# A ​\$257 comma 800 mortgage for

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

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
1. Interest for the first payment

Principal amount x 0.0065 = 257800 x0.0065 = \$1675.7

1. The principal repaid in the first payment

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

1. Total finance charge can be calculated as

Finance charge = total payment– principal amount

Total payment = number ofpayments x PMT = 360 x 1855.83 = \$668098.8

Now finance charge = 668098.8– 257800 = 410298.8

So finance charge =\$410298.8

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