Chapter 7.1, Problem 25E

To determine

The amount invested at each rate by writing and solving a system ofequations,if total annual interest earnedfrom a total of $12,000 invested in two funds paying thegiven rates of simple interest is 500$ at rates 2% and 6%.

Answer to Problem 25E

The amount invested at rate 2% is 3333.33 $ and at rate 5% is 8666.66 $

Explanation of Solution

Given info:

Total amount deposited = 12000 $

Total interest received = 500 $

  $\begin{array}{l}{\text{Interest rate, r}}_1\text{ = }2\%\\ {\text{Interest rate, r}}_2\text{ = 5}\%\end{array}$

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

  $\begin{array}{l}{I}_1={\mathrm{P}}_1{r}_{1}t\\ I_1={\mathrm{P}}_1\times \frac{2}{100}\times 1\\ I_1=0.02{\mathrm{P}}_1\end{array}$

Interest of rate 2

  $\begin{array}{l}{I}_2={\mathrm{P}}_2{r}_{2}t\\ I_2={\mathrm{P}}_2\times \frac{5}{100}\times 1\\ I_2=0.05{\mathrm{P}}_2\end{array}$

Total interest

  $\begin{array}{l}I=500\$\\ I_1+I_2=500\$\\ 0.02{\mathrm{P}}_1+0.05{\mathrm{P}}_2=500\$\\ 2{\mathrm{P}}_1+5{\mathrm{P}}_2=50000\$\end{array}$

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

  $\begin{array}{l}2{\mathrm{P}}_1+5(12000-P_1)=50000\\ 60000-3{\mathrm{P}}_1=50000\\ {\mathrm{P}}_1=\frac{10000}{3}=3333.33\$\end{array}$

And

  $\begin{array}{l}{P}_2=12000-P_1\\ P_2=12000-3333.33=8666.66\$\end{array}$

The amount invested at rate 2% is 3333.33 $ and at rate 5% is 8666.66 $

Conclusion:

Thus,the amount invested at rate 2% is 3333.33 $ and at rate 5% is 8666.66 $