1. **State the problem:** Halima took out a loan with compound interest. We know the amount owed at the end of 2002 (£427.33) and at the end of 2007 (£592.39). We need to find how much interest the loan gathered over the next 3 years after 2007. 2. **Identify the formula:** The compound interest formula is: $$ A = P(1 + r)^t $$ where $A$ is the amount after time $t$, $P$ is the principal amount, $r$ is the annual interest rate, and $t$ is the number of years. 3. **Find the interest rate $r$:** We know: $$ 592.39 = 427.33(1 + r)^5 $$ Divide both sides by 427.33: $$ \frac{592.39}{427.33} = (1 + r)^5 $$ $$ \approx 1.3869 = (1 + r)^5 $$ Take the 5th root: $$ 1 + r = \sqrt[5]{1.3869} $$ $$ 1 + r \approx 1.0683 $$ So, $$ r \approx 0.0683 $$ or 6.83% per year. 4. **Calculate amount after 3 more years (from 2007 to 2010):** $$ A = 592.39(1 + 0.0683)^3 $$ $$ A = 592.39 \times 1.0683^3 $$ Calculate $1.0683^3$: $$ 1.0683^3 \approx 1.219 $$ So, $$ A \approx 592.39 \times 1.219 = 721.88 $$ 5. **Calculate interest gathered over these 3 years:** $$ \text{Interest} = 721.88 - 592.39 = 129.49 $$ **Final answer:** The loan gathered approximately £129.49 in interest over the next 3 years after 2007.