A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,400, and risk factors make it prudent to invest some money in all three investments. What is the minimum amount that will be invested at 7%, and in this case how much will be invested at the other rates?
A)$100,000 at 8%, $0 at 7%, $120,000 at 10%
B)$90,000 at 8%, $110,000 at 7%, $20,000 at 10%
C)$95,000 at 8%, $115000 at 7%, $10,000 at 10%
D)$100,000 at 8%, $120,000 at 7%, $0 at 10%
E)$90,000 at 8%, $130,000 at 7%, $0 at 10%

Question

A trust account manager has $220,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $16,400, and risk factors make it prudent to invest some money in all three investments. What is the minimum amount that will be invested at 7%, and in this case how much will be invested at the other rates? ​
A)$100,000 at 8%, $0 at 7%, $120,000 at 10%
B)$90,000 at 8%, $110,000 at 7%, $20,000 at 10%
C)$95,000 at 8%, $115000 at 7%, $10,000 at 10%
D)$100,000 at 8%, $120,000 at 7%, $0 at 10%
E)$90,000 at 8%, $130,000 at 7%, $0 at 10%

Quizplus · Accepted Answer

Let the amount invested at 8% be x, the amount invested at 7% be y and the amount invested at 10% be z.

We know that:

x + y + z = $220,000 (total amount to be invested)

0.08x + 0.07y + 0.10z = $16,400 (investment goal)

We are also given that some money must be invested in all three investments, which means that y must be greater than zero.

To minimize the amount invested at 7%, we need to maximize the amount invested at 8% and 10%.

Let's try to solve for x and z:

x + y + z = $220,000

z = $220,000 - x - y

0.08x + 0.07y + 0.10z = $16,400

0.08x + 0.07y + 0.10($220,000 - x - y) = $16,400

0.02x - 0.03y = $2,400

y = (2/3)x - $8000

We know that y must be greater than zero, so:

(2/3)x - $8000 > 0

(2/3)x > $8,000

x > $12,000

So the minimum amount that will be invested at 7% is when x = $100,000 and z = $120,000:

$100,000 at 8%, $120,000 at 10%, and $0 at 7%.

Therefore, the answer is D.

A Trust Account Manager Has $220,000 to Be Invested