SOLUTION: Hi tom buys a house for 210,000 dollars pays a 60,000 deposit and then pays off the balance at 950 dollars per month 25 years. find the yearly interest paid and the interest

Question 1110313: Hi
tom buys a house for 210,000 dollars pays a 60,000 deposit and then pays off the balance at 950 dollars per month 25 years.
find the yearly interest paid and the interest rate charged
thanks
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
The monthly payment for a mortgage is calculated using the following formula,
:
M = Pi/[q(1-[1+(i/q)]^-nq)], where M is the monthly payment, P is the principal amount being financed, i is the interest rate, q is the number of payments per year and n is the number of years
:
for this problem, M = $959, P = $210,000 - $60,000 = $150,000, q = 12, n = 25
:
the equation becomes
:
950 = 150000i/[12(1-[1+(i/12)]^-300)]
:
The interest rate i can't be solved for algebraically - it must be found numerically.
:
(1) Try a value i1 for i. A reasonable guess that will be too high is the interest rate for simple interest, i1 = Mnq/P - 1.(i1 = 950(300)/150000 = 0.9
(2) Using that, compute the principal P1 for that rate. If P1 < P, then the interest rate i1 > i, but if P1 > P, then i1 < i.
P1 = M[q(1-[1+(i/q)]^-nq)]/i
P1 = 950[12(1-[1+(0.9/12)]^-300)]/0.9
P1 = 12666.67
(3) Using the fact from (2), try another interest rate i2 and compute the corresponding principal value P2.
let i2 be 0.07
P2 = 950[12(1-[1+(0.07/12)]^-300)]/0.07
P2 = 134412.55
(4) Then try the new rate,
i3 = (i1[P2-P]+i2[P-P1])/(P2-P1)
(5) Replace the worse of the two starting interest rates with this new rate i3.
Repeat steps 4 and 5, always using the two interest rates with the corresponding principals closest to P as i1 and i2. Continue until you have found an interest rate such that the corresponding principal when rounded to the nearest cent gives P. Then i is equal to that interest rate.
:
carry out this series of repetitions
:
Note that i = 5.81% yields a principal of $150141.30
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