Which Of These Expressions Will Give The Unpaid Balance After 5 Years On An $$80,000$ Loan With An APR Of $4.8%$, Compounded Monthly, If The Monthly Payment Is $$ 519.17 519.17 519.17 [/tex]?A. $$80000(1+0.048)^{60} +

Mar 13, 2025 by ADMIN 224 views

**Calculating Unpaid Balance after 5 Years on a Loan**

Understanding the Problem

When it comes to calculating the unpaid balance on a loan, it's essential to consider the interest rate, compounding frequency, and the number of payments made. In this scenario, we have a loan of $80,000 with an APR of 4.8%, compounded monthly, and a monthly payment of $519.17. We need to determine which expression will give us the unpaid balance after 5 years.

The Formula for Unpaid Balance

The formula for calculating the unpaid balance on a loan is given by:

A = P \left(1 + \frac{r}{n}\right)^{nt}

Where:

$A$ is the unpaid balance
$P$ is the principal amount (initial loan amount)
$r$ is the annual interest rate (in decimal form)
$n$ is the number of times interest is compounded per year
$t$ is the number of years

Expression A: Using the Formula

Let's analyze the first expression:

A = 80000(1+0.048)^{60} + 519.17 \times \frac{(1+0.048/12)^{60}-1}{0.048/12}

This expression uses the formula for calculating the unpaid balance, where the principal amount is $80,000, the annual interest rate is 4.8%, and the number of times interest is compounded per year is 12 (monthly compounding). The expression also includes the monthly payment of $519.17, which is used to calculate the total interest paid over the 5-year period.

Expression B: Using the Formula for Total Amount Paid

The second expression is:

A = 80000(1+0.048/12)^{60} + 519.17 \times \frac{(1+0.048/12)^{60}-1}{0.048/12}

This expression uses the formula for calculating the total amount paid, which is the principal amount plus the interest paid over the 5-year period. The monthly payment of $519.17 is used to calculate the total interest paid, and the result is added to the principal amount to get the total amount paid.

Expression C: Using the Formula for Unpaid Balance with Monthly Payments

The third expression is:

A = 80000(1+0.048/12)^{60} - 519.17 \times \frac{(1+0.048/12)^{60}-1}{0.048/12}

This expression uses the formula for calculating the unpaid balance with monthly payments. The monthly payment of $519.17 is subtracted from the total amount paid to get the unpaid balance.

Which Expression is Correct?

To determine which expression is correct, we need to analyze each one and see which one gives us the correct unpaid balance after 5 years.

Expression A: Using the Formula

Let's start by analyzing Expression A:

A = 80000(1+0.048)^{60} + 519.17 \times \frac{(1+0.048/12)^{60}-1}{0.048/12}

This expression uses the formula for calculating the unpaid balance, but it includes the monthly payment of $519.17, which is not necessary to calculate the unpaid balance. The correct formula for calculating the unpaid balance is:

A = P \left(1 + \frac{r}{n}\right)^{nt}

Where:

$A$ is the unpaid balance
$P$ is the principal amount (initial loan amount)
$r$ is the annual interest rate (in decimal form)
$n$ is the number of times interest is compounded per year
$t$ is the number of years

Expression B: Using the Formula for Total Amount Paid

Let's analyze Expression B:

A = 80000(1+0.048/12)^{60} + 519.17 \times \frac{(1+0.048/12)^{60}-1}{0.048/12}

Expression C: Using the Formula for Unpaid Balance with Monthly Payments

Let's analyze Expression C:

A = 80000(1+0.048/12)^{60} - 519.17 \times \frac{(1+0.048/12)^{60}-1}{0.048/12}

This expression uses the formula for calculating the unpaid balance with monthly payments. The monthly payment of $519.17 is subtracted from the total amount paid to get the unpaid balance.

Conclusion

After analyzing each expression, we can conclude that Expression B is the correct one. It uses the formula for calculating the total amount paid, which is the principal amount plus the interest paid over the 5-year period. The monthly payment of $519.17 is used to calculate the total interest paid, and the result is added to the principal amount to get the total amount paid.

The Correct Expression

The correct expression for calculating the unpaid balance after 5 years on a loan of $80,000 with an APR of 4.8%, compounded monthly, and a monthly payment of $519.17 is:

A = 80000(1+0.048/12)^{60} + 519.17 \times \frac{(1+0.048/12)^{60}-1}{0.048/12}

Final Answer

The final answer is:

A = 80000(1+0.048/12)^{60} + 519.17 \times \frac{(1+0.048/12)^{60}-1}{0.048/12}$<br/> **Q&A: Calculating Unpaid Balance after 5 Years on a Loan** =====================================================

Frequently Asked Questions

We've received many questions about calculating the unpaid balance after 5 years on a loan. Here are some of the most common questions and answers:

Q: What is the formula for calculating the unpaid balance on a loan?

A: The formula for calculating the unpaid balance on a loan is: