(**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 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?

**Answer:** 371.063 months or 30.92192 years or 30 years and 11
months. You would have 370 payments of $800 and the last payment
would be $850.40. The interest paid over the term of the mortgage
would be $216,850.40.

**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 9% for r (the annual interest rate)). You are solving for t (the term of the mortgage in years) or 12t( the term of the mortgage in months)

**Step 2:**- Multiply both sides of the above equation by 12:
**Step 3:**- Divide both sides of the equation by
(.09)($100,000):
**Step 4:**- Multiply both sides of the equation by
to get

**Step 5:**- Divide both sides of the equation by
1.06666666667:
**Step 6:**- Subtract 1 from both sides of the above equation:
**Step 7:**- Multiply both sides by - 1:
**Step 8:**- Take the natural log of both sides of the above
equation:
**Step 9:**- Simplify the left side of the above equation
using the third rule of logarithms:
**Step 10:**- Simplify the left side of the above equation:
**Step 11:**- Divide both sides of the above equation by:
-0.0896641780644:
or 371.063064259 or 371 months.

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

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