Chapter 7.3, Problem 90E

To determine

To calculate: The amount borrowed by the company at each rate.

Answer to Problem 90E

The amount borrowed at $8\%$ is $\$300000$ , the amount borrowed at $9\%$ is $\$400000$ and the amount borrowed at $10\%$ is $\$75000$ .

Explanation of Solution

Given information: A small corporation borrowed $\$775000$ to expand its software line. Some of the money was borrowed at $8\%$ , some at $9\%$ and some at $10\%$ . The annual interest is $\$67500$ and the amount borrowed at $8\%$ is four times the amount borrowed at $10\%$ .

Formula used:

Use the formula of interest

  $\text{I}=\frac{\text{PRT}}{100}$ .

where $\text{P}$ is the amount borrowed, $\text{R}$ is interest rate and $\text{T}$ is time period in years.

Calculation:

Assume the amount borrowed at $10\%$ be $x$ dollars.

As per given the amount borrowed at $8\%$ will be $4x$ dollars.

The total amount borrowed is $\$775000$ . Therefore the amount borrowed at $9\%$ will be

  $\$775000-5x$ .

The total interest for all the amounts is the sum of interests for each amount. Therefore,

  $\frac{{\text{P}}_{\text{1}}{\text{R}}_{\text{1}}{\text{T}}_{\text{1}}}{100}+\frac{{\text{P}}_{\text{2}}{\text{R}}_{\text{2}}{\text{T}}_{\text{2}}}{100}+\frac{{\text{P}}_{\text{3}}{\text{R}}_{\text{3}}{\text{T}}_{\text{3}}}{100}$ is equal to the total interest.

Substitute the values in the formula

  $\begin{array}{l}\frac{4x\times 8\times 1}{100}+\frac{\left(775000-5x\right)\times 9\times 1}{100}+\frac{x\times 10\times 1}{100}=\$67500\\ \frac{32x+6975000-45x+10x}{100}=67500\\ 6975000-3x=6750000\end{array}$

Further simplify

  $\begin{array}{c}3x=6975000-6750000\\ 3x=225000\\ x=\frac{225000}{3}\\ =75000\end{array}$ .

Therefore the amount borrowed at $8\%$ is

  $\begin{array}{c}4x=4\left(75000\right)\\ =300000\\ =\$300000\end{array}$

The amount borrowed at $9\%$ is

  $\begin{array}{c}\$775000-5x=\$775000-5\left(75000\right)\\ =\$775000-\$375000\\ =\$400000\end{array}$

The amount borrowed at $10\%$ is

  $x=\$75000$ .

Therefore the amount borrowed at $8\%$ is $\$300000$ , the amount borrowed at $9\%$ is $\$400000$ and the amount borrowed at $10\%$ is $\$75000$ .