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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. Future value = $◻

Becky is 2121 years old and spends $175 \$ 175 on cigarettes per month. If she decides to stop smoking, and instead invests this money at the end of every 33 months into an investment paying 4.25% 4.25 \% compounded quarterly, how much will she have when she turns 6161 ? For full marks your answer(s) should be rounded to the nearest cent.\newlineFuture value =$□ = \$ \square

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Q. Becky is 2121 years old and spends $175 \$ 175 on cigarettes per month. If she decides to stop smoking, and instead invests this money at the end of every 33 months into an investment paying 4.25% 4.25 \% compounded quarterly, how much will she have when she turns 6161 ? For full marks your answer(s) should be rounded to the nearest cent.\newlineFuture value =$□ = \$ \square
  1. Calculate Total Investment: Determine the total investment per quarter.\newlineSince Becky spends $175\$175 per month on cigarettes, she would save and invest this amount every month. To find out how much she invests every quarter (33 months), we multiply the monthly amount by 33.\newline$175×3=$525\$175 \times 3 = \$525 per quarter.
  2. Determine Number of Quarters: Calculate the number of quarters from age 2121 to age 6161. Becky will invest at the end of every quarter for 4040 years (from age 2121 to 6161). There are 44 quarters in a year, so we multiply the number of years by 44 to find the total number of quarters. 4040 years ∗* 44 quarters/year 616100 quarters.
  3. Use Annuity Formula: Use the future value of an annuity formula for compound interest.\newlineThe future value of an annuity formula is:\newlineFV=P×[(1+r)n−1]/rFV = P \times \left[\left(1 + r\right)^n - 1\right] / r\newlineWhere:\newlineFVFV = future value of the annuity\newlinePP = payment per period (quarter)\newlinerr = interest rate per period\newlinenn = total number of periods (quarters)\newlineIn this case:\newlineP=$525P = \$525\newliner=4.25%r = 4.25\% annual interest rate compounded quarterly, so we divide by 44 to get the quarterly rate: 0.0425/4=0.0106250.0425 / 4 = 0.010625\newlinen=160n = 160 quarters
  4. Calculate Future Value: Plug the values into the formula and calculate the future value.\newlineFV=$(525)∗[((1+0.010625)160−1)/0.010625]FV = \$(525) * [((1 + 0.010625)^{160} - 1) / 0.010625]\newlineFirst, calculate (1+0.010625)160(1 + 0.010625)^{160}:\newline(1+0.010625)160≈5.4164(1 + 0.010625)^{160} \approx 5.4164\newlineNow, calculate the future value:\newlineFV=$(525)∗[(5.4164−1)/0.010625]FV = \$(525) * [(5.4164 - 1) / 0.010625]\newlineFV=$(525)∗[4.4164/0.010625]FV = \$(525) * [4.4164 / 0.010625]\newlineFV=$(525)∗415.689FV = \$(525) * 415.689\newlineFV≈$(218,236.725)FV \approx \$(218,236.725)
  5. Round to Nearest Cent: Round the future value to the nearest cent.\newlineThe future value rounded to the nearest cent is $218,236.73\$218,236.73.

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