SOLUTION: You want to buy a $238,000 home. You plan to pay 5% as a down payment, and take out a 30 year loan at 5.5% interest for the rest. How much money in interest will you save if yo

Algebra ->  Customizable Word Problem Solvers  -> Finance -> SOLUTION: You want to buy a $238,000 home. You plan to pay 5% as a down payment, and take out a 30 year loan at 5.5% interest for the rest. How much money in interest will you save if yo      Log On

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Question 1206015: You want to buy a $238,000 home. You plan to pay 5% as a down payment, and take out a 30 year loan at 5.5% interest for the rest.
How much money in interest will you save if you finance for 15 years instead of 30 years?
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: $129,619.80

Work Shown

home value = $238,000
down payment = 5% of home value = 0.05*238000 = $11900

loan amount = home value - down payment
loan amount = $238,000 - $11,900
loan amount = $226,100
Or
loan amount = 95% of home value (since 5% is already paid off)
loan amount = 0.95*($238,000)
loan amount = $226,100


Monthly payment formula
P = (L*i)/( 1-(1+i)^(-n) )
where,
P = monthly payment
L = loan amount
i = monthly interest rate in decimal form
n = number of months

In this case,
L = 226100
i = 0.055/12

If you finance for 15 years, aka 15*12 = 180 months, then we'll have n = 180

Which leads to this monthly payment:
P = (L*i)/( 1-(1+i)^(-n) )
P = (226100*0.055/12)/( 1-(1+0.055/12)^(-180) )
P = 1847.425690898
P = 1847.43

The monthly payment for a 15 year mortgage is $1847.43
You can use an online calculator such as this to confirm.
https://www.calculatorsoup.com/calculators/financial/loan-calculator.php

If you pay $1847.43 per month for 180 months, then you'll pay back a total of $332,537.40 due to the scratch work 180*1847.43 = 332537.40
Let A = 332537.40 since we'll refer to this value later.


Now let's consider a 30 year mortgage.
This is over a span of 12*30 = 360 months.
We'll have n = 360 but every other input stays the same.
P = (L*i)/( 1-(1+i)^(-n) )
P = (226100*0.055/12)/( 1-(1+0.055/12)^(-360) )
P = 1283.770932046
P = 1283.77

This time you pay back $1283.77 per month for 360 months.
The total amount paid back is 360*1283.77 = 462157.20 dollars.
Let's call this value B.

Subtract the two results A and B to find how much is saved.
B - A = 462157.20 - 332537.40 = $129,619.80 which is the final answer.


Or you can compute the interest for each mortgage length and then subtract those interest values to find out how much is saved.