Question

Mr. Lopez borrows $290,000.00 that at 4.7% interest, compounded monthly. He will make payments monthly in the amount of $1,400.00. How many payments will it take for Mr. Lopez to payoff this loan? It will take monthly payments. (Round to two decimal places).​

Answer 1

Using the formula for the monthly payment on a loan, which is:

\[ P = \frac{rPv}{1 - (1 + r)^{-n}} \]

Where:
P = monthly payment ($1,400.00)
r = monthly interest rate (4.7% or 0.047)
Pv = present value of the loan ($290,000.00)
n = number of payments

Plugging in the given values, we get:

\[ $1,400.00 = \frac{0.047 \times $290,000.00}{1 - (1 + 0.047)^{-n}} \]

\[ $1,400.00 = \frac{$13,630.00}{1 - (1.047)^{-n}} \]

\[ 1 - (1.047)^{-n} = \frac{$13,630.00}{$1,400.00} \]

\[ (1.047)^{-n} = 1 - \frac{$13,630.00}{$1,400.00} \]

\[ (1.047)^{-n} = 1 - 9.73214 \]

\[ (1.047)^{-n} = -8.73214 \]

\[ -n \times \log(1.047) = \log(-8.73214) \]

\[ n = \frac{\log(8.73214)}{\log(1.047)} \]

\[ n ≈ 324.84 \]

Thus, it will take approximately 324.84 payments rounded to two decimal places, which is equivalent to 325 monthly payments to payoff the loan.