# What is the amount to be paid yearly to repay a loan of $100,000 given at an interest rate of 10% p.a. and which has to be repaid in 10 years?

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### 1 Answer

Let the amount of loan be P, and the rate of interest be at r%, and the amount of repayment A in n years.

Then after the 1st year P becomes (1+r/100)*P and we repay an amount A.

So the next year this balance of (1+r/100)P-A) grows with interest ((1+r/100)P-A)(1+r/100) and we repay A and the balance for for the 3rd year = (1+r/100)^2*P - A(1+r/100) - A. In this way at the end of n th year the position is :

(1+r/100)^n P - A{(1+r/100)^(n-1)+(1+r/100)(n-2) +..+(1+r/100) + 1} which should be zero.

Or (1+r/100)^nP = A{(1+r/100)^(n -1}/{1+r/100 -1}.

Solving for A , we get:

A = (1+r/100)^n*p}(r/100)/ {1+r/100)^n -1}...(1)

We know P= $100000,n = 10 years and r = 10%= 10/100. So, 1+10/100 = 1.1

Therefore, A = {1.1^10*10^5}(0.1)/{1.1^10-1}

A = $16274.54 (approximated to two decimals).

Therefore , in order to settle the loan of $100000 in 10 years, we have to make the annual repayment of $16274.54.