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Interest

by treker » Tue Sep 15, 2009 7:31 am
Last year Luis invested x dollars for one year, half at 8 percent simple annual interest and the other half at 12 percent simple annual interest. Now he wants to reinvest the x dollars for one year in the same two types of investments, but the lower rate has decreased. If the higher rate is unchanged, what fraction of the x dollars must he reinvest at the 12 percent rate so that the total interest earned from the x dollars will be the same for both years ?

(1) The lower rate is now 6 percent.
(2) The total amount of interest earned from the two investments last year was $3,000.
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Re: Interest

by Ian Stewart » Tue Sep 15, 2009 8:38 am
treker wrote:Last year Luis invested x dollars for one year, half at 8 percent simple annual interest and the other half at 12 percent simple annual interest. Now he wants to reinvest the x dollars for one year in the same two types of investments, but the lower rate has decreased. If the higher rate is unchanged, what fraction of the x dollars must he reinvest at the 12 percent rate so that the total interest earned from the x dollars will be the same for both years ?

(1) The lower rate is now 6 percent.
(2) The total amount of interest earned from the two investments last year was $3,000.
First, the initial setup is just a weighted average; if you invest half your money at 8% and half at 12%, you'll make 10%. From Statement 1, if the higher rate is 12% and the lower rate is now 6%, then to earn 10% interest, you'll need to invest twice as much at 12% as at 6%, by weighted average principles (because 10 is twice as far from 6 as it is from 12), so Statement 1 is sufficient.

If you don't understand weighted averages in this way, you can also see this algebraically. If you have x dollars and invest the fraction p at 12% (so invest px dollars at 12%), then you'd invest the fraction 1 - p at 6 percent (so would invest (1-p)x at 6%). To earn 10% on the two investments, we must have:

0.12*p*x + 0.06*(1-p)*x = 0.1*x
12p + 6 - 6p = 10
6p = 4
p = 2/3

So Statement 1 is sufficient. Statement 2 is irrelevant, since we're only concerned with a ratio here. The answer is A.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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