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Simple and compound interest

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Simple and compound interest

by rahulvsd » Wed Mar 07, 2012 7:47 am
$1200 is invested at a given interest rate for two years. The difference between the simple 2-year non-compounded return at the end of the two years and an annually compounded return is $132. What is the interest rate?

Choices
A 10%
B 11%
C 12%
D 13%
E 14%

[spoiler]OA: A. Source: Grockit. Any suggestions on how to solve this quickly[/spoiler]
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by Mike@Magoosh » Wed Mar 07, 2012 2:50 pm
Hi, there. I'm happy to help with this. :)

Let r be the interest rate, expressed as a decimal (i.e. 5% = 0.05).

Simple interest will increase to

1200 + 1200r + 1200r = 1200*(1 + 2r)

Compound interest will increase to

1200(1 + r)(1 + r) = 1200*(1 + 2r + r^2)

It makes sense that the only term that's different between the two expressions is the r^2 term, which is interest on interest. That, right there, is the big idea of compound interest. In simple interest, you get interest only on the principle. In compound interest, you get interest on the principle and on the previous interest.

1200*(r^2) = 132

Divide both sides by 12

100*(r^2) = 11

r^2 = 11/100 = 0.11

I get r = 0.331662479, which is not an answer choice. Something is funky here.

If the difference between two-years simple vs. two-years compound were $12, not $132, then the answer would be r = 10%, choice A.

If the difference between one year of interest vs. two-years compound interest were $132, then again, the answer would be r = 10%, choice A.

I think something got confused here about what is being asked, either a problem at the source or a problem in miscopying at some point along the way.

Does the approach I used make sense? Do you have any questions?

Mike :)
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by nishnosh » Sun Mar 25, 2012 1:17 am
Could you explain from this line below pls: 1200(1 + r)(1 + r) = 1200*(1 + 2r + r^2)

I got confused at this line, I thought the formula for compound was = P(1+ (r/n)^nt

t=2
n=2?

thanks
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by Mike@Magoosh » Sun Mar 25, 2012 2:42 pm
nishnosh wrote:Could you explain from this line below pls: 1200(1 + r)(1 + r) = 1200*(1 + 2r + r^2)

I got confused at this line, I thought the formula for compound was = P(1+ (r/n)^nt

t=2
n=2?

thanks
Your formula for compound interest is almost correct, but because it is missing a three parentheses, it is wrong. The correct formula is:

A(t) = P(1+ (r/n))^(nt)

Also, I believe you are interpreting it incorrectly. In that formula, n is the number of compounding periods in a year --- for quarterly compounding, n = 4; for monthly compounding, n = 12; for daily compounding, n = 365. Here, the problem states we are compounding annually, only once a year, so n=1.

So, in this problem, t does equal 2, but because we are only compounding annually (i.e. only once a year), then n=1. Of course, P = 1200, so

A(t) = P(1 + (r/n))^(nt) = 1200*(1 + (r/1))^(2*1) = 1200(1 + r)^2

From there, I just wrote out what it means to square something --- to multiply it by itself.

1200(1 + r)^2 = 1200*(1 + r)*(1 + r)

Then, I just did FOIL multiplication of the two binomials:

1200*(1 + r)*(1 + r) = 1200*(1*1 + 1*r + 1*r + r*r) = 1200*(1 + 2r + r^2)

After that, I subtracted what the simple interest formula equaled --- 1200*(1 + 2r) ---- to get the difference, since this problem gives us a dollar amount for the difference.

Does all this make sense? Does it clear up your question? Please let me know if you have any further questions.

Mike :)
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