APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

(Amortization Word Problems)

To solve an exponential or logarithmic word problem, convert the narrative to an equation and solve the equation.
There is a relationship between the mortgage amount, the number of payments, the amount of the payment, how often the payment is made, and the interest rate. The following formulas illustrate the relationship:

where P = the payment, r = the annual rate, M = the mortgage amount, t = the number of years, and n = the number of payments per year.

Example 1: What is the monthly payment on a mortgage of \$75,000 with an 8% interest rate that runs for 20 years, 25 years, 30 years? How much interest is paid over 20 years, 25 years, 30 years?

Answer: 20 years: payment = \$627.33 per month; after 20 years of payments, you will have paid \$150,559.20 (\$75,559.20 in interest)
25 years: payment = \$578.86 per month; after 25 years of payments, you will have paid \$173,658.00 (\$98,658.00 in interest).
30 years payment = \$\$550.32 per month; after 30 years of payments, you will have paid \$198,116.43 (\$123,116.43 in interest).

Solution and Explanations:

20 Years: In the equation

substitute \$75,000 for M (the mortgage amount), 8% for r (the annual interest rate), 20 for t (the number of years), and 12 for n (the number of payments per year. You are solving for P (the monthly payment for the 20 years)

The monthly payment will be \$627.33. After 20 years of payments, you will have paid

Everything over the initial \$75,000 is interest. Therefore, after 20 years, you will have paid

in interest.

25 Years: In the equation

substitute \$75,000 for M (the mortgage amount), 8% for r (the annual interest rate), 25 for t (the number of years), and 12 for n (the number of payments per year. You are solving for P (the monthly payment for the 25 years)

The monthly payment will be \$578.86. After 25 years of payments, you will have paid

Everything over the initial \$75,000 is interest. Therefore, after 25 years, you will have paid

in interest.

30 Years: In the equation

substitute \$75,000 for M (the mortgage amount), 8% for r (the annual interest rate), 30 for t (the number of years), and 12 for n (the number of payments per year. You are solving for P (the monthly payment for the 30 years)

The monthly payment will be \$550.32. After 30 years of payments, you will have paid

Everything over the initial \$75,000 is interest. Everything over the initial \$75,000 is interest. Therefore, after 30 years, you will have paid

in interest.

If you would like to review another example, click on Example.

Problem 1: Find the monthly payments on a \$100,000, 30-year mortgage, with monthly payments at 9.5%. How much interest will you have over the 30 years?

Problem 2: Suppose you wanted to take out a mortgage for \$100,000 with monthly payments at 9%, but you could only afford \$800 per month payments. How long would you have to make payments to pay off the mortgage, and how much interest would you pay for this payment period?

Problem 3: Suppose a bank offers you a 10% interest rate on a 20-year mortgage to be paid back with monthly payments. Suppose the most you can afford to pay in monthly payments is \$700. How much of a mortgage could you afford?

Problem 4: Suppose you need to take out a mortgage of \$100,000. All you can afford for monthly payments is \$800. You will retire in 25 years; therefore, the longest you can make these payments is 25 years. What interest rate would you need to take out a mortgage of \$100,000 and pay it back in 300 monthly payments of \$800

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