A sum of money becomes Rs. 4500 after two years and Rs. 6750 after 4 years on com$ pound Interest. The sum is :

Options

A. Rs. 4000
B. Rs. 2500
C. Rs. 3000
D. Rs. 3050

Answer

To find the principal amount (the sum of money) that becomes Rs. 4500 after 2 years and Rs. 6750 after 4 years on compound interest, we can follow these steps:

### Step 1: Understand the Compound Interest Formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^N \]
where:
- \( A \) = Amount after \( N \) years
- \( P \) = Principal amount (initial sum of money)
- \( R \) = Rate of interest per annum
- \( N \) = Number of years

### Step 2: Set Up the Equations
From the problem, we have two scenarios:
1. After 2 years, the amount is Rs. 4500:
\[ A_1 = P \left(1 + \frac{R}{100}\right)^2 = 4500 \]
2. After 4 years, the amount is Rs. 6750:
\[ A_2 = P \left(1 + \frac{R}{100}\right)^4 = 6750 \]

### Step 3: Divide the Two Equations
To eliminate \( P \), we can divide the second equation by the first:
\[ \frac{A_2}{A_1} = \frac{P \left(1 + \frac{R}{100}\right)^4}{P \left(1 + \frac{R}{100}\right)^2} \]
This simplifies to:
\[ \frac{6750}{4500} = \left(1 + \frac{R}{100}\right)^{4-2} \]
\[ \frac{6750}{4500} = \left(1 + \frac{R}{100}\right)^2 \]
Calculating the left side gives:
\[ 1.5 = \left(1 + \frac{R}{100}\right)^2 \]

### Step 4: Solve for \( 1 + \frac{R}{100} \)
Taking the square root of both sides:
\[ 1 + \frac{R}{100} = \sqrt{1.5} \]
Calculating \( \sqrt{1.5} \):
\[ 1 + \frac{R}{100} \approx 1.2247 \]
Now, solving for \( R \):
\[ \frac{R}{100} \approx 0.2247 \implies R \approx 22.47 \]

### Step 5: Substitute Back to Find \( P \)
Now we can substitute \( R \) back into one of the original equations to find \( P \). Using the first equation:
\[ 4500 = P \left(1 + \frac{22.47}{100}\right)^2 \]
\[ 4500 = P \left(1.2247\right)^2 \]
Calculating \( (1.2247)^2 \):
\[ 4500 = P \times 1.5 \]
Now, solving for \( P \):
\[ P = \frac{4500}{1.5} = 3000 \]

### Final Answer
The sum of money (the principal amount) is Rs. 3000.

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