Mason borrows $41,217 to buy a car. He will make monthly payments for 6 years. The car loan has interest rate of 5.40%. After a 16 months Mason decides to pay off his loan. How much does he owe the loan?

Answer 1

To determine how much Mason owes on his car loan after 16 months, we first need to calculate his monthly payment using the loan details, and then find out how much he has paid so far and the remaining balance after 16 months.

Step 1: Calculate the Monthly Payment

Mason's car loan:

• Principal (P): $41,217
• Annual Interest Rate (r): 5.40% or 0.054
• Loan Term (t): 6 years = 72 months

The monthly interest rate (i) can be calculated as: \[ i = \frac{r}{12} = \frac{0.054}{12} \approx 0.0045 \]

The formula for the monthly payment (M) on an amortizing loan is: \[ M = P \cdot \frac{i(1+i)^n}{(1+i)^n - 1} \] where:

• \( n \) is the total number of payments (72 months).

Substituting the values: \[ M = 41217 \cdot \frac{0.0045(1+0.0045)^{72}}{(1+0.0045)^{72} - 1} \]

Calculating \( (1+0.0045)^{72} \): \[ (1.0045)^{72} \approx 1.34825 \]

Now substituting this back into the formula: \[ M \approx 41217 \cdot \frac{0.0045 \cdot 1.34825}{1.34825 - 1} \] \[ M \approx 41217 \cdot \frac{0.006068125}{0.34825} \] \[ M \approx 41217 \cdot 0.01743 \approx 718.18 \]

So, the monthly payment \( M \) is approximately $718.18.

Step 2: Total Paid After 16 Months

Now, we need to calculate how much Mason has paid after 16 months: \[ \text{Total Paid} = M \times 16 = 718.18 \times 16 \approx 11,53.88 \]

Step 3: Remaining Balance After 16 Months

Next, we calculate the remaining balance after 16 months of payments. This can be calculated using the loan amortization formula:

The remaining balance (B) of a loan after \( k \) payments is given by: \[ B = P \cdot \frac{(1+i)^n - (1+i)^k}{(1+i)^n - 1} \] where:

• \( k \) is the number of payments made (16).

Substituting the values: \[ B = 41217 \cdot \frac{(1.0045)^{72} - (1.0045)^{16}}{(1.0045)^{72} - 1} \] Calculating \( (1.0045)^{16} \): \[ (1.0045)^{16} \approx 1.075642 \] Now substituting into the formula: \[ B = 41217 \cdot \frac{1.34825 - 1.075642}{1.34825 - 1} \] \[ = 41217 \cdot \frac{0.272608}{0.34825} \] \[ \approx 41217 \cdot 0.78232 \approx 32229.01 \]

Answer

Thus, after 16 months, Mason owes approximately $32,229.01 on his loan.

Answer 2

What values do I enter for N, i/y, PV, PMT, and FV to solve on a finance calculator?

Answer 3

To use a finance calculator to determine the remaining balance of Mason's car loan after 16 months, you can set up the values based on the loan parameters as follows:

Key Variables:

• N (Number of Payments): This is the total number of payments Mason would make for the entire loan period. Since the loan term is 6 years, \( N = 6 \times 12 = 72 \) months.

• I/Y (Interest Rate per Year): This is the annual interest rate. Since you want the monthly interest rate, you should divide the annual interest rate by 12. Thus, \( I/Y = \frac{5.40}{12} \approx 0.45 \) (expressed as a percentage, so enter 0.45).

• PV (Present Value): This represents the loan amount (the principal), which is \( PV = -41,217 \). (It is often entered as a negative number because it represents an outgoing payment.)

• PMT (Payment): This is the monthly payment amount calculated previously, which is approximately \( PMT = 718.18 \). (This is also entered as a negative value since it represents a cash outflow.)

• FV (Future Value): This is what we're trying to solve for (the remaining balance after 16 months). Thus, you would leave this value as 0 or simply do not enter anything, depending on the finance calculator you use.

Steps to Set Up on a Financial Calculator:

1. N = 72 (total months for the loan)
2. I/Y = 0.45 (monthly interest rate in percentage)
3. PV = -41,217 (the amount borrowed, as a negative number)
4. PMT = -718.18 (monthly payment, as a negative number)
5. FV = 0 (or leave it blank depending on the calculator)

Solve for:

• FV (future value or remaining balance) after 16 months.

If your calculator requires that you input the number of periods already paid, you might need to use a different approach to find the remaining balance after 16 payments. Make sure to adjust the N or calculate the remaining balance specifically after those 16 payments by adjusting or using the effective remaining payments for future calculations beyond the first 16.

If your calculator allows, after entering the values, you can use the "Compute" function to find the new balance after 16 payments. Some calculators may require you to input:

1. N = 56 (remaining payments after 16)
2. Recalculate based on remaining payment periods or directly using the amortization method to see the balance after 16 payments.