Question

periodic deposit: $? at the end of each year
rate: 3.5% compounded annually
time: 16 years
financial goal: $100,000
a. determine the periodic deposit.
b. how much of the financial goal comes from deposits and how much comes from interest?
a. in order to have $100,000 in 16 years, you should deposit $\square$ each year. (do not round until the final answer. then round up to the nearest dollar.)
b. $\square$ of the $100,000 comes from your deposits and $\square$ comes from interest. (use the answer from part a to find this answer. round to the nearest dollar as needed.)

Explanation:

Step1: Recall future value of annuity formula

The formula for the future value of an ordinary annuity is:
$$FV = P \times \frac{(1 + r)^n - 1}{r}$$
where $FV = 100000$, $r = 0.035$, $n = 16$, and $P$ is the periodic deposit.

Step2: Rearrange to solve for $P$

$$P = \frac{FV \times r}{(1 + r)^n - 1}$$

Step3: Calculate $(1 + r)^n$

$$(1 + 0.035)^{16} \approx 1.733986$$

Step4: Compute denominator

$$1.733986 - 1 = 0.733986$$

Step5: Calculate numerator

$$100000 \times 0.035 = 3500$$

Step6: Solve for $P$

$$P = \frac{3500}{0.733986} \approx 4768.47$$
Round up to the nearest dollar: $P = 4769$

Step7: Calculate total deposits

Total deposits = $P \times n = 4769 \times 16 = 76304$

Step8: Calculate interest amount

Interest = $FV - \text{Total Deposits} = 100000 - 76304 = 23696$

Answer:

a. $\$4769$
b. $\$76304$ of the $\$100,000$ comes from your deposits and $\$23696$ comes from interest.