A sum of Rs. 10000 is borrowed at 8% p.a. compounded annually which is paid back in 3 equal annual installments . What is the amount of each installments ?

Answer

To solve the problem of finding the amount of each installment for a loan of Rs. 10,000 borrowed at an interest rate of 8% per annum compounded annually, paid back in 3 equal annual installments, we can follow these steps:

### Step 1: Understand the formula for calculating the present value of annuities
The formula for the present value of an annuity is given by:

\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]

Where:
- \(PV\) = Present Value (the amount borrowed)
- \(P\) = Amount of each installment
- \(r\) = Interest rate per period (as a decimal)
- \(n\) = Number of periods (installments)

### Step 2: Substitute the known values into the formula
Here, we know:
- \(PV = 10,000\)
- \(r = \frac{8}{100} = 0.08\)
- \(n = 3\)

Substituting these values into the formula gives:

\[ 10,000 = P \times \left( \frac{1 - (1 + 0.08)^{-3}}{0.08} \right) \]

### Step 3: Calculate the annuity factor
Now, we need to calculate the annuity factor:

\[ 1 + r = 1 + 0.08 = 1.08 \]
\[ (1 + r)^{-n} = (1.08)^{-3} \approx 0.7938 \]
\[ 1 - (1 + r)^{-n} = 1 - 0.7938 \approx 0.2062 \]
\[ \frac{0.2062}{0.08} \approx 2.5775 \]

### Step 4: Solve for \(P\)
Now, we can substitute this back into the equation:

\[ 10,000 = P \times 2.5775 \]
\[ P = \frac{10,000}{2.5775} \approx 3880.34 \]

### Step 5: Conclusion
The amount of each installment is approximately Rs. 3880.34.

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