1. **State the problem:** A woman deposits 1500 quarterly from age 50 to 60 into an account with 7% interest compounded quarterly. At age 60, she withdraws the amount and deposits it into a mutual fund with 9% interest compounded monthly, then deposits 500 monthly until age 65. We need to find the total amount at age 65. 2. **Calculate amount at age 60:** - Number of quarters from 50 to 60: $n = 10 \times 4 = 40$ - Quarterly interest rate: $i = \frac{7\%}{4} = 0.0175$ - Use future value of an ordinary annuity formula: $$FV = P \times \frac{(1+i)^n - 1}{i}$$ - Substitute values: $$FV = 1500 \times \frac{(1+0.0175)^{40} - 1}{0.0175}$$ 3. **Calculate $(1+0.0175)^{40}$:** $$ (1.0175)^{40} \approx 2.0061 $$ 4. **Calculate numerator:** $$ 2.0061 - 1 = 1.0061 $$ 5. **Calculate fraction:** $$ \frac{1.0061}{0.0175} \approx 57.49 $$ 6. **Calculate future value at 60:** $$ FV = 1500 \times 57.49 = 86235 $$ 7. **Deposit this amount into mutual fund at age 60:** - Interest rate monthly: $j = \frac{9\%}{12} = 0.0075$ - Number of months from 60 to 65: $m = 5 \times 12 = 60$ 8. **Calculate amount after 5 years without additional deposits:** $$ A = 86235 \times (1 + 0.0075)^{60} $$ 9. **Calculate $(1 + 0.0075)^{60}$:** $$ (1.0075)^{60} \approx 1.5657 $$ 10. **Calculate amount after growth:** $$ A = 86235 \times 1.5657 = 134999.5 $$ 11. **Calculate future value of monthly deposits $500$ for 60 months:** $$ FV_{deposits} = 500 \times \frac{(1+0.0075)^{60} - 1}{0.0075} $$ 12. **Calculate numerator:** $$ 1.5657 - 1 = 0.5657 $$ 13. **Calculate fraction:** $$ \frac{0.5657}{0.0075} = 75.43 $$ 14. **Calculate future value of deposits:** $$ FV_{deposits} = 500 \times 75.43 = 37715 $$ 15. **Calculate total amount at age 65:** $$ Total = 134999.5 + 37715 = 172714.5 $$ 16. **Round to nearest dollar:** $$ \boxed{172715} $$ The amount in the account when she reaches age 65 is $172715.