Write an appropriate system of equations to solve the following problem.

Question

Suppose you receive a total of $1520 a year in interest from three investments. The interest rates for the three investments are 5%, 7%, and 8%. The 5% investment is half of the 75 investment, and the 7% investment is $1500 less than the 8% investment. What is the amount of each investment?

Answer 1

Let's begin by defining our variables:

 

X = amount in the 5% account

Y = amount in the 7% account

Z = amount in the 8% account

 

With three variables, we need three equations.  We will find these equations from the information in the problem.  First, the easiest of them is that they total up to 1520.  So if we add them all together we end up with 1520:

 

X + Y + Z = 1520

 

Next, we know that the 5% account is half of the 7% account:

 

X = 0.5 Y  or:  2X = Y

 

Finally, the 7% investment is 1500 less than 8% investment:

 

Y = Z - 1500

 

So our three equations:

 

X + Y + Z = 1520

X = 0.5 Y

Y = Z-1500 or Z = Y+1500

 

Let's replace the X and Z in the first equation with the information from the second and third equations.

 

(0.5Y) + Y + (Y+1500) = 1520     ---     Now we just add up the Ys together

2.5Y + 1500 = 1520     ---     Subtract 1500 from both sides

2.5Y = 20     ---    Solve for Y

 

Y = 8

 

So now we use Y = 8 to find the other two values.

 

X = (0.5Y)  -->  X = (0.5*8)  --->  X = 4

Z = Y+1500   --->   Z = 1508.

 

Finally, we just test the numbers by adding them together.  We should end up with 1520.

 

1508 + 8 + 4 = 1508 + 12 = 1520.  They add up correctly and our answer is verified.

 

$4 in the 5% account

$8 in the 7% account

$1508 in the 8% account