Chapter 7.1, Problem 28E

To determine

The amount invested at each rate by writing and solving a system ofequations,if total annual interest earnedfrom a total of $12,000 invested in two funds paying thegiven rates of simple interest is 254$ at rates 1.75% and 2.25%.

Answer to Problem 28E

The amount invested at rate 1.75% is 3200 $ and at rate 2.25% is 8800 $

Explanation of Solution

Given info:

Total amount deposited = 12000 $

Total interest received = 254 $

  $\begin{array}{l}{\text{Interest rate, r}}_1\text{ = }1.75\%\\ {\text{Interest rate, r}}_2\text{ = 2}\text{.25}\%\end{array}$

Time = 1 Year

Formula used:

Simple interest

  $I=\mathrm{Pr}t$

where is the P principal, r is the annual interestrate, and t is the time.

Calculation:

Let $P_1$ be the principal of rate $r_1$ and $P_2$ be the principal of rate $r_2$ ,

We have

Total principal

  $\begin{array}{l}{P}_1+P_2=12000\$\\ P_2=12000-P_1\end{array}$

Interest of rate 1

  $\begin{array}{l}{I}_1={\mathrm{P}}_1{r}_{1}t\\ I_1={\mathrm{P}}_1\times \frac{1.75}{100}\times 1\\ I_1=0.0175{\mathrm{P}}_1\end{array}$

Interest of rate 2

  $\begin{array}{l}{I}_2={\mathrm{P}}_2{r}_{2}t\\ I_2={\mathrm{P}}_2\times \frac{2.25}{100}\times 1\\ I_2=0.0225{\mathrm{P}}_2\end{array}$

Total interest

  $\begin{array}{l}I=254\$\\ I_1+I_2=254\$\\ 0.0175{\mathrm{P}}_1+0.0225{\mathrm{P}}_2=254\$\\ 175{\mathrm{P}}_1+225{\mathrm{P}}_2=2540000\$\end{array}$

Substituting $P_2=12000-P_1$ in the above equation

  $\begin{array}{l}175{\mathrm{P}}_1+225(12000-P_1)=2540000\\ \text{2700000}-50{\mathrm{P}}_1=2540000\\ {\mathrm{P}}_1=\frac{\text{160000}}{50}=\text{3200}\$\end{array}$

And

  $\begin{array}{l}{P}_2=12000-\text{3200}\\ P_2=12000-\text{3200}=8800\$\end{array}$

The amount invested at rate 1.75% is 3200 $ and at rate 2.25% is 8800 $

Conclusion:

Thus,the amount invested at rate 1.75% is 3200 $ and at rate 2.25% is 8800 $