A sum of money becomes ₹35,280 after 2 years and ₹37.044 after 3 years when lent on compound interest. Find the principal amount.

Options

A. ₹32,000
B. ₹28,000
C. ₹31,500
D. ₹32,500

Explanation

To find the principal amount when a sum of money becomes ₹35,280 after 2 years and ₹37,044 after 3 years at compound interest, we can follow these steps:

### Step 1: Set Up the Equations
Let the principal amount be \( P \) and the rate of interest be \( R \% \).

According to the compound interest formula:
- After 2 years, the amount \( A_2 \) is given by:
\[ A_2 = P \left(1 + \frac{R}{100}\right)^2 = 35,280 \]

- After 3 years, the amount \( A_3 \) is given by:
\[ A_3 = P \left(1 + \frac{R}{100}\right)^3 = 37,044 \]

### Step 2: Write the Equations
From the above, we can write two equations:
1. \( P \left(1 + \frac{R}{100}\right)^2 = 35,280 \) (Equation 1)
2. \( P \left(1 + \frac{R}{100}\right)^3 = 37,044 \) (Equation 2)

### Step 3: Divide the Equations
To eliminate \( P \), we can divide Equation 2 by Equation 1:
\[ \frac{P \left(1 + \frac{R}{100}\right)^3}{P \left(1 + \frac{R}{100}\right)^2} = \frac{37,044}{35,280} \]
This simplifies to:
\[ 1 + \frac{R}{100} = \frac{37,044}{35,280} \]

### Step 4: Calculate the Right Side
Now, we calculate \( \frac{37,044}{35,280} \):
\[ \frac{37,044}{35,280} = \frac{21}{20} \]
Thus, we have:
\[ 1 + \frac{R}{100} = \frac{21}{20} \]

### Step 5: Solve for \( R \)
Subtract 1 from both sides:
\[ \frac{R}{100} = \frac{21}{20} - 1 = \frac{21}{20} - \frac{20}{20} = \frac{1}{20} \]
Now, multiply by 100 to find \( R \):
\[ R = 100 \times \frac{1}{20} = 5 \]

### Step 6: Substitute \( R \) Back to Find \( P \)
Now that we have \( R = 5 \), we can substitute it back into Equation 1 to find \( P \):
\[ P \left(1 + \frac{5}{100}\right)^2 = 35,280 \]
This simplifies to:
\[ P \left(1.05\right)^2 = 35,280 \]
Calculating \( (1.05)^2 \):
\[ (1.05)^2 = 1.1025 \]
So we have:
\[ P \times 1.1025 = 35,280 \]
Now, solve for \( P \):
\[ P = \frac{35,280}{1.1025} \]
Calculating \( P \):
\[ P = 32,000 \]

### Final Answer
The principal amount is ₹32,000.

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