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 4: Suppose you need to take out a mortgage of $80,000. All you can afford for monthly payments is $700. 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 $80,000 and pay it back in 300 monthly payments of $700.

Answer: 9.52%

Solution and Explanations:

Step 1: In the equation

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substitute 700 for P (the monthly payment), 12 for n (the number of payments per year, 25 for t (the number of years), and $80,000 for M (the mortgage amount). You are solving for r (the annual interest)

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Step 2: We are going to solve for the rate using iteration (trial and error): Try a 9% interest rate by substituting .09 in the right side of the above equation and comparing your answer with 700

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This amount is less than $700;, therefore the rate of 9% is too low.

Step 3: Try 10%:

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This amount is more than $700; therefore, the rate of 10% is too high. The actual amount is between 9% and 10%.

Step 4: Try 9.5%:

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This amount is less than $700; therefore, the rate of 9.5% is too low. The actual rate is between 9.5% and 10%, and the rate is closer to 9.5% than 10% because $698.96 is closer to $700 than $726.86

Step 5: Try 9.52%

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This amount is less than $700; therefore, the rate of 9.52% is very close, but still too high. The actual rate is between 9.5% and 9.52%, and the rate is closer to 9.52% than 9.5% because $700.07 is closer to $700 than $698.96.

Step 6: If you want to get even closer, try 9.519%:

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This is about as close as you will get because the interest rate will most likely rounded to 9.52%.

If you would like to work a problem and review the answer and solution, click on Problem.

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