# Neeta takes out a 25-year mortgage of $40 000 to buy her house. Compound interest is charged on the loan at a rate of 8% per annum. She has to pay off the mortgage with 25 equal payments, the first...

Neeta takes out a 25-year mortgage of $40 000 to buy her house. Compound interest is charged on the loan at a rate of 8% per annum. She has to pay off the mortgage with 25 equal payments, the first of which is to be one year after the loan is taken out. What is the annual payment?

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### 1 Answer

Neeta has taken a mortgage of $40000 to buy her house. The rate of interest applicable is 8% per annum. The mortgage is for 25 years and has to be repaid in 25 equal installments starting one year after it was taken.

Let the amount that has to be repaid be represented by P. To estimate the value of P, use the concept of present value. As the rate of interest is 8%, a repayment after n years is equivalent to a value `P/(1 +0.08)^n` today. The sum of the present value of all the repayments is equal to the amount borrowed or $40000. This gives:

`P/(1.08) + P/(1.08)^2 + ...P/(1.08)^25 = 40000`

=> `P(1/(1.08) + 1/(1.08)^2 + ...1/(1.08)^25) = 40000`

`1/(1.08) + 1/(1.08)^2 + ...1/(1.08)^25` is a geometric series with the first term `1/1.08` and the common ratio `1/1.08` . The sum of 25 terms of such a series is `(1/1.08)*((1 - (1/1.08)^25)/(1 - 1/1.08)) ~~ 10.674`

P = 40000/10.674 = 3747.15

**The amount to be repaid every year is $3747.15**