SOLUTION: A person purchased a ​$198741 home 10 years ago by paying ​20% down and signing a​ 30-year mortgage at 11.4​% compounded monthly. Interest rates have dropped and the owner

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Question 1189735: A person purchased a ​$198741 home 10 years ago by paying ​20% down and signing a​ 30-year mortgage at 11.4​% compounded monthly. Interest rates have dropped and the owner wants to refinance the unpaid balance by signing a new 20​-year mortgage at compounded 6.6% monthly. How much interest will refinancing​ save?
Money Saved: $

Please Help! Thank you :)
Found 3 solutions by ikleyn, MathTherapy, math_tutor2020:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
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Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

A person purchased a ​$198741 home 10 years ago by paying ​20% down and signing a​ 30-year mortgage at 11.4​% compounded monthly. Interest rates have dropped and the owner wants to refinance the unpaid balance by signing a new 20​-year mortgage at compounded 6.6% monthly. How much interest will refinancing​ save?
Money Saved: $

Please Help! Thank you :)
Amount of mortgage loan: .8(198,741) = $158,992.80

Each monthly payment for 1st 10 years (to date): $1,562.37

Using the formula for the balance on the mortgage loan, we find the unpaid balance after 10 years of monthly 
payments (120 periods) to be: $147,456.55 

Over the next 20 years, and at a 6.6% annual interest rate, each monthly payment to pay off balance ($147,456.55), is $1,108.09

INITIAL payoff amount for 30 years: $562,453.20 (360 * 1,562.37)

INITIAL 30-year interest amount: $562,453.20 - 158,992.80 = $403,460.40 

Total amount paid over 10 years (120 months): 120(1,562.37) = $187,484.40

Initial principal/mortgage-loan: $158,992.80
Principal/mortgage-loan balance after 10 years (120 months): $147,456.55
Amount applied to initial principal/mortgage-loan in 10 years: $158,992.80 - 147,456.55 = 11,536.25
Amount applied to mortgage interest in 10 years: $187,484.40 - 11,536.25 = $175,948.15

Unpaid interest at the 10-year mark: $403,460.40 - 175,948.15 = $227,512.25

Total amount to be paid in 240 monthly periods, or in 20 years, at the NEW rate: 240(1,108.09) = $265,941.60
Interest to be paid from total paid in 240 months, or 20 years: $265,941.60 - 147,456.55 = $118,485.05

Interest saved: $227,512.25 - 118,485.05 = $109,027.20
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

I'm using the formulas found on this page
https://www.mtgprofessor.com/formulas.htm

The home value is $198,741.
20% of it is paid off through the downpayment, so 80% is remaining
0.80*198741 = 158,992.80

This is the loan amount L
L = 158,992.80

For the original mortgage at 11.4%, the monthly rate c is
c = r/12 = 0.114/12 = 0.0095
and the number of months is n = 360 because 30*12 = 360.

So,
L = 158,992.80
c = 0.0095
n = 360
Plug those values into the first formula mentioned in that link above. You should get a monthly payment of P = 1562.37

Plug those values into the second formula along with lowercase p = 120 (since 10 years = 120 months) to find that B = 147,456.55
Unfortunately that link uses uppercase P to represent payment and lowercase p to denote the number of months paid into; the letter choice may seem a bit confusing.

Anyways, we get B = 147,456.55 which is the unpaid balance at year 10. This is the starting point, and we'll have two branching options

----------------------------------------------------

starting balance = 147,456.55

Mortgage A (interest rate = 11.4%)
monthly payment = 1562.37
total amount paid back = (240 months)*(1562.37 per month) = 374,968.80
m = Total interest paid
m = 374,968.80 - 147,456.55
m = 227,512.25

Mortgage B (interest rate = 6.6%)
monthly payment = 1108.09
total amount paid back = (240 months)*(1108.09 per month) = 265,941.60
n = Total interest paid
n = 265,941.60 - 147,456.55
n = 118,485.05

m-n = amount of interest saved by going with mortgage B
m-n = 227,512.25 - 118,485.05
m-n = 109,027.20


Answer: $109,027.20
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