Solve the problem.
-You have $50,000 and wish to invest part at 10% and the rest at 6%. How much should be invested at each rate to produce the same return as if it all had been invested at 9%?
A) $37,000 at 10%, $13,000 at 6%
B) $37,500 at 10%, $12,500 at 6%
C) $37,500 at 6%, $12,500 at 10%
D) $37,000 at 6%, $13,000 at 10%

Question

Quizplus · Accepted Answer

Let x be the amount invested at 10% and y be the amount invested at 6%. We know that:
x + y = 50,000   (equation 1)
0.1x + 0.06y = 0.09(50,000)   (equation 2)
Solving for y in equation 1, we get y = 50,000 - x. Substituting this into equation 2, we get:
0.1x + 0.06(50,000-x) = 4,500
0.04x + 3,000 = 4,500
0.04x = 1,500
x = 37,500
Substituting x back into y = 50,000 - x, we get y = 12,500. Therefore, the optimal allocation is $37,500 invested at 10% and $12,500 invested at 6%. This will produce a return of:
0.1(37,500) + 0.06(12,500) = 0.09(50,000) = 4,500
which is the same as if the entire amount had been invested at 9%. The weighted average of the two investment rates should be 9%, since that's the rate we're trying to match. Let x be the amount invested at 10%. Then, we know that $50,000 - x is the amount invested at 6%. Using the weighted average formula, we can set up the following equation:
0.1x + 0.06($50,000 - x) = 0.09($50,000)
Solving for x, we get x = $37,500, which means $12,500 is invested at 6%. Therefore, the answer is B.

Solve the Problem. -You Have $50,000 and Wish to Invest Part at 10