1. **Problem statement:** Caleb made monthly deposits into a savings account for 5 years. The account earns 5.10% interest compounded quarterly and the balance after 5 years is 10625. We need to find the monthly deposit amount. 2. **Given:** - Annual nominal interest rate $r = 0.051$ (5.10%) compounded quarterly. - Number of years $t = 5$. - Future value $FV = 10625$. - Deposits made monthly, so number of deposits $n = 5 \times 12 = 60$. 3. **Find the effective monthly interest rate:** Quarterly interest rate $i_q = \frac{0.051}{4} = 0.01275$. Effective monthly interest rate $i_m = (1 + i_q)^{\frac{1}{3}} - 1 = (1 + 0.01275)^{\frac{1}{3}} - 1$. Calculate: $$i_m = (1.01275)^{0.3333} - 1 \approx 1.00424 - 1 = 0.00424$$ 4. **Use the future value of an ordinary annuity formula:** $$FV = P \times \frac{(1 + i_m)^n - 1}{i_m}$$ Where $P$ is the monthly deposit. Rearranged to solve for $P$: $$P = \frac{FV \times i_m}{(1 + i_m)^n - 1}$$ 5. **Calculate $P$:** Calculate $(1 + i_m)^n = (1.00424)^{60} \approx e^{60 \times \ln(1.00424)} \approx e^{0.254} \approx 1.289$. Then: $$P = \frac{10625 \times 0.00424}{1.289 - 1} = \frac{45.05}{0.289} \approx 155.92$$ So, the monthly deposit is approximately $155.92$. --- 6. **Part b: How long to grow $10,625$ to $40,530$ with same deposits and interest rate?** Given: - Future value $FV = 40530$. - Monthly deposit $P = 155.92$. - Monthly interest rate $i_m = 0.00424$. - Initial balance is zero, but deposits continue. Use the annuity formula: $$FV = P \times \frac{(1 + i_m)^n - 1}{i_m}$$ Solve for $n$: $$\frac{FV \times i_m}{P} = (1 + i_m)^n - 1$$ $$ (1 + i_m)^n = 1 + \frac{FV \times i_m}{P}$$ $$ n = \frac{\ln\left(1 + \frac{FV \times i_m}{P}\right)}{\ln(1 + i_m)}$$ Calculate: $$1 + \frac{40530 \times 0.00424}{155.92} = 1 + \frac{171.77}{155.92} = 1 + 1.101 = 2.101$$ $$n = \frac{\ln(2.101)}{\ln(1.00424)} = \frac{0.742}{0.00423} \approx 175.4 \text{ months}$$ Convert months to years and months: $$175.4 \div 12 = 14 \text{ years and } 7.4 \text{ months}$$ Approximately 14 years and 8 months. --- **Final answers:** - a) Monthly deposit = $155.92$ - b) Time to reach $40,530 = 14$ years and 8 months