APPLICATIONS OF EXPONENTIAL AND APPLICATIONS OF EXPONENTIAL AND
(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:
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)
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.
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%.
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%.
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%.
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%.
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%.
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%.
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%.
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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