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:** Compound interest is calculated using the formula: $$ A = P(1 + r)^t $$ where $A$ is the amount owed after $t$ years, $P$ is the principal amount, $r$ is the annual interest rate, and $t$ is the time in years. 3. **Find the interest rate $r$:** From 2002 to 2007 is 5 years. We have: $$ 592.39 = 427.33(1 + r)^5 $$ Divide both sides by 427.33: $$ \frac{592.39}{427.33} = (1 + r)^5 $$ Calculate the left side: $$ 1.3869 \approx (1 + r)^5 $$ Take the 5th root of both sides: $$ 1 + r = \sqrt[5]{1.3869} $$ Calculate: $$ 1 + r \approx 1.068 \Rightarrow r \approx 0.068 $$ So the annual interest rate is approximately 6.8%. 4. **Calculate amount owed after next 3 years (2007 to 2010):** $$ A = 592.39(1 + 0.068)^3 $$ Calculate: $$ A = 592.39 \times 1.068^3 = 592.39 \times 1.2187 \approx 721.56 $$ 5. **Calculate interest gathered over these 3 years:** Interest = Amount after 3 years - Amount at 2007 $$ 721.56 - 592.39 = 129.17 $$ **Final answer:** The loan gathered approximately £129.17 in interest over the next 3 years.