Becky is $21$ years old and spends $\$ 175$ on cigarettes per month. If she decides to stop smoking, and instead invests this money at the end of every $3$ months into an investment paying $4.25 \%$ compounded quarterly, how much will she have when she turns $61$ ? For full marks your answer(s) should be rounded to the nearest cent. $\newline$ Future value $= \$ \square$

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Algebra 1

Exponential growth and decay: word problems

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Q. Becky is

21

years old and spends

\$ 175

on cigarettes per month. If she decides to stop smoking, and instead invests this money at the end of every

3

months into an investment paying

4.25 \%

compounded quarterly, how much will she have when she turns

61

? For full marks your answer(s) should be rounded to the nearest cent.

\newline

Future value

= \$ \square

Calculate Total Investment: Determine the total investment per quarter. $\newline$ Since Becky spends $\$175$ per month on cigarettes, she would save and invest this amount every month. To find out how much she invests every quarter ( $3$ months), we multiply the monthly amount by $3$ . $\newline$ $\$175 \times 3 = \$525$ per quarter.
Determine Number of Quarters: Calculate the number of quarters from age $21$ to age $61$ . Becky will invest at the end of every quarter for $40$ years (from age $21$ to $61$ ). There are $4$ quarters in a year, so we multiply the number of years by $4$ to find the total number of quarters. $40$ years $*$ $4$ quarters/year $61$ $0$ quarters.
Use Annuity Formula: Use the future value of an annuity formula for compound interest. $\newline$ The future value of an annuity formula is: $\newline$ $FV = P \times \left[\left(1 + r\right)^n - 1\right] / r$ $\newline$ Where: $\newline$ $FV$ = future value of the annuity $\newline$ $P$ = payment per period (quarter) $\newline$ $r$ = interest rate per period $\newline$ $n$ = total number of periods (quarters) $\newline$ In this case: $\newline$ $P = \$525$ $\newline$ $r = 4.25\%$ annual interest rate compounded quarterly, so we divide by $4$ to get the quarterly rate: $0.0425 / 4 = 0.010625$ $\newline$ $n = 160$ quarters
Calculate Future Value: Plug the values into the formula and calculate the future value. $\newline$ $FV = \$(525) * [((1 + 0.010625)^{160} - 1) / 0.010625]$ $\newline$ First, calculate $(1 + 0.010625)^{160}$ : $\newline$ $(1 + 0.010625)^{160} \approx 5.4164$ $\newline$ Now, calculate the future value: $\newline$ $FV = \$(525) * [(5.4164 - 1) / 0.010625]$ $\newline$ $FV = \$(525) * [4.4164 / 0.010625]$ $\newline$ $FV = \$(525) * 415.689$ $\newline$ $FV \approx \$(218,236.725)$
Round to Nearest Cent: Round the future value to the nearest cent. $\newline$ The future value rounded to the nearest cent is $\$218,236.73$ .