Full solution

Q. The Food Store is planning a major expansion for

4

years from today. In preparation for this, the company is setting aside

\$42,000

& each quarter, starting today, for the next

4

years. How much money will the firm have when it is ready to expand if it can earn an interest rate of

8.0\%

p.a. on its savings?

Identify Quarters in $4$ Years: First, let's identify the number of quarters in $4$ years since the company is saving money each quarter. $4$ years $*$ $4$ quarters/year $= 16$ quarters.
Calculate Future Value: Now, we need to calculate the future value of an annuity due to the fact that the company is making payments at the beginning of each period (quarter). $\newline$ The formula for the future value of an annuity due is $FV = P \times [(1 + r)^n - 1] \times (1 + r) / r$ , where $P$ is the payment per period, $r$ is the interest rate per period, and $n$ is the number of periods.
Calculate Quarterly Interest Rate: The annual interest rate is $8.0\%$ , but we need the quarterly interest rate since the payments are quarterly. $\newline$ Quarterly interest rate = $8.0\% / 4 = 2.0\%$ or $0.02$ in decimal form.
Plug Values into Formula: Now we can plug the values into the formula. $\newline$ P = $\$42,000$ , r = $0.02$ , and n = $16$ . $\newline$ FV = $\$42,000 \times [(1 + 0.02)^{16} - 1] \times (1 + 0.02) / 0.02$ .
Calculate $(1.02)^{16}$ : Let's calculate the future value. $\newline$ $FV = (\$42,000) \times \left[(1.02)^{16} - 1\right] \times \frac{1.02}{0.02}$ .
Subtract $1$ from $1$ . $3686$ : First, calculate $(1.02)^{16}$ . $(1.02)^{16} = 1.3686$ (rounded to four decimal places).
Multiply by $1$ . $02$ : Now, subtract $1$ from $1.3686$ . $\newline$ $1.3686 - 1 = 0.3686$ .
Divide by $0$ . $02$ : Multiply $0.3686$ by $1.02$ . $\$0.3686 \times 1.02 = 0.3760$ (rounded to four decimal places).
Divide by $0$ . $02$ : Multiply $0.3686$ by $1.02$ . $\newline$ $0.3686 \times 1.02 = 0.3760$ (rounded to four decimal places).Finally, divide by $0.02$ . $\newline$ $0.3760 / 0.02 = 18.8$ .