Question: Use the appropriate formula to determine the periodic deposit.
b. How much of the financial goal comes from deposits and how much comes from interest?
Periodic Deposit
Rate
Time
Financial Goal
$? at the end of each month
6.75% compounded monthly
50 years
$1,250,000
1. **State the problem:**
We want to find the periodic deposit amount $P$ made at the end of each month to reach a financial goal of $1,250,000$ after $50$ years with an interest rate of $6.75\%$ compounded monthly.
2. **Formula used:**
The future value of an ordinary annuity formula is:
$$ A = P \times \frac{(1 + r)^n - 1}{r} $$
where:
- $A$ is the future value ($1,250,000$),
- $P$ is the periodic deposit (unknown),
- $r$ is the monthly interest rate,
- $n$ is the total number of deposits.
3. **Calculate parameters:**
- Annual interest rate = $6.75\% = 0.0675$
- Monthly interest rate $r = \frac{0.0675}{12} = 0.005625$
- Number of months $n = 50 \times 12 = 600$
4. **Solve for $P$:**
Rearranging the formula:
$$ P = A \times \frac{r}{(1 + r)^n - 1} $$
Substitute values:
$$ P = 1,250,000 \times \frac{0.005625}{(1 + 0.005625)^{600} - 1} $$
Calculate $(1 + 0.005625)^{600}$:
$$ (1.005625)^{600} \approx 23.286 $$
So:
$$ P = 1,250,000 \times \frac{0.005625}{23.286 - 1} = 1,250,000 \times \frac{0.005625}{22.286} $$
Calculate fraction:
$$ \frac{0.005625}{22.286} \approx 0.0002524 $$
Therefore:
$$ P = 1,250,000 \times 0.0002524 = 315.5 $$
5. **Interpretation:**
The periodic deposit is approximately $315.50$ at the end of each month.
6. **Calculate total deposits and interest earned:**
- Total deposits = $P \times n = 315.5 \times 600 = 189,300$
- Interest earned = Financial goal $- $ Total deposits = $1,250,000 - 189,300 = 1,060,700$
**Final answer:**
- Periodic deposit: $\boxed{315.50}$ per month
- Total deposits: $189,300$
- Interest earned: $1,060,700$