1. **Stating the problem:** Haleema invests 3700 into a bank account that compounds interest monthly at a rate that makes it equal to another account compounding yearly at 5% after one year. We need to find the amount in her account after 2 years and 11 months. 2. **Formula for compound interest:** 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 amount), - $r$ is the annual interest rate (decimal), - $n$ is the number of compounding periods per year, - $t$ is the time in years. 3. **Find the monthly interest rate:** Given yearly rate $r = 0.05$ and monthly compounding $n=12$, the monthly rate $i$ satisfies: $$\left(1 + i\right)^{12} = 1 + 0.05$$ 4. **Calculate monthly interest rate $i$:** $$1 + i = (1.05)^{\frac{1}{12}}$$ $$i = (1.05)^{\frac{1}{12}} - 1$$ 5. **Calculate $i$ numerically:** $$i \approx 1.05^{0.0833333} - 1 \approx 1.004074 - 1 = 0.004074$$ 6. **Calculate total time in months:** 2 years 11 months = $2 \times 12 + 11 = 35$ months. 7. **Calculate amount after 35 months:** $$A = 3700 \times (1 + 0.004074)^{35}$$ 8. **Calculate the power:** $$ (1.004074)^{35} \approx e^{35 \times \ln(1.004074)} \approx e^{35 \times 0.004066} = e^{0.1423} \approx 1.153$$ 9. **Calculate final amount:** $$A \approx 3700 \times 1.153 = 4266.1$$ 10. **Round to nearest penny:** £4266.10 --- **Next problem:** Center $g$ is increased by 26% and $h$ is decreased by 11%. Find the percentage change in $$\frac{g^2}{2h}$$. 11. **Original expression:** $$\frac{g^2}{2h}$$ 12. **New values:** $$g_{new} = g \times 1.26$$ $$h_{new} = h \times 0.89$$ 13. **New expression:** $$\frac{(1.26g)^2}{2 \times 0.89h} = \frac{1.26^2 g^2}{2 \times 0.89 h} = \frac{1.5876 g^2}{1.78 h}$$ 14. **Simplify ratio of new to old:** $$\frac{\frac{1.5876 g^2}{1.78 h}}{\frac{g^2}{2h}} = \frac{1.5876}{1.78} \times 2 = \frac{1.5876 \times 2}{1.78} = \frac{3.1752}{1.78} \approx 1.783$$ 15. **Percentage change:** $$ (1.783 - 1) \times 100\% = 78.3\%$$ increase. **Final answers:** - Amount in account after 2 years 11 months: £4266.10 - $\frac{g^2}{2h}$ has increased by 78% (to nearest 1%)