1. **State the problem:** We are given the initial value of a house $P=265000$, the final value after 4 years $F=370000$, and the time period $t=4$ years. We need to find the annual percentage increase rate $r$ using the formula: $$F = P \left(1 + \frac{r}{100}\right)^t$$ 2. **Write the formula with given values:** $$370000 = 265000 \left(1 + \frac{r}{100}\right)^4$$ 3. **Isolate the growth factor:** Divide both sides by 265000: $$\frac{370000}{265000} = \left(1 + \frac{r}{100}\right)^4$$ Intermediate step showing cancellation: $$\frac{\cancel{370000}}{\cancel{265000}} = \left(1 + \frac{r}{100}\right)^4$$ Calculate the fraction: $$1.396226415 = \left(1 + \frac{r}{100}\right)^4$$ 4. **Take the fourth root of both sides to solve for $1 + \frac{r}{100}$:** $$\sqrt[4]{1.396226415} = 1 + \frac{r}{100}$$ Calculate the root: $$1.086 = 1 + \frac{r}{100}$$ 5. **Solve for $r$:** $$\frac{r}{100} = 1.086 - 1 = 0.086$$ Multiply both sides by 100: $$r = 0.086 \times 100 = 8.6$$ 6. **Final answer:** The annual percentage increase rate $r$ is **8.6%** (correct to 1 decimal place).