Respuesta :
Answer:
The balance be after he has made exactly half of his monthly payments is $56881.4.
Step-by-step explanation:
Given : Dean took out a 10-year loan for $40,000 at an APR of 4% compounded monthly.
To find : What will his balance be after he has made exactly half of his monthly payments?
Solution :
Formula of monthly payment ,
[tex]M=\frac{\text{Amount}}{\text{Discount factor}}[/tex]
Discount factor [tex]D=\frac{1-(1+i)^{-n}}{i}[/tex]
Where, Amount = $40,000
Rate r= 4% compounded monthly
[tex]i=\frac{4}{100}=0.04[/tex]
Time = 10 years
[tex]n=10\times12=120[/tex]
Now, put all the values we get,
[tex]D=\frac{1-(1+i)^{-n}}{i}[/tex]
[tex]D=\frac{1-(1+0.04)^{-120}}{0.04}[/tex]
[tex]D=\frac{1-(1.04)^{-120}}{0.04}[/tex]
[tex]D=\frac{1-0.00903}{0.04}[/tex]
[tex]D=\frac{0.9909}{0.04}[/tex]
[tex]D=24.7725[/tex]
[tex]M=\frac{\text{Amount}}{\text{Discount factor}}[/tex]
[tex]M=\frac{40000}{24.7725}[/tex]
[tex]M=1614.69[/tex]
Half of the monthly payment is $807.345
Payment for 10 years is [tex]807.345\times 120=96881.4[/tex]
The balance is $96881.4-$40000=$56881.4
Therefore, The balance be after he has made exactly half of his monthly payments is $56881.4.
