Susan invested $5,000 in a savings account that paid quarterly interest. After six years, the money had accumulated to $6,539.96. What was the annual interest rate?

Steps to Solve

1. Identify the variables in the compound interest formula

The compound interest formula is given by:

$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$

Where:

• ( A = 6539.96 ) (the amount after time)
• ( P = 5000 ) (the principal amount)
• ( r ) (the annual interest rate, which we need to find)
• ( n = 4 ) (the number of compounding periods per year, quarterly)
• ( t = 6 ) (the number of years)

2. Rearranging the formula to solve for ( r )

We need to isolate ( r ). Start by dividing both sides by ( P ):

$$ \frac{A}{P} = \left(1 + \frac{r}{n}\right)^{nt} $$

This gives us:

$$ \frac{6539.96}{5000} = \left(1 + \frac{r}{4}\right)^{4 \cdot 6} $$

Calculate ( \frac{6539.96}{5000} ):

$$ \frac{6539.96}{5000} = 1.307992 $$

So we have:

$$ 1.307992 = \left(1 + \frac{r}{4}\right)^{24} $$

3. Taking the 24th root

Now we take the 24th root on both sides to solve for ( 1 + \frac{r}{4} ):

$$ 1 + \frac{r}{4} = (1.307992)^{\frac{1}{24}} $$

Calculate ( (1.307992)^{\frac{1}{24}} ):

Using a calculator, we find:

$$ (1.307992)^{\frac{1}{24}} \approx 1.011710 $$

Thus:

$$ 1 + \frac{r}{4} \approx 1.011710 $$

4. Isolating ( r )

Now, subtract 1 from both sides:

$$ \frac{r}{4} \approx 1.011710 - 1 $$

Which simplifies to:

$$ \frac{r}{4} \approx 0.011710 $$

Now, multiply both sides by 4 to find ( r ):

$$ r \approx 0.04684 $$

5. Converting ( r ) into a percentage

To express ( r ) as a percentage, multiply by 100:

$$ r \approx 0.04684 \times 100 = 4.684% $$