At the beginning of 2011, Albert invests $15,000 at 10% simple annual

I guess your reasoning is backed by logic that 3 and 7 are factors of 21000 but you need to note that 4, 6 and 8 also are factors of 21000. So would you please elaborate why can't the answer be 1/4 or 1/6 or 1/8???

Please Let me know if my guess about your reasoning is wrong and you have some different reasoning.

10/100*15000 + 7/100*6000 + 8/100*x = 9/100 * (15000+6000+x)
Solving for x: x = 3000
Required Ratio = 3000 : 21000 = 1 : 7

15000*10**1/100 + 6000*7*1/100 + x*8*1/100 = 9*(21000+x)*1/100
x= 3000
ratio = 3000/21000 = 1:7

D

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.

Hi Bunuel,
Can u plz explain the alternative approach using the weighted average? Why have u ignored the 8% principal amount in this calculation?
Thanks!

Realise that this is a weighted average or mixture problem.

10*15
7*6
8*x
9 (21+x)

150+42+8x=9x+189
x=3

3:21

1:7

Given: 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.

Asked: 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

Total Interest = 1500 + 420 + 8%x = 1920 + 8%x
Total Principal = 21000 + x

1920 + 8%x = 9% (21000 +x)
1920 + 8%x = 1890 + 9% x
30 = 1%x
x = $3000

Ratio of $x to the sum of his two other investments is = \(\frac{3000}{(15000+6000)}= \frac{3}{21} = \frac{1}{7}\)

IMO D

Solution:

Using the formula Interest = Rate x Time, we can create the equation for the total interest earned by the 3 investments, letting x equal the amount invested at 8% simple interest:

15,000 * 0.1 + 6,000 * 0.07 + x * 0.08 = (15,000 + 6,000 + x) * 0.09

1,500 + 420 + 0.08x = 1,890 + 0.09x

30 = 0.01x

3,000 = x

Therefore, the desired ratio is 3,000 : (15,000 + 6,000) = 3,000 : 21,000 = 1:7.

Answer: D

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