question 9\nsuppose you want to have $800,000 for retirement in 30 years. your account earns 8%…

question 9\nsuppose you want to have $800,000 for retirement in 30 years. your account earns 8% interest.\na) how much would you need to deposit in the account each month?\n$\n\nb) how much interest will you earn?\n$

question 9\nsuppose you want to have $800,000 for retirement in 30 years. your account earns 8% interest.\na) how much would you need to deposit in the account each month?\n$\n\nb) how much interest will you earn?\n$

Answer

Explanation:

Step1: Convert annual - rate and years to monthly values

The annual interest rate $r = 8%=0.08$, so the monthly interest rate $i=\frac{0.08}{12}$. The number of years $n = 30$, so the number of months $t=30\times12 = 360$. Let the monthly deposit be $P$. The future - value of an ordinary annuity formula is $F = P\times\frac{(1 + i)^{t}-1}{i}$. We know $F=$800000$.

Step2: Solve for the monthly deposit $P$

We can re - arrange the future - value of an ordinary annuity formula to solve for $P$: [P=\frac{F\times i}{(1 + i)^{t}-1}] Substitute $F = 800000$, $i=\frac{0.08}{12}$, and $t = 360$ into the formula: [i=\frac{0.08}{12}\approx0.00667] [(1 + i)^{t}=(1+\frac{0.08}{12})^{360}\approx10.9357] [P=\frac{800000\times\frac{0.08}{12}}{(1+\frac{0.08}{12})^{360}-1}=\frac{800000\times0.00667}{10.9357 - 1}=\frac{5336}{9.9357}\approx537.05]

Step3: Calculate the total amount deposited

The total amount deposited over 360 months is $T = P\times t=537.05\times360 = 193338$.

Step4: Calculate the interest earned

The interest earned $I=F - T$. Substitute $F = 800000$ and $T = 193338$ into the formula: [I=800000 - 193338=606662]

Answer:

a) $537.05 b) $606662