Question

Phyllis invested \(12000, a portion earning a simple interest rate of$4\frac{1}{2}\%$ per year and the rest earning a rate of 4% per year. After 1 year the total interest earned on these investments was \)525. How much money did she invest at each rate?

Short answer

$9,000 invested at$4\frac{1}{2}\%$ and $3,000 invested at 4%.

Step-by-step solution

Step 1. Definition of simple interest formula.

Simple interest is a method of calculating interest based on the original amount invested or loaned called the principal. The annual interest is given bywhere P is the principal amount invested or borrowed, r is the interest rate andt is the number of years elapsed.

Step 2. Make the equation.

Let x be the amount invested at $4\frac{1}{2}\%$.

Use the total of 12,000 to write the amount invested at 4% in terms of x.

$12,000-x$

Interest earned at $4\frac{1}{2}\%$:

$x\left(0.045\right)\left(1\right)=0.045x$

Interest earned at 4%:

$\left(12000-x\right)\left(0.04\right)\left(1\right)=480-0.04x$

Add the interest earned from both investments and set it equal to the total interest earned.

$\text{Totalinterest}=\left(\text{Interestearnedof4}\frac{1}{2}\%\right)+\left(\text{Interestearnedat4\%}\right)$

$525=0.045x+\left(480-0.04x\right)$

Step 3. Solve the equation.

Solve the equation to find the amount invested at $4\frac{1}{2}\%$:

$\begin{array}{l}525=0.045x+480-0.04x\\ 525=0.005x+480\end{array}$

Subtract 480 from both sides:

$\begin{array}{l}525-480=0.005x+480-480\\ 0.005x=45\end{array}$

Divide both sides by 0.005:

$\begin{array}{l}\frac{0.005x}{0.005}=\frac{45}{0.005}\\ x=9,000\end{array}$

Now, subtract 9000 from 12,000 to find the amount invested at 4%

$12,000-9,000=3,000$.

Hence, $9,000 invested at$4\frac{1}{2}\%$ and $3,000 invested at 4%.