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C.I., S.I

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C.I., S.I

by kanha81 » Thu Apr 09, 2009 1:03 pm
This is quite an interesting problem that I came across on 4gmat.com forum. This problem makes you solve backward to find the answer.

Shawn invested one half of his savings in a bond that paid simple interest for 2 years and received $ 550 as interest. He invested the remaining in a bond that paid compound interest, interest being compounded annually, for the same 2 years at the same rate of interest and received $605 as interest. What was the value of his total savings before investing in these two bonds?

A. $ 5500
B. $ 11000
C. $ 22000
D. $ 2750
E. $ 44000

OA [spoiler][D][/spoiler]

Are there any efficient ways to solve this problem under 2 mins?
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by Jose Ferreira » Thu Apr 09, 2009 5:48 pm
Hi,

This question can be approached with algebra as well, though it is quite tricky.

Call P the amount invested in each bond. His savings, the amount we are looking for, is 2P.
Call r the interest rate, which is the same in each case.

For simple interest, the amount earned per year is Pr. So 2Pr = 550, or Pr = 275.

For compound interest, since we are compounding annually, we multiply each year's principal by (1 + r) to get next year's principal. So, the total amount after two years is P(1 + r)^2. Since this is the total amount, we must remember to subtract P, the original amount, to leave just the interest earned.
So P(1 + r)^2 - P = 605.

Simplifying this equation, we get P(1 + 2r + r^2) - P = 605, or P + 2Pr + Pr^2 - P = 605, or 2Pr + Pr^2 = 605.

We know that 2Pr = 550, so we can further simplify to 550 + Pr^2 = 605, or Pr^2 = 55.

From here, we can solve for P = 55/r^2.

Plug this into Pr = 275: 55/r^2 × r = 275, or 55/r = 275, or [spoiler]1/r = 5, or r = 1/5[/spoiler].

Plug this into Pr = 275: [spoiler]P × 1/5 = 275, or P = 1375[/spoiler]. Remember that we want 2P, which is 2750.


An advanced way to think about this: the thing that differentiates compound interest from simple interest is that in compound, one earns interest on previous interest earned, while in simple, one does not.

In the first year, the same amount of interest will be earned with either type of interest, since there is no previous interest. [The money will grow from P to P(1 + r) = P + Pr.]

In the second year, the amount by which the bond with compound interest exceeds the bond with simple interest is equal to the interest on the first year's interest.

So, in the first year, they both earn 550/2 = 275.
The additional 55 earned by the bond with compound interest all comes in the second year. It is the interest on the 275 in interest from the first year.

In other words, r × 275 = 55, or [spoiler]r = 1/5[/spoiler]. From here, you can proceed as above.
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by relic » Thu Apr 09, 2009 6:06 pm
This is a brutal question. Can it be done under 2 minutes? Not sure, but let's try...

If x is the total savings, then each investment is x/2.

The interest on the simple bond at interest rate, r for 2 years

550 = 2(x/2)*r, so r = 550/x

The interest on the compound bond at interest rate, r for 2 years

605 = x/2(1+r)^2 - x/2 = x/2((1+r)^2 - 1) = x/2(2r + r^2) = x*r + (x*r^2)/2

substitute for r

605 = x*(550/x) + x*(550/x)^2)/2 = 550 + (550^2)/(2x)

clean it up a bit

55 = (550^2)/(2x)

110x = 550^2

x= 550*550/110 = 550*5 = 2750

I don't know... If you see a question this hard you might not have to worry.
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Re: C.I., S.I

by dtweah » Thu Apr 09, 2009 6:26 pm
kanha81 wrote:This is quite an interesting problem that I came across on 4gmat.com forum. This problem makes you solve backward to find the answer.

Shawn invested one half of his savings in a bond that paid simple interest for 2 years and received $ 550 as interest. He invested the remaining in a bond that paid compound interest, interest being compounded annually, for the same 2 years at the same rate of interest and received $605 as interest. What was the value of his total savings before investing in these two bonds?

A. $ 5500
B. $ 11000
C. $ 22000
D. $ 2750
E. $ 44000

OA [spoiler][D][/spoiler]

Are there any efficient ways to solve this problem under 2 mins?
This problem is testing whether you know the formula for finding a compounded interest without the principal attached. That is the trickiest part of the question. Beyond that everything is simple equations which can be solved easily.

(x/2) 2 r% =550

x/2 (2r%+(r%)^2)=605 Formula is A(nr%+ (r%)^n)

550/2r% = 605/ (2r%+(r%)^2)
275= 605/2+r% (Efficiency in solving equations matters)
550 +275r%=605

r%= 55/275

r% = .2

X = 550/r%= 550/.2= 2750
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