Question: Mara and Braylen are both age $75$. When Mara was $23$ years old, she began depositing $1300$ per year into a savings account. She made deposits for the first $10$ years, at which point she was forced to stop making deposits. However, she left her money in the account, where it continued to earn interest for the next $42$ years. Braylen didn't start saving until he was $45$ years old, but for the next $30$ years he made annual deposits of $1300$. Assume that both accounts earned an average annual return of $5\%$ (compounded once a year). Complete parts (a) through (d) below.

1. **State the problem:** We want to find the future values of Mara's and Braylen's savings accounts given their deposit schedules and interest rate. 2. **Formulas and rules:** The future value of an annuity (series of equal payments) compounded annually is given by: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where $P$ is the annual payment, $r$ is the annual interest rate (as a decimal), and $n$ is the number of payments. For money left to grow without additional deposits, the future value is: $$FV = PV \times (1 + r)^t$$ where $PV$ is the present value at the start of the growth period, and $t$ is the number of years. 3. **Mara's deposits and growth:** - Deposits: $P = 1300$ - Interest rate: $r = 0.05$ - Number of deposits: $n = 10$ Calculate the value of the annuity after $10$ years: $$FV_{10} = 1300 \times \frac{(1 + 0.05)^{10} - 1}{0.05}$$ Calculate: $$(1.05)^{10} = 1.628894626777442$$ So: $$FV_{10} = 1300 \times \frac{1.628894626777442 - 1}{0.05} = 1300 \times \frac{0.628894626777442}{0.05} = 1300 \times 12.57789253554884 = 16351.2602962135$$ 4. **Mara's money grows for 42 more years without deposits:** $$FV_{52} = FV_{10} \times (1.05)^{42}$$ Calculate: $$(1.05)^{42} = 7.039988$$ So: $$FV_{52} = 16351.2602962135 \times 7.039988 = 115063.68$$ 5. **Braylen's deposits:** - Starts at age $45$, deposits for $30$ years - Same $P = 1300$, $r = 0.05$, $n = 30$ Calculate: $$FV_{30} = 1300 \times \frac{(1.05)^{30} - 1}{0.05}$$ Calculate: $$(1.05)^{30} = 4.321942375$$ So: $$FV_{30} = 1300 \times \frac{4.321942375 - 1}{0.05} = 1300 \times \frac{3.321942375}{0.05} = 1300 \times 66.4388475 = 86370.50$$ 6. **Summary:** - Mara's account value at age 75: approximately $115063.68$ - Braylen's account value at age 75: approximately $86370.50$ 7. **Interpretation:** Mara's early start and long growth period without deposits results in a larger final amount despite fewer total deposits. **Number of distinct questions found:** 4 (parts a through d) but only the first problem solved here.