The difference, after two years, between compound interest and simple

Question

The difference, after two years, between compound interest and simple interest on a certain sum of money invested at the same rate of interest, is $18. If the simple interest accumulated on the sum after two years is $180, what is the rate of interest at which the sum of money was invested?

(A) 36%
(B) 30%
(C) 25%
(D) 20%
(E) 10%

(A) 36%
(B) 30%
(C) 25%
(D) 20%
(E) 10%

pushpitkc · Accepted Answer

Given data : Compound Interest - Simple Interest(on the same sum of money) = 18$
Also, the period of interest : 2 yrs
Simple interest accumulated over 2 years = 180$

We know that  the simple interest = compound interest(for the first year, for any amount for same rate of interest) 
So, the difference in interests would have occurred in the second year.
 Simple interest is going to be same for all years for any sum at 
a fixed rate of interest (making the simple interest of year 1 = 90$)
Since the compound interest increases by 18$, the rate of interest must be 20%(because 20% of 90 is 18)

Hence, rate of interest :  20%(Option D)

roadrunner · Answer

In compounding interest, interest earned in year t0 is reinvested for year t1 i.e. interest is earned on interest payments. In simple interest, fixed interest is paid each year (no interest on last year's payment)

==> simple interest each year = 180/2 = 90$

As total period is 2 years only, difference between compounding interest payment and simple interest payment arises from the interest earned on 1st year interest under CI.

ROI = 18/90 = 1/5 = 20%

Ans. is D

ydmuley · Answer

This is a very good question. Thanks Bunuel At least I am able to brush up the concepts for SI and CI.

Simple Interest For First Year = Compound Interest For the First Year Simple Interest for 2 Years = $ 180 Simple Interest Per Year = $\frac{180}{2} = $90 Considering the difference between the Compound interest and Simple Interest for the second year = $18 We know that the compound interest add $18 in the $90 amount of the first year. This can give us the interest rate = \frac{18}{90} = \frac{6}{30}= \frac{1}{5} = 20% Hence, Anwer is D

anuj11 · Answer

Hi..

can someone explain why are we doing (18/90) ?? As 90 is the simple interest and 18 is the difference between the two interests ??

TimeTraveller · Answer

SI for 1st year = CI for 1st year =  $90 .

Let CI for the 2nd year =  x .

So,  x - 90 = 18

x = $108 .

90 + 20\%(90) = 108 , so Ans - D.

sarugiri · Answer

Hi Anuj11,

We know SI is 180$ for 2yrs, making it 90$ for each year.

The difference b/w Simple and Compound interest starts from 2nd year.

Hence the interest amount of 2nd year  is compounding, and the difference b/w SI and CI as per the stem is 18$. The question is asking 18$ difference is what % of the interest amount.

18 = (x/100)90
x = (1/5) = 20%

Hope it answers your doubt.

DavidTutorexamPAL · Answer

We'll use the underlying logic behind interest to create a simple equation, without bothering with variables.
This is a Logical approach.

The only difference between the compound and simple interest is that the compound interest was calculated twice - once at the end of the first year and once at the end of the second year.
So, the $18 difference must be due to the second year's interest earned from the first year's interest.
Since 2 years of simple interest is $180, one year is $90 so 90*rate = 18 meaning that our interest rate is 18/90 = 2/10 = 20%

(D) is our answer.

Note that understanding the underlying logic created an extremely simple calculation: 18 divided by (180/2), as opposed to something full of variables.

US09 · Answer

If simple interest was 180$, that means simple interest must have been 90$ per year. So, the difference in 18$ must have ocurred due to the interest in interest, i.e. (90*x)/100 = 18, x = 20%.

IMO D.

vipulshahi · Answer

urvashis09

Can you please elaborate the explaination which you mentioned above.

I have on concern , $18 is for 2 year , how you can take $90*x/100 = $18

US09 · Answer

Sure, when you calculate simple interest and compound interest for any given amount, the interest for the first year remains the same. Example, you have 100$ on both SI and CI for 2 years @ 10%, for the first year both will earn an interest of 10$. But for the second year, SI will again earn 10$, but CI will earn 11$. Why? because now the amount for CI becomes the amount initially invested plus the interest earned i.e. 100$ + 10$ = 110$. Since the interest earned is not taken out but added in the principal amount invested for CI, the difference starts coming from second year onwards.

Hope that helps.

ScottTargetTestPrep · Answer

Let’s let P = the principal and r = the interest rate., We use the formula for simple interest: I = P x r x t and the compound interest formula (for annual compounding): I = P x  , and we have:

P x r x 2 = 180 (from the simple interest formula)

P x   = 198 (from the compound interest formula for annual compounding

Dividing the first equation by the second equation, we have:

(r x 2)/ = 180/198

198(r x 2) = 180

198(2r) = 180(1 + 2r + r^2 - 1)

396r = 360r + 180r^2

180r^2 - 36r = 0

36r(5r - 1) = 0

r = 0

or

5r - 1 = 0

r = 1/5

Since r can’t be 0, r = 1/5 = 20%.

Answer: D

Elleo · Answer

Hello,
Could you someone explain why above question doesn't mentioned the total amount of compound interest for 2 years?

If simple interest 90$ =compound interest 90$ . 
It means 90+90=180 for first year.
If compound interest 108(90$x/100%=18$; 18+90)=90 simple interest. 
It means 90+108=198 for the second year.

So totall sum of interest,as I understand, shall be 180+198= 378$ for two years.
Or did I miss something?

Posted from my mobile device

exc4libur · Answer

ar2=180…ar=90 
 (a(1+r)^2-a)-ar2=18 
  -ar2=18… -ar2=18…ar^2=18 
 ar=90…ar^2=18…90/r=18/r^2…90r^2=18r…r=18/90=1/5=0.2

Ans. (D)

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

The difference, after two years, between compound interest and simple

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The difference, after two years, between compound interest and simple

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards