Question

Karen has a $140,000 home financed with a 30-year loan at 5%. If she wished to pay one extra monthly payment each year, how many months will it take her to pay off her mortgage?

Answer 1

332 months approx.

Explanation:

EMI formula is :

`(p* r*(1+r)^(n))/((1+r)^(n)-1)`

Substituting p = 140000

r =
`5/12/100=0.004166`

n =
`30*12=360`

`(140000*0.004166*(1+0.004166)^(360))/((1+0.004166)^(360)-1)`

Solving this we get;

EMI = $751.48

So, total amount that is to re paid in 360 installments =
`751.48*360=270532.80` dollars

Now, if one extra payment is made each year that means 13 payments per year.

So, amount paid per year =
`751.48*13=9769.24` dollars

Now, $270532.80 will be repaid in =
`(270532.80)/(9769.24)`

= 27.69 years

This will become approx
`27.69*12=332.28` months

= 332 months.

Answer 2

Karen take 330 months to pay off her mortgage.

Explanation:

Given : Karen has a $140,000 home financed with a 30-year loan at 5%. If she wished to pay one extra monthly payment each year.

To find : How many months will it take her to pay off her mortgage?

Solution : First we find the monthly payment

Formula of monthly payment

Monthly payment,
`M=\frac{\text{Amount}}{\text{Discount factor}}`

Discount factor
`D=(1-(1+i)^(-n))/(i)`

Where, Amount = $140,000

Rate r= 5%=0.05 compounded quarterly

`i=(0.05)/(12)=0.00416`

Time = 30 years

`n=30*12=360`

Now, put all the values we get,

`D=(1-(1+i)^(-n))/(i)`

`D=(1-(1+0.00416)^(-360))/(0.00416)`

`D=(1-(1.00416)^(-360))/(0.00416)`

`D=(1-0.2243)/(0.00416)`

`D=(0.7757)/(0.00416)`

`D=186.46`

Monthly payment,
`M=\frac{\text{Amount}}{\text{Discount factor}}`

`M=(140000)/(186.46)`

`M=750.8`

Approximately,monthly payment is $751

Extra payment for each year is = 751 × 30 = $22530

She wished to pay one extra monthly payment each year

i.e, Yearly payment is 751 × 12 = $9012

Years she paid extra payment is
`(22530)/(9012)=2.48`

Total years is 30 years , Extra year is 2.5 years

Time she take off to pay her mortgage is 30-2.5 = 27.5 years

1 year = 12 months

27.5 years = 27.5 × 12 = 330 months

Therefore, Karen take 330 months to pay off her mortgage.