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

Answer:

Karen take 330 months to pay off her mortgage.

Step-by-step 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 $D=\frac{1-(1+i)^{-n}}{i}$

Where, Amount = $140,000

Rate r= 5%=0.05 compounded quarterly

$i=\frac{0.05}{12}=0.00416$

Time = 30 years  

$n=30\times12=360$

Now, put all the values we get,

$D=\frac{1-(1+i)^{-n}}{i}$

$D=\frac{1-(1+0.00416)^{-360}}{0.00416}$

$D=\frac{1-(1.00416)^{-360}}{0.00416}$

$D=\frac{1-0.2243}{0.00416}$

$D=\frac{0.7757}{0.00416}$  

$D=186.46$  

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

$M=\frac{140000}{186.46}$

$M=750.8$

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 $\frac{22530}{9012}=2.48$

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.

Answer 2

Answer:

332 months approx.

Step-by-step explanation:

EMI formula is :

$\frac{p\times r\times(1+r)^{n}}{(1+r)^{n}-1}$

Substituting p = 140000

r = $5/12/100=0.004166$

n = $30\times12=360$

$\frac{140000\times0.004166\times(1+0.004166)^{360}}{(1+0.004166)^{360}-1}$

Solving this we get;

EMI = $751.48

So, total amount that is to re paid in 360 installments = $751.48\times360=270532.80$ dollars

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

So, amount paid per year = $751.48\times13=9769.24$ dollars

Now, $270532.80 will be repaid in = $\frac{270532.80}{9769.24}$

= 27.69 years

This will become approx $27.69\times12=332.28$ months

= 332 months.