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You can afford a $250 per month car payment. You've found a 3 year loan at 5% interest. How big of a loan can you afford? Enter an inteser or decimal number [more..] Add Work Check Answer

You can afford a $250 \$ 250 per month car payment. You've found a 33 year loan at 5% 5 \% interest. How big of a loan can you afford?\newlineEnter an inteser or decimal number [more..]\newlineAdd Work\newlineCheck Answer

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Q. You can afford a $250 \$ 250 per month car payment. You've found a 33 year loan at 5% 5 \% interest. How big of a loan can you afford?\newlineEnter an inteser or decimal number [more..]\newlineAdd Work\newlineCheck Answer
  1. Formula Explanation: 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 context. The formula is:\newlineP=PMT×[1−(1+r)−nr]P = PMT \times \left[\frac{1 - (1 + r)^{-n}}{r}\right]\newlineWhere:\newlinePP = Present value of the annuity (loan amount)\newlinePMTPMT = Periodic payment amount ($250\$250 per month)\newlinerr = Periodic interest rate (monthly interest rate)\newlinenn = Total number of payments (number of months)\newlineFirst, we need to convert the annual interest rate to a monthly interest rate and calculate the total number of payments.\newlineAnnual interest rate = 5%5\%\newlineMonthly interest rate = 5%12\frac{5\%}{12} months = 0.00416670.0041667 (approximately)\newlineTotal number of payments = 33 years PP00 months/year = PP11 months
  2. Interest Rate Conversion: Now we can plug these values into the formula to calculate the loan amount.\newlineP=$250×[1−(1+0.0041667)−360.0041667]P = \$250 \times \left[\frac{1 - (1 + 0.0041667)^{-36}}{0.0041667}\right]\newlineBefore we calculate this, let's simplify the expression inside the brackets.\newline(1+0.0041667)−36(1 + 0.0041667)^{-36} is the same as 1(1+0.0041667)36\frac{1}{(1 + 0.0041667)^{36}}
  3. Calculate Total Payments: We calculate (1+0.0041667)36(1 + 0.0041667)^{36} using a calculator.\newline(1+0.0041667)36≈1.15762(1 + 0.0041667)^{36} \approx 1.15762 (rounded to five decimal places)\newlineNow we take the reciprocal of this value to get the present value factor.\newline1/1.15762≈0.863851 / 1.15762 \approx 0.86385 (rounded to five decimal places)
  4. Calculate Present Value Factor: We can now complete the calculation for the present value of the annuity (loan amount).\newlineP=$250×[1−0.863850.0041667]P = \$250 \times \left[\frac{1 - 0.86385}{0.0041667}\right]\newlineP=$250×[0.136150.0041667]P = \$250 \times \left[\frac{0.13615}{0.0041667}\right]\newlineP=$250×32.66739P = \$250 \times 32.66739 (rounded to five decimal places)\newlineP≈$8166.85P \approx \$8166.85

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