Question

Suppose you take out a home mortgage for ​$180.000 at a monthly interest rate of 0.4​%. If you make payments of ​$1000​/month, after how many months will the loan balance be​ zero? Estimate the answer by graphing the sequence of loan balances and then obtain an exact answer.

Answer 1

Answer:

time = 318.77

It will be after 318 months

Explanation:

We are asked to find the time of an annuity of 1,000 monthly payment

which present value is 180,000

$C * \frac{1-(1+r)^{-time} }{rate} = PV$

C = 1000

rate = 0.004

time ??

PV = 180,000

$1,000 \times \frac{1-(1+0.004)^{-time} }{0.004} = 180,000$

We clear out the dividend:

$1-(1+0.004)^{-time} = \frac{180,000\times 0.004}{1,000}$

Then we clear the power up, notice it is negative, so we have to multiply by (-1)

$(-1) \times (-(1+0.004)^{-time}) = (-1) \times (0.72 - 1)}$

$1.004^{-time} = 0.28$

We now use logarithmics to solve for time

$log_{1.004}0.28 = \frac{log0.28}{log1.004} = -318.87 =-time$

time = 319