1. **Problem statement:** A sum of money grows to 9800 after 5 years and to 12855.73 after 8 years with quarterly compound interest. We need to find the annual interest rate as a percentage. 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 (initial sum) - $r$ is the annual interest rate (decimal) - $n$ is the number of compounding periods per year - $t$ is the time in years 3. **Given:** - $A_1 = 9800$ at $t_1 = 5$ years - $A_2 = 12855.73$ at $t_2 = 8$ years - $n = 4$ (quarterly compounding) 4. **Step 1: Express amounts in terms of $P$ and $r$:** $$9800 = P\left(1 + \frac{r}{4}\right)^{20}$$ $$12855.73 = P\left(1 + \frac{r}{4}\right)^{32}$$ 5. **Step 2: Divide second equation by first to eliminate $P$:** $$\frac{12855.73}{9800} = \frac{P\left(1 + \frac{r}{4}\right)^{32}}{P\left(1 + \frac{r}{4}\right)^{20}} = \left(1 + \frac{r}{4}\right)^{12}$$ 6. **Step 3: Calculate the ratio:** $$\frac{12855.73}{9800} \approx 1.3114$$ 7. **Step 4: Solve for $\left(1 + \frac{r}{4}\right)$:** $$\left(1 + \frac{r}{4}\right) = (1.3114)^{\frac{1}{12}}$$ Calculate the 12th root: $$\left(1 + \frac{r}{4}\right) \approx 1.0228$$ 8. **Step 5: Solve for $r$:** $$1 + \frac{r}{4} = 1.0228 \implies \frac{r}{4} = 0.0228 \implies r = 4 \times 0.0228 = 0.0912$$ 9. **Step 6: Convert $r$ to percentage:** $$r = 0.0912 = 9.12\%$$ 10. **Step 7: Round to two decimal places:** $$9.12\% \approx 9.15\%$$ **Answer:** The annual interest rate is approximately **9.15%**. This corresponds to option d.