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Find the monthly payment for a 3030 year loan of 100000100000 with an APR of 4%4\%?

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Full solution

Q. Find the monthly payment for a 3030 year loan of 100000100000 with an APR of 4%4\%?
  1. Convert APR to monthly rate: Convert the annual percentage rate (APR) to a monthly interest rate.\newlineThe APR is given as 4%4\%, which needs to be converted to a monthly rate since we are calculating a monthly payment. To do this, divide the APR by 1212 (the number of months in a year).\newlineMonthly interest rate = APR / 1212\newline= 4%4\% / 1212\newline= 0.040.04 / 1212\newline= 0.003333…0.003333\ldots
  2. Convert rate to decimal: Convert the monthly interest rate to a decimal.\newlineTo use the monthly interest rate in calculations, it needs to be in decimal form rather than a percentage.\newlineMonthly interest rate (in decimal) = 0.003333…0.003333\ldots
  3. Determine number of payments: Determine the number of monthly payments over the life of the loan. Since the loan is for 3030 years and payments are monthly, multiply the number of years by 1212. Number of monthly payments = 3030 years ×12\times 12 months/year =360= 360 payments
  4. Use payment formula: Use the formula for calculating the monthly payment on an installment loan.\newlineThe formula is:\newlineMonthly payment = P×(r(1+r)n((1+r)n−1))P \times \left(\frac{r(1+r)^n}{((1+r)^n - 1)}\right)\newlineWhere:\newlinePP = principal amount (the initial amount of the loan)\newlinerr = monthly interest rate (in decimal form)\newlinenn = total number of payments
  5. Plug values into formula: Plug the values into the formula to calculate the monthly payment.\newlineP=$100,000P = \$100,000\newliner=0.003333...r = 0.003333...\newlinen=360n = 360\newlineMonthly payment = $100,000×(0.003333...(1+0.003333...)360)/((1+0.003333...)360−1)\$100,000 \times (0.003333...(1+0.003333...)^{360}) / ((1+0.003333...)^{360} - 1)
  6. Calculate monthly payment: Calculate the monthly payment using the values provided.\newlineThis step involves complex calculations that are typically done with a calculator or financial software. For the sake of this example, we will simplify the calculation process.\newlineMonthly payment ≈\approx $100,000×(0.003333...×(1.003333...)360)/((1.003333...)360−1)\$100,000 \times (0.003333... \times (1.003333...)^{360}) / ((1.003333...)^{360} - 1)
  7. Perform final calculation: Perform the calculation to find the monthly payment. Using a calculator or financial software to handle the exponentiation and division, we find: Monthly payment ≈\approx $477.42\$477.42

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