Barbara Katzman bought an income property for $\$ 27,000$ three years ago. She has held the property for the three years without renting it. If she rents the property out now, what should be the size of the monthly rent payment due in advance if money is worth $7 \%$ compounded annually? $\newline$ The size of the monthly rent payment due should be $\$$ $\square$ $\newline$ (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

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Q. Barbara Katzman bought an income property for

\$ 27,000

three years ago. She has held the property for the three years without renting it. If she rents the property out now, what should be the size of the monthly rent payment due in advance if money is worth

7 \%

compounded annually?

\newline

The size of the monthly rent payment due should be

\$

\square

\newline

(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

Identify Formula: Identify the formula to calculate the monthly rent payment. $\newline$ We need to use the formula for the future value of an annuity due, which is used to calculate the future value of a series of equal payments made at the beginning of each period. The formula is: $\newline$ $FV = Pmt \times \left[(1 + r/n)^{(nt)} - 1\right] \times (n/r)$ $\newline$ Where: $\newline$ $FV$ = future value of the annuity $\newline$ $Pmt$ = payment per period $\newline$ $r$ = annual interest rate $\newline$ $n$ = number of times the interest is compounded per year $\newline$ $t$ = number of years $\newline$ Since the payments are monthly and the interest is compounded annually, $n$ will be $1$ .
Calculate Future Value: Calculate the future value (FV) of the property after three years. $\newline$ The property was bought for $\$27,000$ , and we need to find its value after three years with a $7\%$ annual interest rate compounded annually. $\newline$ $FV = PV \times (1 + r)^t$ $\newline$ $FV = \$27,000 \times (1 + 0.07)^3$ $\newline$ $FV = \$27,000 \times (1.07)^3$ $\newline$ $FV = \$27,000 \times 1.225043$ $\newline$ $FV = \$33,076.16$
Set Up Formula: Set up the annuity due formula with the known values. $\newline$ We have the future value (FV) from Step $2$ , the annual interest rate $r$ is $7\%$ , and the number of times the interest is compounded per year $n$ is $1$ . The number of years $t$ is $3$ . We need to find the monthly payment $Pmt$ . $\newline$ $FV = Pmt \times \left[(1 + \frac{r}{n})^{(n\times t)} - 1\right] \times \left(\frac{n}{r}\right)$ $\newline$ $\$33,076.16 = Pmt \times \left[(1 + 0.07/1)^{(1\times3)} - 1\right] \times \left(\frac{1}{0.07}\right)$
Solve for Payment: Solve for the monthly payment (Pmt).
$\$33,076.16 = \text{Pmt} \times [(1.07)^3 - 1] \times (1/0.07)$
$\$33,076.16 = \text{Pmt} \times [1.225043 - 1] \times (1/0.07)$
$\$33,076.16 = \text{Pmt} \times 0.225043 \times 14.285714$
$\$33,076.16 = \text{Pmt} \times 3.214203$
$\text{Pmt} = \$33,076.16 / 3.214203$
$\text{Pmt} = \$10,289.47$
This calculation is incorrect because we have not accounted for the fact that the payments are monthly, not annual. We need to adjust the formula to reflect monthly payments.