1. **State the problem:** Benny wants to give each child 34,000 when they turn 21. He invests monthly at 4.2% annual interest compounded monthly. We need to find how much more he must invest monthly for Stephanie than for Seamus so both get 34,000 at age 21. 2. **Identify variables:** - Interest rate per month: $i = \frac{4.2\%}{12} = 0.0035$ - Seamus's current age: 3 years 4 months = $3 + \frac{4}{12} = 3.3333$ years - Stephanie's current age: 6 years 6 months = $6 + \frac{6}{12} = 6.5$ years - Time until 21 for Seamus: $21 - 3.3333 = 17.6667$ years - Time until 21 for Stephanie: $21 - 6.5 = 14.5$ years - Number of months for Seamus: $n_S = 17.6667 \times 12 = 212$ - Number of months for Stephanie: $n_{St} = 14.5 \times 12 = 174$ 3. **Formula for future value of an ordinary annuity:** $$FV = P \times \frac{(1+i)^n - 1}{i}$$ where $P$ is the monthly payment, $i$ is monthly interest rate, $n$ is number of months. 4. **Set up equations for each child:** For Seamus: $$34000 = P_S \times \frac{(1+0.0035)^{212} - 1}{0.0035}$$ For Stephanie: $$34000 = P_{St} \times \frac{(1+0.0035)^{174} - 1}{0.0035}$$ 5. **Calculate annuity factors:** Calculate $(1+0.0035)^{212}$: $$ (1.0035)^{212} \approx e^{212 \times \ln(1.0035)} \approx e^{212 \times 0.003494} = e^{0.740} \approx 2.096$$ Calculate $(1+0.0035)^{174}$: $$ (1.0035)^{174} \approx e^{174 \times 0.003494} = e^{0.607} \approx 1.835$$ 6. **Calculate annuity denominators:** For Seamus: $$\frac{2.096 - 1}{0.0035} = \frac{1.096}{0.0035} \approx 313.14$$ For Stephanie: $$\frac{1.835 - 1}{0.0035} = \frac{0.835}{0.0035} \approx 238.57$$ 7. **Solve for monthly payments:** For Seamus: $$P_S = \frac{34000}{313.14} \approx 108.56$$ For Stephanie: $$P_{St} = \frac{34000}{238.57} \approx 142.48$$ 8. **Find how much more Stephanie must invest monthly:** $$142.48 - 108.56 = 33.92$$ **Final answer:** Benny must invest approximately **33.92** more per month for Stephanie than for Seamus to ensure both receive 34,000 at age 21.