Bunuel wrote:
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
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION: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. Can u plz explain the alternative approach using the weighted average? Why have u ignored the 8% principal amount in this calculation?