The Formula for Calculating Loan Payments

Oct 13, 2025 by talynt.org 53 views

Understanding the Problem

Jacob took out a loan of $\$950$ at an annual percentage rate (APR) of $13.2\%$ , compounded monthly. He plans to make monthly payments of $\$62.50$ to pay off the loan. The question is, which expression could be used to calculate the number of months it will take Jacob to pay off the loan.

Key Concepts

Before we dive into the solution, let's review some key concepts:

Annual Percentage Rate (APR): The APR is the interest rate charged on a loan over a year. In this case, the APR is $13.2\%$ .
Compounded Monthly: This means that the interest is applied monthly, rather than annually.
Monthly Payment: Jacob will make monthly payments of $\$62.50$ to pay off the loan.

The Formula for Calculating Loan Payments

The formula for calculating loan payments is:

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

where:

$M$ is the monthly payment
$P$ is the principal loan amount (the initial amount borrowed)
$r$ is the monthly interest rate (the APR divided by 12)
$n$ is the number of payments (the number of months it will take to pay off the loan)

Rearranging the Formula to Solve for n

We want to solve for $n$ , the number of payments. To do this, we can rearrange the formula as follows:

n = \frac{\log \left( \frac{M(1+r)}{P} \right)}{\log(1+r)}

Plugging in the Values

Now that we have the formula, let's plug in the values:

$M = 62.50$
$P = 950$
$r = 0.012$ (the APR divided by 12)

Substituting these values into the formula, we get:

n = \frac{\log \left( \frac{62.50(1+0.012)}{950} \right)}{\log(1+0.012)}

Simplifying the Expression

To simplify the expression, we can use a calculator to evaluate the logarithms:

n = \frac{\log \left( \frac{62.50(1.012)}{950} \right)}{\log(1.012)}

n = \frac{\log \left( \frac{62.80}{950} \right)}{\log(1.012)}

n = \frac{\log(0.066)}{\log(1.012)}

n = \frac{-1.176}{0.0143}

n = 82.1

Conclusion

Therefore, the expression that could be used to calculate the number of months it will take Jacob to pay off the loan is:

n = \frac{\log \left( \frac{M(1+r)}{P} \right)}{\log(1+r)}

where:

$M$ is the monthly payment
$P$ is the principal loan amount (the initial amount borrowed)
$r$ is the monthly interest rate (the APR divided by 12)

Q&A: Calculating Loan Payments

Q: What is the formula for calculating loan payments?

A: The formula for calculating loan payments is:

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

where:

$M$ is the monthly payment
$P$ is the principal loan amount (the initial amount borrowed)
$r$ is the monthly interest rate (the APR divided by 12)
$n$ is the number of payments (the number of months it will take to pay off the loan)

Q: How do I calculate the number of payments (n) using the formula?

A: To calculate the number of payments (n), you can rearrange the formula as follows:

n = \frac{\log \left( \frac{M(1+r)}{P} \right)}{\log(1+r)}

Q: What is the monthly interest rate (r)?

A: The monthly interest rate (r) is the APR divided by 12. For example, if the APR is 13.2%, the monthly interest rate would be:

r = \frac{0.132}{12} = 0.011

Q: How do I calculate the monthly payment (M) using the formula?

A: To calculate the monthly payment (M), you can use the formula:

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

Q: What is the principal loan amount (P)?

A: The principal loan amount (P) is the initial amount borrowed. For example, if Jacob borrowed $\$950$ , the principal loan amount would be:

P = 950

Q: How do I use the formula to calculate the number of payments (n) if I know the monthly payment (M), principal loan amount (P), and APR?

A: To use the formula to calculate the number of payments (n), you can plug in the values as follows:

$M = 62.50$
$P = 950$
$r = 0.012$ (the APR divided by 12)

Substituting these values into the formula, you get:

n = \frac{\log \left( \frac{62.50(1+0.012)}{950} \right)}{\log(1+0.012)}

Q: What is the significance of the logarithm in the formula?

A: The logarithm in the formula is used to simplify the calculation of the number of payments (n). By using the logarithm, you can avoid having to calculate the exponentiation of the monthly interest rate (r) to the power of n.

Q: Can I use a calculator to evaluate the logarithms in the formula?

A: Yes, you can use a calculator to evaluate the logarithms in the formula. This will make it easier to calculate the number of payments (n) without having to perform the calculations manually.

Q: What is the final answer to the problem?

A: The final answer to the problem is:

n = 82.1

This means that it will take Jacob approximately 82.1 months to pay off the loan, assuming he makes monthly payments of $\$62.50$ and the APR is 13.2%.