SOLUTION:
Step 1: List the given values.
Since the interest is compounded quarterly, the value of n is 4.
[tex]\begin{aligned} & A = 12,500 \\ & P = 9,000 \\ & n = 4 \\ & t = \text{5 years} \end{aligned}[/tex]
Step 2: Calculate the rate by using the formula for compound interest.
Note that we must multiply the calculated rate by 100 to convert it to percent because rate is usually expressed as percent.
[tex]\begin{aligned} A & = P \left(1 + \frac{r}{n} \right)^{nt} \\ 12,500 & = 9,000 \left(1 + \frac{r}{4} \right)^{4(5)} \\ 12,500 & = 9,000 \left(1 + \frac{r}{4} \right)^{20} \\ \frac{12,500}{9,000} & = \frac{9,000}{9,000} \left(1 + \frac{r}{4} \right)^{20} \\ 1.388889 & = \left(1 + \frac{r}{4} \right)^{20} \\ \sqrt[20]{1.388889} & = \sqrt[20]{\left(1 + \frac{r}{4} \right)^{20}} \\ 1.016561 & = 1 + \frac{r}{4} \\ \frac{r}{4} & = 1.016561 - 1 \\ \frac{r}{4} & = 0.016561 \\ r & = 4(0.016561) \\ r & = 0.066244 \\ r & = 0.066244 \times 100 \\ r & = 6.6244\% \\ & \approx \boxed{6.62\%} \end{aligned}[/tex]
Hence, the rate is 6.62%.
