An investor has $450,000 to invest in two types of investments.Type A pays 6% annually and type B pays 7% annually.To have a well-balanced portfolio,the investor imposes the following conditions.At least one-third of the total portfolio is to be allocated to type A investments and at least one-third of the portfolio is to be allocated to type B investments.What is the optimal amount that should be invested in each investment?
A)$160,000 in type A (6%),$290,000 in type B (7%)
B)$0 in type A (6%),$450,000 in type B (7%)
C)$450,000 in type A (6%),$0 in type B (7%)
D)$300,000 in type A (6%),$150,000 in type B (7%)
E)$150,000 in type A (6%),$300,000 in type B (7%)

Question

An investor has $450,000 to invest in two types of investments.Type A pays 6% annually and type B pays 7% annually.To have a well-balanced portfolio,the investor imposes the following conditions.At least one-third of the total portfolio is to be allocated to type A investments and at least one-third of the portfolio is to be allocated to type B investments.What is the optimal amount that should be invested in each investment? 
A)$160,000 in type A (6%),$290,000 in type B (7%) 
B)$0 in type A (6%),$450,000 in type B (7%) 
C)$450,000 in type A (6%),$0 in type B (7%) 
D)$300,000 in type A (6%),$150,000 in type B (7%) 
E)$150,000 in type A (6%),$300,000 in type B (7%)

Quizplus · Accepted Answer

Let x be the amount of money invested in type A, and y be the amount of money invested in type B.
From the problem statement, we have two inequalities:
x ≥ 1/3(450,000) = 150,000
y ≥ 1/3(450,000) = 150,000
We also know that the total amount invested is $450,000, so x + y = 450,000.
We want to maximize the total return on investment, which is given by:
0.06x + 0.07y
Using the constraint x + y = 450,000, we can rewrite this as:
0.06x + 0.07(450,000 - x) = 0.07(450,000) - 0.01x
Differentiating with respect to x, we get:
-0.01
This is negative, which means that the total return on investment decreases as x increases. Therefore, we want to invest as little as possible in type A, while still satisfying the inequality x ≥ 150,000. The optimal amount of money to invest in each investment is:
$150,000 in type A (6%), $300,000 in type B (7%).

An Investor Has $450,000 to Invest in Two Types of Investments.Type