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

time = 318.77

It will be after 318 months

Step-by-step explanation:

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

which present value is 180,000

`C * (1-(1+r)^(-time) )/(rate) = PV\\`

C = 1000

rate = 0.004

time ??

PV = 180,000

`1,000 * (1-(1+0.004)^(-time) )/(0.004) = 180,000\\`

We clear out the dividend:

`1-(1+0.004)^(-time) = (180,000* 0.004)/(1,000)\\`

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

`(-1) * (-(1+0.004)^(-time)) = (-1) * (0.72 - 1)}\\`

`1.004^(-time) = 0.28`

We now use logarithmics to solve for time

`log_(1.004)0.28 = (log0.28)/(log1.004) = -318.87 =-time`

time = 319