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.
Part 1
a. How much money does Mara have in her account at age 75? At age 75, Mara has enter your response here in her account. (Round to the nearest cent as needed.)
1. **Problem Statement:** Calculate how much money Mara has in her account at age 75.
2. **Given:**
- Mara starts depositing at age 25.
- Deposits: $1200 per year for 10 years.
- After 10 years, no more deposits but money earns interest for 40 more years.
- Interest rate: 6% compounded annually.
3. **Step 1: Calculate the future value of the annuity after 10 years of deposits.**
The formula for the future value of an annuity is:
$$FV = P \times \frac{(1 + r)^n - 1}{r}$$
where:
- $P = 1200$ (annual deposit)
- $r = 0.06$ (annual interest rate)
- $n = 10$ (number of deposits)
Calculate:
$$FV = 1200 \times \frac{(1 + 0.06)^{10} - 1}{0.06} = 1200 \times \frac{(1.06)^{10} - 1}{0.06}$$
Calculate $(1.06)^{10}$:
$$ (1.06)^{10} = 1.790847 $$
So:
$$FV = 1200 \times \frac{1.790847 - 1}{0.06} = 1200 \times \frac{0.790847}{0.06} = 1200 \times 13.18078 = 15816.94$$
4. **Step 2: Calculate the value of this amount after 40 more years of interest without deposits.**
Use compound interest formula:
$$A = P (1 + r)^t$$
where:
- $P = 15816.94$
- $r = 0.06$
- $t = 40$
Calculate:
$$A = 15816.94 \times (1.06)^{40}$$
Calculate $(1.06)^{40}$:
$$ (1.06)^{40} = 10.2857 $$
So:
$$A = 15816.94 \times 10.2857 = 162592.20$$
5. **Final answer:**
At age 75, Mara has approximately **$162592.20** in her account.