Answer:
The APR is 4.38%.
Step-by-step explanation:
Given : Matt Johnson takes out a mortgage for $240,000. There is a loan of 30 years at $1,200 per month. This gives a total interest of $192,000.
To find : What is the APR using the formula?
Solution : 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 = $240,000
Monthly payment M=$1200
APR i =?
Time = 30 years
[tex]n=30\times12=360[/tex]
Monthly payment, [tex]M=\frac{\text{A}}{\frac{1-(1+i)^{-n}}{i}}[/tex]
[tex]M=\frac{A\times i}{1-(1+i)^{-n}}[/tex]
Substitute value,
[tex]1200=\frac{240000\times i}{1-(1+i)^{-360}}[/tex]
[tex]\frac{1-(1+i)^{-360}}{i}=\frac{240000}{1200}[/tex]
[tex]\frac{1-(1+i)^{-360}}{i}=200[/tex]
Solving using calculator,
[tex]i=0.003655[/tex]
Now, In percentage and in months
[tex]i=0.003655\times 100\times 12[/tex]
[tex]i=4.38\%[/tex]
The APR is 4.38%.
