(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:
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)
This amount is less than $700;, therefore the rate of 9% is too
low.
This amount is more than $700; therefore, the rate of 10% is too
high. The actual amount is between 9% and 10%.
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
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.
This is about as close as you will get because the interest rate will most likely rounded to 9.52%.
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Author: Nancy Marcus