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).
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.