Suppose you want to have $\$ 300,000$ for retirement in $25$ years. Your account earns $6 \%$ interest. $\newline$ a) How much would you need to deposit in the account each month? $\newline$ $\$$ $\newline$ b) How much interest will you earn? $\newline$ $\$$

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Q. Suppose you want to have

\$ 300,000

for retirement in

25

years. Your account earns

6 \%

interest.

\newline

a) How much would you need to deposit in the account each month?

\newline

\$

\newline

b) How much interest will you earn?

\newline

\$

Calculate Monthly Deposit: We need to use the formula for the future value of an annuity to find out the monthly deposit required. The formula is: $\newline$ $FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)$ $\newline$ where: $\newline$ FV = future value of the annuity (the amount you want to have saved by retirement) $\newline$ P = monthly payment (what we're solving for) $\newline$ r = monthly interest rate (annual rate divided by $12$ ) $\newline$ n = total number of payments (months in $25$ years) $\newline$ First, we need to calculate the monthly interest rate and the total number of payments.
Calculate Monthly Interest Rate: The annual interest rate is $6$ %, so the monthly interest rate is: $\newline$ $r = \frac{6\%}{12} = \frac{0.06}{12} = 0.005$
Calculate Total Payments: There are $25$ years and $12$ months in a year, so the total number of payments is: $\newline$ $n = 25 \times 12 = 300$
Solve for Monthly Payment: Now we can rearrange the formula to solve for P, the monthly payment: $\newline$ $P = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right)}$ $\newline$ Substitute the known values into the formula: $\newline$ $P = \frac{300000}{\left( \frac{(1 + 0.005)^{300} - 1}{0.005} \right)}$
Calculate Denominator: Calculate the denominator of the fraction: $\newline$ $(1 + 0.005)^{300} - 1$ $\newline$ This requires a calculator or a spreadsheet to compute accurately.
Calculate Monthly Payment: Using a calculator, we find: $\newline$ $(1 + 0.005)^{300} - 1 \approx 2.396.83$ $\newline$ Now we can calculate the monthly payment: $\newline$ $P = \frac{300000}{2.396.83}$
Calculate Total Amount Paid: Using a calculator, we find: $\newline$ $P \approx \frac{300000}{2.396.83} \approx 125.23$ $\newline$ So, you would need to deposit approximately $$125$.$23$ each month.
Calculate Interest Earned: To find out how much interest you will earn, we need to calculate the total amount paid over the $25$ years and subtract the final amount you want to have.$\newline$Total amount paid = monthly payment * number of payments$\newline$\[ Total\ amount\ paid = 125.23 \times 300 \]
Calculate Interest Earned: To find out how much interest you will earn, we need to calculate the total amount paid over the $25$ years and subtract the final amount you want to have.$\newline$Total amount paid = monthly payment * number of payments$\newline$\[ Total\ amount\ paid = 125.23 \times 300 \]Using a calculator, we find:$\newline$\[ Total\ amount\ paid = 125.23 \times 300 \approx 37569 \]
Calculate Interest Earned: To find out how much interest you will earn, we need to calculate the total amount paid over the $25$ years and subtract the final amount you want to have.$\newline$Total amount paid = monthly payment * number of payments$\newline$\[ Total\ amount\ paid = 125.23 \times 300 \]Using a calculator, we find:$\newline$\[ Total\ amount\ paid = 125.23 \times 300 \approx 37569 \]Now, calculate the interest earned by subtracting the final amount from the total amount paid:$\newline$\[ Interest\ earned = Total\ amount\ paid - Final\ amount \]$\newline$\[ Interest\ earned = 37569 - 300000 \]