutors, who will respond to you via email! As a Magoosh premium subscriber, you can click on the ‘Help’ tab on the left side of your screen or email help@magoosh.com for these types of questions in the future!

Thanks!

Jessica

Oops, that’s a typo. I’ll take care of it. Thanks for pointing that out :).

Hi Amy :).

Yes, that’s definitely a series of typos. Thanks for catching that!

As for r, it can stand for ‘compounded annually’ as long as there is the n (number of times compounded annually). Therefore 10% compounded semi-annually works out to 5%: (10/100(2)) = 1/20. So either way works.

Hey Chris,

Is it the word ‘ compounded ‘ that decides.the fate of the question or some other words are also included in the.pool..?

For compounded interest questions, ‘compounded’ determines the fate of the question.