Question

. An individual wishes to invest $5000 over the next year in two types of investment: Investment A yields 5%, and investment B yields 8%. Market research recommends an allocation of at least 25% in A and at most 50% in B. Moreover, investment in A should be at least half the investment in B. How should the fund be allocated to the two investments?

Answer 1

Answer:

The amount invested at Investment A must be greater than or equal to $2,750

The amount invested at Investment B must be less than or equal to $2,250

Step-by-step explanation:

Let

x -----> the amount invested at Investment A yields 5%

y -----> the amount invested at Investment B yields 8%

we know that

$x+y=5,000$

$x=5,000-y$  -----> equation A

$x \geq 0.25(5,000)$

$x \geq \ 1,250$ -----> inequality B

$y \leq 0.50(5,000)$

$y \leq \ 2,250$ -----> inequality C

$x \geq \frac{1}{2}y$ -----> inequality D

Substitute equation A in the inequality D and solve for y

$5,000-y \geq \frac{1}{2}y$

Multiply by 2 both sides

$10,000-2y \geq y$

Multiply by -1 both sides

$-10,000+2y \leq -y$

Adds y both sides

$-10,000+2y+y \leq 0$

$-10,000+3y \leq 0$

adds 10,000 both sides

$3y \leq 10,000$

Divide by 3 both sides

$y \leq \ 3,333.33$ -----> inequality E

therefore

Solve for y

we have

$y \leq \ 2,250$ -----> inequality C

$y \leq \ 3,333.33$ -----> inequality E

The solution of inequality C and inequality E is

$y \leq \ 2,250$

For y=2,250

x=5,000-y ----> x=5,000-2,250=2,750

so

$x \geq \ 2,750$ -----> inequality F

Solve for x

we have

$x \geq \ 1,250$ -----> inequality B

$x \geq \ 2,750$ -----> inequality F

The solution of inequality B and inequality F is

$x \geq \ 2,750$

therefore

The amount invested at Investment A must be greater than or equal to $2,750

The amount invested at Investment B must be less than or equal to $2,250