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.

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

Answer: 8.42%

Solution and Explanations:

Step 1: In the equation

substitute $100,000 for M (the mortgage amount), 12 for n (the number of payments per year, $800 for P (the monthly payment), and 25 for t (the term of the mortgage in years). You are solving for r (the annual interest rate)

Step 2: We are going to solve this problem by iteration (trial and error). Let's start with an interest rate of 9%. Substitute .09 for r in the right side of the above equation and compare it to the $800 on the left side of the equation.
If, after the substitution, the value of the right side is lower than $800, it means that your guess of 9% was too low. If, after the substitution, the value of the right side is higher than $800, it means that your guess of 9% was too high.

The value of the right side of the equation, using r = 9%, is higher than $800. This means that your guess of 9% for the rate was too high.

Step 3: Try 8%:

The value of the right side of the equation, using r = 8%, is lower than $800. This means that your guess of 8% for the rate was too low. Since 9% was too high and 8% was too low, the real rate is between 8% and 9%.

Step 4: Try 8.5%:

The value of the right side of the equation, using r = 8.5%, is a littler higher than $800. This means that your guess of 8.5% for the rate was a little high. Since 8.5% was too high and 8% was too low, the real rate is between 8% and 8.5%. Since $805.23 is very close to $800, the real rate is close to 8.5%.

Step 5: Try 8.4%:

The value of the right side of the equation, using r = 8.4%, is a little lower than $800. This means that your guess of 8.4% for the rate was too low. Since 8.5% was too high and 8.4% was too low, the real rate is between 8.4% and 8.5%.

Step 6: Try 8.45%:

The value of the right side of the equation, using r = 8.45%, is a little higher than $800. This means that your guess of 8.45% for the rate was a little high. Since 8.45 was too high and 8.4% was too low, the real rate is between 8.4% and 8.45%.

Step 7: Try 8.42%:

The value of the right side of the equation, using r = 8.42%, is lower than $800. This means that your guess of 8.42% for the rate was close but still too low. Since 8.45% was too high and 8.42% was too low, the real rate is between 8.42% and 8.45%.

Step 8: Try 8.43%:

The value of the right side of the equation, using r = 8.43%, is very close but still higher than $800. This means that your guess of 8.43% for the rate was too low. Since 8.42% was too high and 8.43% was too low, the real rate is between 8.42% and 8.43%.

Step 9: Try 8.422%:

The value of the right side of the equation, using r = 8.422%, is very close but still lower than $800. This means that your guess of 8.422 % for the rate is very close but still a little . Since 8.43% was too high and 8.422% was too low, the real rate is between 8.422% and 8.43%.

Step 10: Try 8.423%:

The value of the right side of the equation, using r = 8.423%, is higher than $800. This means that your guess of 8.423% for the rate was too high. Since 8.423% was too high and 8.422% was too low, the real rate is between 8.422% and 8.423%.

We could keep on until you reached the degree of accuracy you desired. However, most banks round to four decimals. Since both 8.422% and 8.423% rounded to 8.42%, that is our answer.

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