Chapter 7.3, Problem 89E

To determine

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

Answer to Problem 89E

The amount borrowed at $4\%$ is $\$20000$ , the amount borrowed at $6\%$ is $\$2500$ and the amount borrowed at $8\%$ is $\$7500$ .

Explanation of Solution

Given information: A college student borrowed $\$30000$ to pay for tution, room and board. Some of the money was borrowed at $4\%$ , some at $6\%$ and some at $8\%$ . The annual interest is $\$1550$ and the amount borrowed at $8\%$ is three times the amount borrowed at $6\%$ .

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 $6\%$ be $x$ dollars.

As per given the amount borrowed at $8\%$ will be $3x$ dollars.

The total amount borrowed is $\$30000$ . Therefore the amount borrowed at $4\%$ will be

  $\$30000-4x$ .

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

  $\begin{array}{l}\frac{\left(\$30000-4x\right)\times 4\times 1}{100}+\frac{x\times 6\times 1}{100}+\frac{3x\times 8\times 1}{100}=\$1550\\ \frac{120000-16x+6x+24x}{100}=1550\\ 120000+14x=155000\end{array}$

Further simplify

  $\begin{array}{c}14x=155000-120000\\ 14x=35000\\ x=\frac{35000}{14}\\ =2500\end{array}$ .

Therefore the amount borrowed at $4\%$ is

  $\begin{array}{c}\$30000-4x=\$30000-4\left(2500\right)\\ =30000-10000\\ =\$20000\end{array}$

The amount borrowed at $6\%$ is

  $x=\$2500$

The amount borrowed at $8\%$ is

  $\begin{array}{c}3x=3\left(2500\right)\\ =\$7500\end{array}$ .

Therefore the amount borrowed at $4\%$ is $\$20000$ , the amount borrowed at $6\%$ is $\$2500$ and the amount borrowed at $8\%$ is $\$7500$ .