Question

A ​$257 comma 800 mortgage for 30 years for a new home is obtained at the rate of 7.8​% compounded monthly. Find​ (a) the monthly​ payment, (b) the interest in the first ​payment, (c) the principal repaid in the first​ payment, and​ (d) the finance charge. ​(a) The monthly payment on the mortgage is ​$ nothing. ​(Round the final answer to two decimal places as needed. Round all intermediate values to six decimal places as​ needed.) ​(b) The interest in the first payment is ​$ nothing. ​(Round the final answer to two decimal places as needed. Round all intermediate values to six decimal places as​ needed.) ​(c) The principal repaid in the first payment is ​$ nothing. ​(Round the final answer to two decimal places as needed. Round all intermediate values to six decimal places as​ needed.) ​(d) The finance charge is ​$ nothing.

Answer 1

According to given information principal amount P = $257800

Interest rate r =7.8% = 7.8/100 = 0.078

Compounding frequency is monthly so r = 0.078/12 = 0.0065

Now number of payments t = 30 x 12 = 360

Now we can use the below formula to find the payment (PMT)

PMT = [ p x r x (1+r)^t ] / [(1+r)^t-1]

PMT = [257800 x 0.0065 x (1+0.005)³⁶⁰ ] / [(1+0.0065)³⁶⁰-1]

PMT = [1675.7 x (1.0065)³⁶⁰/ [(1.0065)³⁶⁰ - 1]

PMT = [1675.7 x 10.302924] / [10.302924 - 1]

PMT = [17264.609746] / [9.302924]

PMT = $1855.826162 ~ $1855.83

1. So the monthly payment = $1855.83

2. Interest for the first payment

Principal amount x 0.0065 = 257800 x 0.0065 = $1675.7

3. The principal repaid in the first payment

Monthly PMT – interest amount = 1855.83 – 1675.7 = $180.13

4. Total finance charge can be calculated as

Finance charge = total payment – principal amount

Total payment = number of payments x PMT = 360 x 1855.83 = $668098.8

Now finance charge = 668098.8 – 257800 = 410298.8

So finance charge = $410298.8