Question

Question 10

A bank loaned out $29,000, part of it at the rate of 13% annual interest, and the rest at 4% annual interest.

The total interest earned for both loans was $2,195.00. How much was loaned at each rate?

Answer 1

$11,500 was invested at 13%.

$17,500 was invested at 4%

Explanation:

This is a simple interest problem.

The simple interest formula is given by:

`E = P*I*t`

In which E is the amount of interest earned, P is the principal(the initial amount of money), I is the interest rate(yearly, as a decimal) and t is the time.

After t years, the total amount of money is:

`T = E + P`

In this question:

Loans totaling 29,000.

P was invested at 13%

29000 - P was invested at 4%.

First investment:

Principal P.

Interest 13% = 0.13.

One year, so t = 1.

So

`E_(1) = P*0.13*1`

`E_(1) = 0.13P`

Second investment:

Principal 29000 - P.

Interest 4% = 0.04.

One year, so t = 1.

So

`E_(2) = (29000-P)*0.04`

The total interest earned for both loans was $2,195.00.

This means that
`E_(1) + E_(2) = 2195`

So

`E_(2) = 2195 - E_(1)`

So we solve the following system:

`E_(1) = 0.13P`

`E_(2) = (29000-P)*0.04`

`2195 - E_(1) = (29000-P)*0.04`

`2195 - 0.13P = 1160 - 0.04P`

`0.09P = 2195 - 1160`

`P = (2195 - 1160)/(0.09)`

`P = 11500`

$11,500 was invested at 13%.

29000 - 11500 = 17500

$17,500 was invested at 4%