1. **Problem Statement:** We have an initial sum invested that grows to 9800 after 5 years and to 12855.73 after 8 years at the same quarterly compounded interest rate. We need to find the annual interest rate $i$ as a percentage. 2. **Formula for compound interest compounded quarterly:** $$A = P\left(1 + \frac{i}{4}\right)^{4t}$$ where $A$ is the amount after $t$ years, $P$ is the principal, $i$ is the annual interest rate (decimal), and interest is compounded quarterly (4 times a year). 3. **Using the given data:** Let $P$ be the initial sum. At $t=5$ years: $$9800 = P\left(1 + \frac{i}{4}\right)^{20}$$ At $t=8$ years: $$12855.73 = P\left(1 + \frac{i}{4}\right)^{32}$$ 4. **Divide the second equation by the first to eliminate $P$:** $$\frac{12855.73}{9800} = \frac{P\left(1 + \frac{i}{4}\right)^{32}}{P\left(1 + \frac{i}{4}\right)^{20}} = \left(1 + \frac{i}{4}\right)^{12}$$ Calculate the left side: $$\frac{12855.73}{9800} \approx 1.3114$$ So: $$\left(1 + \frac{i}{4}\right)^{12} = 1.3114$$ 5. **Solve for $1 + \frac{i}{4}$:** Take the 12th root: $$1 + \frac{i}{4} = (1.3114)^{\frac{1}{12}}$$ Calculate: $$ (1.3114)^{\frac{1}{12}} \approx 1.0228$$ 6. **Find $i$:** $$1 + \frac{i}{4} = 1.0228 \implies \frac{i}{4} = 0.0228 \implies i = 4 \times 0.0228 = 0.0912$$ 7. **Convert to percentage and round:** $$i = 0.0912 = 9.12\%$$ **Final answer:** The annual interest rate is **9.12%**.