Questions: Given a loan amount P, an annual interest rate r, and the length of the loan in years, find the monthly payment R necessary to pay off the loan by completing parts a through c. Represent the number of monthly payments by n. Amount Rate Time 110,000 9% 15 years A=422,184.76 (Round to two decimal places as needed.) b. Calculate ((1+r/12)^n-1)/(r/12) and call this number B. 378.41 (Round to two decimal places as needed.) C. Let R=A/B The monthly payment necessary to pay off the loan is R= (Round up to the nearest cent.)

Given a loan amount P, an annual interest rate r, and the length of the loan in years, find the monthly payment R necessary to pay off the loan by completing parts a through c. Represent the number of monthly payments by n.

Amount Rate Time
110,000 9% 15 years

A=422,184.76 (Round to two decimal places as needed.)
b. Calculate

((1+r/12)^n-1)/(r/12)

and call this number B.
378.41 (Round to two decimal places as needed.)
C. Let R=A/B

The monthly payment necessary to pay off the loan is R=
(Round up to the nearest cent.)

Transcript text: Given a loan amount $P$, an annual interest rate $r$, and the length of the loan in years, find the monthly payment $R$ necessary to pay off the loan by completing parts a through $c$. Represent the number of monthly payments by $n$. \begin{tabular}{|c|c|c|} \hline Amount & Rate & Time \\ \hline$\$ 110,000$ & $9 \%$ & 15 years \\ \hline \end{tabular} $A=422,184.76$ (Round to two decimal places as needed.) b. Calculate \[ \left(\frac{\left(1+\frac{r}{12}\right)^{n}-1}{\frac{r}{12}}\right) \] and call this number $B$. 378.41 (Round to two decimal places as needed.) C. Let $R=\frac{A}{B}$ The monthly payment necessary to pay off the loan is $R=\$$ $\square$ Round up to the nearest cent.)

Solution

Solution Steps

To solve this problem, we need to follow these steps:

Calculate the total number of monthly payments, $ n $, which is the number of years multiplied by 12.
Calculate the monthly interest rate by dividing the annual interest rate by 12.
Compute the value of $ B $ using the given formula.
Calculate the monthly payment $ R $ by dividing the total amount $ A $ by $ B $.

Step 1: Calculate the Number of Monthly Payments

The total number of monthly payments $ n $ is calculated as follows: \[ n = \text{years} \times 12 = 15 \times 12 = 180 \]

Step 2: Calculate the Monthly Interest Rate

The monthly interest rate $ r_{monthly} $ is given by: \[ r_{monthly} = \frac{r}{12} = \frac{0.09}{12} = 0.0075 \]

Step 3: Calculate $ B $

Using the formula for $ B $: \[ B = \frac{\left(1 + r_{monthly}\right)^{n} - 1}{r_{monthly}} = \frac{\left(1 + 0.0075\right)^{180} - 1}{0.0075} \approx 378.4058 \]

Step 4: Calculate the Monthly Payment $ R $

The monthly payment $ R $ is calculated as: \[ R = \frac{A}{B} = \frac{422184.76}{378.4058} \approx 1115.69 \]