Question

The controller​ (money manager) for a small company puts some money in a bank account paying 3​% per year. He uses some additional​ money, amounting to one third the amount placed in the​ bank, to buy bonds paying 4​% per year. With the balance of the​ funds, he buys a 10​% certificate of deposit. The first year the investments bring a return of ​$865. If the total of the investments is​ $10,000 how much is invested at each​ rate?

Answer 1

Answer:

$1,500 is invested in a bank account paying 3% per year.
$500 is used to buy bonds paying 4% per year.
$8,000 is used to buy 10% certificate of deposit   

Explanation:

Let 

x = money used to invest in a bank account paying 3% per year
y = money used to buy 10% certificate of deposit

Because the money used to buy bonds (that pays 4% per year) is equal to one-third of the money used to invest in a bank account paying 3% per year,

x/3 = money used to buy bonds (that pays 4% per year)

Because the total investment is $10,000, we add all the variables that represents the amount of money for each investments. So,

$x + \frac x 3 + y = 10,000$    
$\frac {4x} 3 + y = 10,000$
$4x + 3y = 30,000 \text{ (multiply both sides by 3)}$      (1)

To compute the return of each investment, we change the rate to decimal and multiply it to the amount of investment because the problem only states the return of first year of investment. Then,

0.03x = return from a bank account paying 3% per year
0.04(x/3) = return from a bond paying 4% per year
0.1y = return from 10% certificate of deposit

Since the total return for the first year of investment is $865, we just add the returns of each investment and create the following equation:

$0.03x + 0.04(\frac x 3) + 0.1y = 865 \\ 0.03x + \frac{0.04x}{3} + 0.1y = 865 \\ 0.09x + 0.04x + 0.3y = 2,595 \text{ (multiply both sides by 3)} \\ 0.13x + 0.3y = 2,595 \\ 1.3x + 3y =25,950 \text{ (multiply both sides by 10) (2)}$

Note: In equation (2), we want to have an equation with 3y so that when we subtract equation (1) to equation (2), we only have an equation involving x, which is easier to solve.

By subtracting equation (1) to equation (2),

(4x + 3y) - (1.3x + 3y) = 30,000 - 25,950
2.7x = 4,050
x = $1,500 = amount deposited in a bank account paying 3% per year

So,

x/3 = 1500/3
x/3 = $500 = amount used to buy bonds paying 4% per year

Using the value of x = 1,500 and replacing the value of x in equation (1), 

4x + 3y = 30,000
4(1,500) + 3y = 30,000
6,000 + 3y = 30,000
3y = 24,000
y = $8,000 = money used to buy 10% certificate of deposit