an interest rate question-- have no clue..?

Question

Jeff and Darleen bought a house for $74,000. After making a $4000 down payment, they took out a $70,000 mortgage at 7% compounded semiannually for 30 years. How large are their semiannual payments to cover the principal and interst? and How much total moeny will they wind up paying fo th house?

I need to know also the formula ussed if appli.

Answer 1

Let each payment = $S.
Amount borrowed = $70,000.
Interest rate = 7% pa = 3.5% per half year = 0.035.
Time = 30 years = 60 half years (ie 60 installments)

After 1st installment
Amount owing (in $) = A1 = 70000 (1 + 0.035) - S

After 2nd installment
Amount owing (in $) = A2 = A1 (1 + 0.035) - S
= (70000 (1 + 0.035) - S)(1 + 0.035) - S
= 70000 (1 + 0.035)² - S(1 + 0.035) - S

After 3rd installment
Amount owing (in $) = A3 = A2 (1 + 0.035) - S
=(70000 (1 + 0.035)² - S(1 + 0.035) - S)(1 + 0.035) - S
=70000 (1 + 0.035)³ - S(1 + 0.035)² - S(1 + 0.035) - S
........

After 60th installment
Amount owing (in $) = A60 = A59 (1 + 0.035) - S
=70000(1 + 0.035)^60 - S(1 + 0.035)^59 - ... - S(1 + 0.035)³ - S(1 + 0.035)² - S(1 + 0.035) - S
= 0 (pay off in 60 installments!!)

Thus 70000(1 + 0.035)^60 = S(1 + 0.035)^59 + ...+ S(1 + 0.035)³ + S(1 + 0.035)² + S(1 + 0.035) + S
= S(1 + 1.035 + 1.032² + 1.035³ + .... + 1.035^59)
= S (1.035^60 - 1)/(1.035 - 1)

So S = 70000(1 .035)^60*0.035/(1.035^60 - 1)
Divide top and bottom by (1 .035)^60

S = 0.035 * 70000/(1 - (1.035)^(-60))
= 2806.20 (nearest cent)
So they pay $2806.20 every half year (nearest cent).

Total amount repaid = 60 x $2806.20
= $168,372

So interest paid over the loan period = $(168,372 - 70.000)
=$98,672

Answer 2

I doubt that your instructor expects you to to these without the formulae, and the formula for this one looks pretty formidable:
let
P = $70,000 (the amount financed)
r = 7% (annual rate)
i = 0.07/2 + 0.035 (semi-annual rate)
n = 30*2 = 60 (number of payments)
p = semi-annual payment
then
p = iP/(1-1/(1+i)^n) (the formula in calculation form)
p = 0.035*$70,000 / (1 - 1 / (1.035)^60)
p = 0.035*$70,000 / (1 - 1 / 7.8980909)
p = 0.035*$70,000 / (1 - 0.1269343)
p = 0.035*$70,000 / 0.6530657
p = 0.035*$70,000*1.145389

p = $2,806.20 (rounded)

A = np = 60*$2,806.20 =$168,372.00

Answer 3

The payment will be $3805.99 per 6 months.
They will have paid $228,359.59 when they are done.

I used a little spreadsheet I wrote.

Answer 4

Check out the PMT() function in excel

Answer 5

not sure not a bank person