1. **State the problem:** We want to find the nominal interest rate compounded quarterly that grows $22,000 to $37,984.34 in 20 years. 2. **Formula used:** The compound interest formula is $$A = P \left(1 + \frac{r}{n}\right)^{nt}$$ where: - $A$ is the amount after time $t$ - $P$ is the principal amount - $r$ is the nominal annual interest rate (decimal) - $n$ is the number of compounding periods per year - $t$ is the number of years 3. **Given values:** - $A = 37984.34$ - $P = 22000$ - $n = 4$ (quarterly compounding) - $t = 20$ 4. **Plug values into the formula:** $$37984.34 = 22000 \left(1 + \frac{r}{4}\right)^{4 \times 20} = 22000 \left(1 + \frac{r}{4}\right)^{80}$$ 5. **Isolate the compound factor:** $$\frac{37984.34}{22000} = \left(1 + \frac{r}{4}\right)^{80}$$ $$1.72656 = \left(1 + \frac{r}{4}\right)^{80}$$ 6. **Take the 80th root of both sides:** $$\sqrt[80]{1.72656} = 1 + \frac{r}{4}$$ 7. **Calculate the 80th root:** $$1 + \frac{r}{4} = 1.007\quad (rounded)$$ 8. **Solve for $r$:** $$\frac{r}{4} = 1.007 - 1 = 0.007$$ $$r = 0.007 \times 4 = 0.028$$ 9. **Convert to percentage:** $$r = 0.028 \times 100 = 2.8\%$$ **Final answer:** The nominal interest rate compounded quarterly is **2.8%**.