Question

Please post a tentative solution to the problem below following the mortgage formula and using exponential equations and logarithms (as appropriate) to solve for time.

Mortgage Formula:


`1-(1+(r)/(n) )^-^n^t =(rM)/(Pn)`

where
P = monthly payment
r = annual rate
n = number of payments per year (typically 12)
t = # years of mortgage

Suppose you wanted to take out a mortgage for $100,000 with monthly payments at 9%, but you can only afford $800 monthly payments. How long will you have to make payments to pay off the mortgage?

How much interest would you pay for this period (hint: subtract the total amount paid over the time of the mortgage and the original mortgage amount)?

Answer 1

The amount of interest paid over the time of the mortgage is $176,800.

Explanation:

To solve this problem, we will use the mortgage formula to find the time it will take to pay off the mortgage.

First, we need to convert the annual rate to a monthly rate by dividing by 12:

r = 9% / 12 = 0.75%

Next, we can plug in the values we know into the mortgage formula and solve for t:

$800 = P = 100000[r(1 + r)^nt]/[(1 + r)^nt - 1]

$800 = 100000[(0.0075)(1 + 0.0075)^12t]/[(1 + 0.0075)^12t - 1]

Multiplying both sides by [(1 + 0.0075)^12t - 1], we get:

(1 + 0.0075)^12t - 1 = 100000(0.0075)(1 + 0.0075)^12t

Dividing both sides by 100000(0.0075), we get:

(1 + 0.0075)^12t - 1 / 100000(0.0075) = 1

Now we can use logarithms to solve for t:

log(1 + 0.0075)^12t - 1 / 100000(0.0075) = log(1)

[(12t)log(1 + 0.0075) - log(1 - $800/100000(0.0075))] / 12log(1 + 0.0075) = 0

[(12t)log(1.0075) - 0.23074] / 12log(1.0075) = 0

12t = 0.23074 / (log(1.0075))

t = 0.23074 / (12log(1.0075))

t ≈ 346 months

Therefore, it will take approximately 346 months, or 28.83 years, to pay off the mortgage.

To find the amount of interest paid over this period, we can subtract the total amount paid from the original mortgage amount:

Total amount paid = $800 x 346 = $276,800

Interest paid = $276,800 - $100,000 = $176,800

Therefore, the amount of interest paid over the time of the mortgage is $176,800.