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Suppose that you are thinking about buying a car and have narrowed down your choices to two options.



The new-car option: The new car costs \$29,000 and can be financed with a four-year loan at 6.43\%.

The used-car option: A three-year-old model of the same car costs \$18,000 and can be financed with a three-year loan at 6.84\%.



What is the difference in monthly payments between financing the new car and financing the used car? Use

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



(Round to the nearest cent as needed.)

Verified Answer :

  • Determine loan parameters for both options: New car (\$29,000, 4 years, 6.43%) and Used car (\$18,000, 3 years, 6.84%).
  • Use the monthly payment formula \(PMT = \frac{P(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}]}\) for both scenarios.
  • Calculate the monthly payment for the new car (\(\$686.80\)) and the used car (\(\$554.47\)).
  • Find the difference between the two payments, resulting in \(\boxed{\$132.33}\).

Explanation

  1. Identifying the Loan Parameters

To determine the difference in monthly payments between the new car and the used car, we first identify the parameters for each loan option.

For the new car:

  • Principal amount \(\$P_n = \$29,000\)
  • Loan term \(t_n = 4\) years
  • Annual interest rate \(r_n = 6.43\% = 0.0643\)
  • Compounding/payment frequency \(n = 12\) times per year

For the used car:

  • Principal amount \(\$P_u = \$18,000\)
  • Loan term \(t_u = 3\) years
  • Annual interest rate \(r_u = 6.84\% = 0.0684\)
  • Compounding/payment frequency \(n = 12\) times per year

We use the monthly payment formula:
\[PMT = \frac{P \left( \frac{r}{n} \right)}{1 - \left( 1 + \frac{r}{n} \right)^{-nt}}\]

  1. Calculating Monthly Payments

Now, we calculate the monthly payment for the new car (\(PMT_n\)):

  • Monthly interest rate \(i_n = \frac{0.0643}{12} \approx 0.00535833\)
  • Number of payments \(n_n = 12 \times 4 = 48\)

\[PMT_n = \frac{29000 \times \left( \frac{0.0643}{12} \right)}{1 - \left( 1 + \frac{0.0643}{12} \right)^{-48}} \approx \$686.79775\]

Next, we calculate the monthly payment for the used car (\(PMT_u\)):

  • Monthly interest rate \(i_u = \frac{0.0684}{12} = 0.0057\)
  • Number of payments \(n_u = 12 \times 3 = 36\)

\[PMT_u = \frac{18000 \times \left( \frac{0.0684}{12} \right)}{1 - \left( 1 + \frac{0.0684}{12} \right)^{-36}} \approx \$554.47191\]

  1. Calculating the Difference and Conclusion

Finally, we find the absolute difference in the monthly payments:
\[\text{Difference} = |PMT_n - PMT_u|\]
\[\text{Difference} = |686.79775 - 554.47191| = 132.32584\]

Rounding to the nearest cent, the difference is \(\$132.33\). Thus, the difference in monthly payments between financing the new car and financing the used car is \(\boxed{\$132.33}\).

Examples

Understanding how to calculate monthly loan payments is crucial for personal financial planning. For instance, when buying a vehicle, home, or appliance, comparing different loan terms and interest rates using this formula helps in evaluating affordability and managing monthly cash flow effectively. This mathematical approach allows you to make informed decisions and better understand the long-term impact of financial commitments.

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