Manhattan GMAT Challenge Problem of the Week - 5 July 2011

by Manhattan Prep, Jul 5, 2011

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Question

At the beginning of 2011, Albert invests $15,000 at 10% simple annual interest, $6,000 at 7% simple annual interest, and $x at 8% simple annual interest. If, by the end of 2011, Albert receives interest totaling 9% of the sum of his three investments, then the ratio of $x to the sum of his two other investments is

(A) 1 : 3

(B) 1 : 4

(C) 1 : 6

(D) 1 : 7

(E) 1 : 8

Answer

One straightforward way to attack this problem is to compute the interest on each investment separately, then add up these separate bits of interest and set that sum equal to 9% of the total investment. We can write a word equation as an intermediate step:

Interest on investment #1 + #2 + #3 = 9% of all invested dollars

10%($15,000) + 7%($6,000) + 8%($x) = 9%($15,000 + $6,000 + $x)

Drop dollar signs and convert percents to decimals:

1,500 + 420 + 0.08x = 0.09(21,000 + x)

Multiply through by 100:

192,000 + 8x = 9(21,000 + x) = 189,000 + 9x

3,000 = x

We are asked for the ratio of $x to the sum of the other two investments, i.e., $15,000 + $6,000 = $21,000. $3,000 : $21,000 is equivalent to the ratio 1 : 7.

Alternatively, the overall interest rate of 9% can be seen as a weighted average of 7%, 8%, and 10%, with each interest rate weighted by the amount invested at that rate. To have an overall rate of 9%, any dollar invested at 7% must be balanced by two dollars invested at 10%. Thus, the $6,000 invested at 7% are balanced by $12,000 invested at 10%. This leaves $3,000 left over invested at 10%, which must be balanced by an equal amount invested at 8% (again, to make the overall average equal to 9%). Thus, x equals 3,000, and the desired ratio is 1 : 7.

The correct answer is D.

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