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You can afford a $\$250250 per month car payment. You've found a 33 year loan at 55%\% interest. How big of a loan can you afford? \newlineEnter an integer or decimal number.

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Full solution

Q. You can afford a $\$250250 per month car payment. You've found a 33 year loan at 55%\% interest. How big of a loan can you afford? \newlineEnter an integer or decimal number.
  1. Use formula for present value: To solve this problem, we need to use the formula for calculating the present value of an annuity, which is the formula for the loan amount in this case. The formula is:\newlineP=PMT×[1−(1+r)−nr]P = \text{PMT} \times \left[\frac{1 - (1 + r)^{-n}}{r}\right]\newlinewhere PP is the present value or the loan amount, PMT\text{PMT} is the monthly payment, rr is the monthly interest rate, and nn is the total number of payments.\newlineFirst, we need to convert the annual interest rate to a monthly interest rate by dividing by 1212 (since there are 1212 months in a year).\newlineMonthly interest rate = Annual interest rate / 1212
  2. Calculate monthly interest rate: Calculate the monthly interest rate:\newlineMonthly interest rate = 5%12\frac{5\%}{12}\newlineMonthly interest rate = 0.0512\frac{0.05}{12}\newlineMonthly interest rate = 0.00416666670.0041666667
  3. Calculate total number of payments: Next, we need to calculate the total number of payments. Since the loan term is 33 years and there are 1212 months in a year, we multiply the number of years by 1212.\newlinen=3 years×12 months/yearn = 3 \text{ years} \times 12 \text{ months/year}
  4. Plug values into formula: Calculate the total number of payments:\newlinen=3×12n = 3 \times 12\newlinen=36n = 36
  5. Calculate loan amount: Now we can plug the values into the formula to calculate the present value (loan amount).\newlineP=250×[1−(1+0.0041666667)−360.0041666667]P = 250 \times \left[\frac{1 - (1 + 0.0041666667)^{-36}}{0.0041666667}\right]
  6. Round loan amount: Calculate the loan amount:\newlineP=250×[1−(1+0.0041666667)−360.0041666667]P = 250 \times \left[\frac{1 - (1 + 0.0041666667)^{-36}}{0.0041666667}\right]\newlineP=250×[1−(1.0041666667)−360.0041666667]P = 250 \times \left[\frac{1 - (1.0041666667)^{-36}}{0.0041666667}\right]\newlineP=250×[1−(1.0041666667)−360.0041666667]P = 250 \times \left[\frac{1 - (1.0041666667)^{-36}}{0.0041666667}\right]\newlineP≈250×[1−0.86070797640.0041666667]P \approx 250 \times \left[\frac{1 - 0.8607079764}{0.0041666667}\right]\newlineP≈250×[0.13929202360.0041666667]P \approx 250 \times \left[\frac{0.1392920236}{0.0041666667}\right]\newlineP≈250×33.439547485P \approx 250 \times 33.439547485\newlineP≈8359.88687125P \approx 8359.88687125
  7. Round loan amount: Calculate the loan amount:\newlineP=250×[(1−(1+0.0041666667)−36)/0.0041666667]P = 250 \times \left[(1 - (1 + 0.0041666667)^{-36}) / 0.0041666667\right]\newlineP=250×[(1−(1.0041666667)−36)/0.0041666667]P = 250 \times \left[(1 - (1.0041666667)^{-36}) / 0.0041666667\right]\newlineP=250×[(1−(1.0041666667)−36)/0.0041666667]P = 250 \times \left[(1 - (1.0041666667)^{-36}) / 0.0041666667\right]\newlineP≈250×[(1−0.8607079764)/0.0041666667]P \approx 250 \times \left[(1 - 0.8607079764) / 0.0041666667\right]\newlineP≈250×[0.1392920236/0.0041666667]P \approx 250 \times [0.1392920236 / 0.0041666667]\newlineP≈250×33.439547485P \approx 250 \times 33.439547485\newlineP≈8359.88687125P \approx 8359.88687125Since the question asks for an integer or decimal number, we can round the loan amount to the nearest dollar if necessary.\newlineLoan amount ≈$8359.89\approx \$8359.89

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