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Which of these expressions will give the unpaid balance after 7 years on a [tex]$60,000[/tex] loan with an APR of 8.4%, compounded monthly, if the monthly payment is [tex]\$516.90[/tex]?

A. [tex]$60000(1+0.007)^{84} + \$516.90\left[\frac{1-(1+0.007)^{84}}{0.007}\right][/tex]

B. [tex]$60000(1+0.084)^7 + \$516.90\left[\frac{1-(1+0.084)^7}{0.084}\right][/tex]

C. [tex]$60000(1+0.007)^7 + \$516.90\left[\frac{1-(1+0.007)^7}{0.007}\right][/tex]

D. [tex]$60000(1+0.084)^{84} + \$516.90\left[\frac{1-(1+0.084)^{84}}{0.084}\right][/tex]

Answer :

To find the unpaid balance on a loan, we consider the formula for calculating the remaining balance on a loan after making regular payments for a certain period. This involves the principal, the monthly interest rate, and the number of payments.

Given:
- Principal loan amount: \[tex]$60,000
- APR (Annual Percentage Rate): 8.4%
- Monthly payment: \$[/tex]516.90
- Loan term we're considering: 7 years
- Compounding: Monthly

To find the correct expression for the unpaid balance after 7 years, we need to calculate using monthly compounding:

1. Monthly Interest Rate:
- The annual rate is 8.4%, so the monthly rate is [tex]\( \frac{8.4\%}{12} = 0.007 \)[/tex].

2. Number of Compounding Periods:
- For 7 years with monthly payments, the number of periods is [tex]\( 7 \times 12 = 84 \)[/tex].

3. Unpaid Balance Formula:
- The standard formula for the remaining balance of an amortizing loan is:

[tex]\[
B = P(1 + r)^n - PMT \left[\frac{(1 + r)^n - 1}{r}\right]
\][/tex]

Here, [tex]\( P \)[/tex] is the principal, [tex]\( r \)[/tex] is the monthly interest rate, [tex]\( n \)[/tex] is the number of payments, and [tex]\( PMT \)[/tex] is the monthly payment.

4. For each choice:

- Option A:
[tex]\[
\$60,000(1 + 0.007)^{84} + \$516.90\left[\frac{1 - (1 + 0.007)^{84}}{0.007}\right]
\][/tex]
This expression is structured correctly for the remaining balance due to monthly compounding, matching the formula.

- Option B:
[tex]\[
\$60,000(1 + 0.084)^7 + \$516.90\left[\frac{1 - (1 + 0.084)^7}{0.084}\right]
\][/tex]
This uses the annual rate compounded annually, which is not correct since we're dealing with monthly compounding.

- Option C:
[tex]\[
\$60,000(1 + 0.007)^7 + \$516.90\left[\frac{1 - (1 + 0.007)^7}{0.007}\right]
\][/tex]
This uses an incorrect number of periods (7) instead of the correct 84.

- Option D:
[tex]\[
\$60,000(1 + 0.084)^{84} + \$516.90\left[\frac{1 - (1 + 0.084)^{84}}{0.084}\right]
\][/tex]
This uses an incorrect rate and periods setup.

Based on the calculations and structure of each option, Option A correctly represents the unpaid balance after 7 years of making monthly payments with monthly compounding.

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