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? $\newline$ $\$ 472.06$ $\newline$ b) How much interest will you earn? $\newline$ $\$ 722,019.86$ $\newline$ Add Work

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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

\$ 472.06

\newline

b) How much interest will you earn?

\newline

\$ 722,019.86

\newline

Add Work

Calculate Monthly Interest Rate: We need to use the formula for the future value of an annuity to find out the monthly deposit required to reach $\$300,000$ in $25$ years with an interest rate of $6\%$ per year. The formula is: $\newline$ $FV = P \times \left(\left(1 + r\right)^{nt} - 1\right) / r$ $\newline$ Where: $\newline$ FV = future value of the annuity (the amount you want to have) $\newline$ P = monthly payment (what we're solving for) $\newline$ r = monthly interest rate (annual rate divided by $12$ ) $\newline$ n = number of times the interest is compounded per year ( $12$ for monthly) $\newline$ t = number of years $\newline$ First, we need to convert the annual interest rate to a monthly rate by dividing by $12$ . $\newline$ $r = 6\%$ per year $= 0.06$ per year $= 0.06 / 12$ per month
Calculate Number of Periods: Now, we calculate the monthly interest rate: $\newline$ $r = \frac{0.06}{12}$ $\newline$ $r = 0.005$ per month
Plug Values into Formula: Next, we calculate the number of periods (months) over $25$ years: $\newline$ $n = 12$ months/year $\newline$ $t = 25$ years $\newline$ $nt = n \times t = 12 \times 25 = 300$ months
Calculate Term Inside Brackets: Now we can plug the values into the formula and solve for $P$ : $FV = \$300,000$ $r = 0.005$ $nt = 300$ $\$300,000 = P \times \left(\left(1 + 0.005\right)^{300} - 1\right) / 0.005$ We need to calculate the term inside the brackets first.
Solve for Monthly Deposit: Calculate the term inside the brackets: $\newline$ $((1 + 0.005)^{300} - 1) / 0.005$ $\newline$ First, calculate $(1 + 0.005)^{300}$ : $\newline$ $(1 + 0.005)^{300} \approx 4.291870$ $\newline$ Now subtract $1$ : $\newline$ $4.291870 - 1 \approx 3.291870$ $\newline$ Finally, divide by $0.005$ : $\newline$ $3.291870 / 0.005 \approx 658.374$ $\newline$ Now we have the term inside the brackets.
Calculate Total Amount Deposited: We can now solve for $P$ : $\newline$ $\$300,000 = P \times 658.374$ $\newline$ $P = \frac{\$300,000}{658.374}$ $\newline$ $P \approx \$455.59$ $\newline$ This is the monthly deposit required to reach $\$300,000$ in $25$ years at a $6\%$ annual interest rate.
Calculate Total Amount Deposited: We can now solve for $P$ : $\newline$ $\$300,000 = P \times 658.374$ $\newline$ $P = \frac{\$300,000}{658.374}$ $\newline$ $P \approx \$455.59$ $\newline$ This is the monthly deposit required to reach $\$300,000$ in $25$ years at a $6\%$ annual interest rate.To find out how much interest will be earned, we need to calculate the total amount deposited and subtract it from the future value. $\newline$ Total amount deposited = $P \times nt$ $\newline$ Total amount deposited = $\$455.59 \times 300$ $\newline$ Total amount deposited \approx $\$136,677$ $\newline$ Now, subtract the total amount deposited from the future value to find the interest earned: $\newline$ Interest earned = $\$300,000 = P \times 658.374$ $0$ Total amount deposited $\newline$ Interest earned = $\$300,000 = P \times 658.374$ $1$ $\newline$ Interest earned \approx $\$300,000 = P \times 658.374$ $2$ $\newline$ However, this calculation is incorrect because the monthly deposit amount was calculated incorrectly in the previous step. The correct monthly deposit amount is $\$300,000 = P \times 658.374$ $3$ , not $\$300,000 = P \times 658.374$ $4$ . Therefore, we need to recalculate the total amount deposited and the interest earned using the correct monthly deposit amount.