Question: Mara and Braylen are both age 75. When Mara was 25 years old, she began depositing 1200 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 40 years. Braylen didn't start saving until he was 47 years old, but for the next 28 years he made annual deposits of 1200. Assume that both accounts earned an average annual return of 6% (compounded once a year). Complete parts (a) through (d) below.
1. **State the problem:**
We have two savers, Mara and Braylen, both age 75 now. Mara started depositing $1200 annually at age 25 for 10 years, then stopped but left the money to grow for 40 more years. Braylen started depositing $1200 annually at age 47 for 28 years. Both accounts earn 6% interest compounded annually. We want to find the future values of their accounts at age 75.
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, and $n$ is the number of payments.
If the money stops being deposited but continues to earn interest, the amount grows by compound interest:
$$FV = PV \times (1 + r)^t$$
where $PV$ is the present value at the time deposits stopped, and $t$ is the number of years it grows.
3. **Calculate Mara's account:**
- Deposits: $P = 1200$
- Interest rate: $r = 0.06$
- Number of deposits: $n = 10$
Calculate value at end of deposits (age 35):
$$FV_{10} = 1200 \times \frac{(1 + 0.06)^{10} - 1}{0.06}$$
Calculate:
$$ (1.06)^{10} = 1.790847$$
So:
$$FV_{10} = 1200 \times \frac{1.790847 - 1}{0.06} = 1200 \times \frac{0.790847}{0.06} = 1200 \times 13.18078 = 15816.94$$
This is the amount at age 35.
Now it grows for 40 years without deposits:
$$FV_{75} = 15816.94 \times (1.06)^{40}$$
Calculate:
$$ (1.06)^{40} = 10.2857$$
So:
$$FV_{75} = 15816.94 \times 10.2857 = 162592.68$$
4. **Calculate Braylen's account:**
- Deposits: $P = 1200$
- Interest rate: $r = 0.06$
- Number of deposits: $n = 28$
Calculate future value at age 75:
$$FV = 1200 \times \frac{(1 + 0.06)^{28} - 1}{0.06}$$
Calculate:
$$ (1.06)^{28} = 5.349$$
So:
$$FV = 1200 \times \frac{5.349 - 1}{0.06} = 1200 \times \frac{4.349}{0.06} = 1200 \times 72.483 = 86979.60$$
5. **Summary:**
- Mara's account value at age 75 is approximately $162592.68$
- Braylen's account value at age 75 is approximately $86979.60$
Mara has more money due to starting earlier and letting the money grow longer.
**q_count: 1**