Answer:
11.5 years
Step-by-step explanation:
The formula for the monthly payment of a loan is:
[tex] A = P\dfrac{r(1 + r)^n}{(1 + r)^n - 1} [/tex]
where
A = monthly payment
r = interest rate per period
n = number of periods
Assuming a monthly payment, we have
P = 17500
r = 0.065/12
The total amount of payments is the monthly payment times the number of payments, so
An = 24893.75
We can try 10 years. n = 10 * 12 = 120
[tex] A = 17500\dfrac{\frac{0.065}{12}(1 + \frac{0.065}{12})^{120}}{(1 + \frac{0.065}{12})^{120} - 1} = 198.71 [/tex]
An = 198.71 * 120 = 23845.08
The total amount of payments is too low.
Let's try 11 years.
n = 11 * 12 = 132
[tex] A = 17500\dfrac{\frac{0.065}{12}(1 + \frac{0.065}{12})^{132}}{(1 + \frac{0.065}{12})^{132} - 1} = 185.92 [/tex]
An = 185.92 * 132 = 24540.90
The total amount of payments is still too low.
Let's try 12 years.
n = 12 * 12 = 144
[tex] A = 17500\dfrac{\frac{0.065}{12}(1 + \frac{0.065}{12})^{144}}{(1 + \frac{0.065}{12})^{144} - 1} = 175.34 [/tex]
An = 185.92 * 144 = 25248.41
The total amount of payments is too high.
11 years is too little, and 12 years is too much.
Let's try 11.5 years.
n = 11.5 * 12 = 138
[tex] A = 17500\dfrac{\frac{0.065}{12}(1 + \frac{0.065}{12})^{138}}{(1 + \frac{0.065}{12})^{138} - 1} = 180.39 [/tex]
An = 180.39 * 138 = 24893.20
11.5 gives a total of payments of $24,893.20 which is very close to $23.893.75.
Answer: 11.5 years
The total amount of payments is too high.
