Sapna borrowed some money on compound interest and returned it in three years in equal annual instalments. If the rate of interest is `15%` per annum and annual instalment is ₹ 486680, find the sum borrowed :

Options

A. 1112220
B. 1111200
C. 1122000
D. none of these

Answer

To find the sum borrowed by Sapna, we will use the formula for calculating the present value of an annuity (equal annual installments) under compound interest. The formula for the present value \( PV \) of an annuity is given by:

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

Where:
- \( P \) = annual payment (installment)
- \( r \) = rate of interest per period (as a decimal)
- \( n \) = number of periods (years)

Given:
- Annual installment \( P = ₹ 486680 \)
- Rate of interest \( r = 15\% = 0.15 \)
- Number of years \( n = 3 \)

### Step 1: Calculate the present value using the formula

Substituting the values into the formula:

\[ PV = 486680 \times \frac{(1 - (1 + 0.15)^{-3})}{0.15} \]

### Step 2: Calculate \( (1 + r)^{-n} \)

First, calculate \( (1 + 0.15)^{-3} \):

\[ (1 + 0.15)^{-3} = (1.15)^{-3} \approx 0.6575 \]

### Step 3: Substitute back into the formula

Now substitute this value back into the present value formula:

\[ PV = 486680 \times \frac{(1 - 0.6575)}{0.15} \]

### Step 4: Calculate \( (1 - 0.6575) \)

Calculate \( (1 - 0.6575) \):

\[ 1 - 0.6575 = 0.3425 \]

### Step 5: Calculate the fraction

Now calculate the fraction:

\[ \frac{0.3425}{0.15} \approx 2.2833 \]

### Step 6: Calculate the present value

Now, multiply this result by the annual installment:

\[ PV = 486680 \times 2.2833 \approx 1111111.11 \]

### Step 7: Round off to the nearest whole number

Thus, the sum borrowed is approximately:

\[ PV \approx ₹ 1111111 \]

### Final Answer

The sum borrowed by Sapna is ₹ 1111111.

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