A certain sum amounts to Rs. 4500 in `2(1)/(2)` years at 20% per annum simple interest. Find the sum (in Rs. ).

Options

A. 3000
B. 2400
C. 2700
D. 3600

Explanation

To find the sum (principal) that amounts to Rs. 4500 in 2.5 years at a rate of 20% per annum simple interest, we can follow these steps:

### Step 1: Understand the formula for Simple Interest
The formula for the total amount (A) in simple interest is given by:
\[ A = P + SI \]
Where:
- \( A \) = Total amount (final amount)
- \( P \) = Principal (initial sum)
- \( SI \) = Simple Interest

The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Convert the time period into years
The time period given is \( 2 \frac{1}{2} \) years. We convert this into a decimal:
\[ T = 2 + \frac{1}{2} = 2.5 \text{ years} \]

### Step 3: Substitute the values into the formula
We know:
- \( A = 4500 \)
- \( R = 20\% \)
- \( T = 2.5 \)

Substituting these values into the simple interest formula:
\[ A = P + \frac{P \times R \times T}{100} \]
This can be rewritten as:
\[ 4500 = P + \frac{P \times 20 \times 2.5}{100} \]

### Step 4: Simplify the equation
We can simplify the equation:
\[ 4500 = P + \frac{P \times 50}{100} \]
\[ 4500 = P + \frac{P}{2} \]
This means:
\[ 4500 = P + 0.5P \]
\[ 4500 = 1.5P \]

### Step 5: Solve for P
To find \( P \), we can rearrange the equation:
\[ P = \frac{4500}{1.5} \]
Calculating this gives:
\[ P = 3000 \]

### Final Answer
The sum (principal) is Rs. 3000.

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