Question

a. 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	6.5% compounded monthly	15 years	$180,000

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a. the periodic deposit is $\square
(do not round until the final answer. then round up to the nearest dollar.)

Explanation:

Step1: Identify the formula for periodic deposit

The formula for the future value of a periodic deposit (ordinary annuity) is $F = P \times \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}}$, where $F$ is the future value (financial goal), $P$ is the periodic deposit, $r$ is the annual interest rate (in decimal), $n$ is the number of compounding periods per year, and $t$ is the number of years. We need to solve for $P$.

Rearranging the formula for $P$: $P = F \times \frac{\frac{r}{n}}{(1 + \frac{r}{n})^{nt} - 1}$

Step2: Identify the values

• $F = 180000$
• $r = 6.5\% = 0.065$
• $n = 12$ (compounded monthly)
• $t = 15$ years

Step3: Calculate the exponent $nt$

$nt = 12 \times 15 = 180$

Step4: Calculate $(1 + \frac{r}{n})^{nt}$

$\frac{r}{n} = \frac{0.065}{12} \approx 0.0054167$

$(1 + 0.0054167)^{180} \approx (1.0054167)^{180}$. Using a calculator, this is approximately $2.701484$

Step5: Calculate the denominator $(1 + \frac{r}{n})^{nt} - 1$

$2.701484 - 1 = 1.701484$

Step6: Calculate the numerator $\frac{r}{n}$

$\frac{0.065}{12} \approx 0.0054167$

Step7: Calculate $P$

$P = 180000 \times \frac{0.0054167}{1.701484}$

First, calculate $\frac{0.0054167}{1.701484} \approx 0.003183$

Then, $180000 \times 0.003183 \approx 572.94$

Since we need to round up to the nearest dollar, $P \approx 573$

Answer:

a. The periodic deposit is $\$573$

(For part b, total deposits = $P \times n \times t = 573 \times 12 \times 15 = 573 \times 180 = 103140$. Interest = Financial Goal - Total Deposits = $180000 - 103140 = 76860$)