Chapter 12.4, Problem 19E

Mortgage Dr. Gupta is considering a 30-year mortgage at 6% interest. She can make payments of $3500 a month. What size loan can she afford?

To determine

The amount of loan that GT can afford.

Answer to Problem 19E

The loan amount is $583,770.65.

Explanation of Solution

Loan amount is an amount which can be borrowed for specific period of time and is expected to be paid back with interest.

Formula to calculate loan amount:

$\text{Monthly}\text{payment}\text{=}\frac{i\times \left(\text{Loan amount}\right)}{1-{\left(1+i\right)}^{-n}}$

Where,

i= interest amount

$A_p$ = loan amount

$n$= period

Given:

Monthly payment:  $3,500

Period: 30 year

Total number of payment: $30\times 12=360\text{times}$

Interest rate: 6%

Calculation:

Calculate the loan amount that GT can afford:

$\begin{array}{c}\text{Monthly}\text{payment}\text{=}\frac{i\times \left(A_p\right)}{1-{\left(1+i\right)}^{-n}}\\ \$3,500=\frac{\frac{6\%}{12}\times \left(A_p\right)}{1-{\left(1+\frac{6\%}{12}\right)}^{-30\times 12}}\\ \$3,500=\frac{0.005\times \left(A_p\right)}{1-{\left(1+0.005\right)}^{-30\times 12}}\\ \$3,500=\frac{0.005\times \left(A_p\right)}{0.83395}\end{array}$

$\begin{array}{c}0.005\times \left(A_p\right)=\$2,918.853\\ \left(A_p\right)=\frac{\$2,918.853}{0.005}\\ \left(A_p\right)=\$583,770.65\end{array}$

Therefore, the loan amount is $583,770.65.