Question

A $4,700 loan at 8.7% compounded monthly was settled by a single payment of $5,700 including accrued interest. How long after the initial loan was the $5,700 payment made? For the purpose of determining the number of days in a partial month, assume that a full month has 30 days. (Do not round intermediate calculations. Round your answer to the nearest day.)

years
months
days

Answer 1

6 months and 17 days after the initial loan.

Explanation:

To determine how long after the initial loan was the $5,700 payment made, we need to calculate the time it took for the loan to accrue enough interest to reach that amount.

First, let's calculate the initial loan amount using the formula for compound interest:

Principal = Final Amount / (1 + (interest rate / number of compounding periods))^number of compounding periods

In this case, the final amount is $5,700, the interest rate is 8.7%, and the loan is compounded monthly. So we have:

$4,700 = $5,700 / (1 + (0.087 / 12))^n

Next, we need to solve for n, the number of compounding periods (months). To do this, we can use logarithms:

(1 + (0.087 / 12))^n = $5,700 / $4,700

n * log(1 + (0.087 / 12)) = log($5,700 / $4,700)

n = log($5,700 / $4,700) / log(1 + (0.087 / 12))

Using a calculator, we find that n is approximately 6.562 months.

Since a full month has 30 days, we can convert the fractional part of the months to days:

0.562 months * 30 days/month = approximately 16.86 days