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, [tex]M=\frac{\text{Amount}}{\text{Discount factor}}[/tex]
Discount factor [tex]D=\frac{1-(1+i)^{-n}}{i}[/tex]
Where, Amount = $140,000
Rate r= 5%=0.05 compounded quarterly
[tex]i=\frac{0.05}{12}=0.00416[/tex]
Time = 30 years
[tex]n=30\times12=360[/tex]
Now, put all the values we get,
[tex]D=\frac{1-(1+i)^{-n}}{i}[/tex]
[tex]D=\frac{1-(1+0.00416)^{-360}}{0.00416}[/tex]
[tex]D=\frac{1-(1.00416)^{-360}}{0.00416}[/tex]
[tex]D=\frac{1-0.2243}{0.00416}[/tex]
[tex]D=\frac{0.7757}{0.00416}[/tex]
[tex]D=186.46[/tex]
Monthly payment, [tex]M=\frac{\text{Amount}}{\text{Discount factor}}[/tex]
[tex]M=\frac{140000}{186.46}[/tex]
[tex]M=750.8[/tex]
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 [tex]\frac{22530}{9012}=2.48[/tex]
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:
332 months approx.
Step-by-step explanation:
EMI formula is :
[tex]\frac{p\times r\times(1+r)^{n}}{(1+r)^{n}-1}[/tex]
Substituting p = 140000
r = [tex]5/12/100=0.004166[/tex]
n = [tex]30\times12=360[/tex]
[tex]\frac{140000\times0.004166\times(1+0.004166)^{360}}{(1+0.004166)^{360}-1}[/tex]
Solving this we get;
EMI = $751.48
So, total amount that is to re paid in 360 installments = [tex]751.48\times360=270532.80[/tex] dollars
Now, if one extra payment is made each year that means 13 payments per year.
So, amount paid per year = [tex]751.48\times13=9769.24[/tex] dollars
Now, $270532.80 will be repaid in = [tex]\frac{270532.80}{9769.24}[/tex]
= 27.69 years
This will become approx [tex]27.69\times12=332.28[/tex] months
= 332 months.
