question 9 nan and neal are twins. nan invests $5,000 per year, starting at age 25. neal decides to wait…

question 9 nan and neal are twins. nan invests $5,000 per year, starting at age 25. neal decides to wait until 30 to start saving for retirement. suppose both can earn a compound annual interest rate of 7% on their investments. both twins will retire at age 60 and do not withdraw any funds prior to retirement. to have the same amount of savings by age 60, how much should neal invest every year? (round your answer to the nearest dollar). a $5,000 b $5,833 c $7,013 d $7,317 e none of these are correct

question 9 nan and neal are twins. nan invests $5,000 per year, starting at age 25. neal decides to wait until 30 to start saving for retirement. suppose both can earn a compound annual interest rate of 7% on their investments. both twins will retire at age 60 and do not withdraw any funds prior to retirement. to have the same amount of savings by age 60, how much should neal invest every year? (round your answer to the nearest dollar). a $5,000 b $5,833 c $7,013 d $7,317 e none of these are correct

Answer

Explanation:

Step1: Calculate Nan's future - value of annuity

Nan starts investing at age 25 and retires at age 60, so the number of periods $n_1=60 - 25=35$ years, and the annual investment $P = 5000$, the interest rate $r=0.07$. The future - value of an ordinary annuity formula is $FVA = P\times\frac{(1 + r)^n-1}{r}$. Substitute the values: $FVA_N=5000\times\frac{(1 + 0.07)^{35}-1}{0.07}$. First, calculate $(1 + 0.07)^{35}\approx10.67658$. Then $(1 + 0.07)^{35}-1\approx9.67658$. And $\frac{(1 + 0.07)^{35}-1}{0.07}\approx\frac{9.67658}{0.07}\approx138.2369$. So $FVA_N=5000\times138.2369 = 691184.5$.

Step2: Calculate Neal's number of periods and solve for his annual investment

Neal starts investing at age 30 and retires at age 60, so the number of periods $n_2=60 - 30=30$ years. Let Neal's annual investment be $x$. We want $FVA_N=FVA_M$ (where $M$ represents Neal). Using the future - value of an ordinary annuity formula $FVA = x\times\frac{(1 + r)^{n_2}-1}{r}$, and we know $FVA = 691184.5$, $r = 0.07$, $n_2=30$. We have the equation $691184.5=x\times\frac{(1 + 0.07)^{30}-1}{0.07}$. First, calculate $(1 + 0.07)^{30}\approx7.61226$. Then $(1 + 0.07)^{30}-1\approx6.61226$. And $\frac{(1 + 0.07)^{30}-1}{0.07}\approx\frac{6.61226}{0.07}\approx94.4609$. So $x=\frac{691184.5}{94.4609}\approx7317$.

Answer:

D. $7,317$