If the difference between the interest and discount in a certain sum of money for 6 months at `6%` be `2cdot25`. Find the sum-

Options

A. Rs. 7525
B. Rs. 2255
C. Rs. 2575
D. Rs. 2755

Answer

To solve the problem step by step, we will use the relationship between Simple Interest (SI), True Discount (TD), and the given difference between them.

### Step 1: Understand the relationship
The difference between Simple Interest (SI) and True Discount (TD) can be expressed as:
\[ \text{SI} - \text{TD} = \text{Difference} \]
Given that the difference is \( 2.25 \).

### Step 2: Identify the variables
Let:
- \( P \) = Principal (the sum we need to find)
- \( R \) = Rate of interest = \( 6\% \)
- \( T \) = Time = \( 6 \text{ months} = \frac{6}{12} \text{ years} = \frac{1}{2} \text{ years} \)

### Step 3: Calculate Simple Interest (SI)
The formula for Simple Interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ \text{SI} = \frac{P \times 6 \times \frac{1}{2}}{100} = \frac{3P}{100} \]

### Step 4: Calculate True Discount (TD)
The True Discount can be calculated using the formula:
\[ \text{TD} = \frac{\text{SI} \times R \times T}{100} \]
Using the same values:
\[ \text{TD} = \frac{P \times 6 \times \frac{1}{2}}{100} = \frac{3P}{100} \]

### Step 5: Set up the equation
From the relationship established in Step 1:
\[ \text{SI} - \text{TD} = 2.25 \]
Substituting the values we calculated:
\[ \frac{3P}{100} - \frac{3P}{100} = 2.25 \]
This simplifies to:
\[ \frac{3P}{100} - \frac{3P}{100} = 2.25 \]

### Step 6: Solve for \( P \)
We can rewrite the equation:
\[ \text{SI} - \text{TD} = 2.25 \]
This means:
\[ \frac{3P}{100} - \frac{3P}{100} = 2.25 \]
This indicates that we need to find the correct relationship.

### Step 7: Correcting the equation
We know that:
\[ \text{SI} - \text{TD} = \frac{P \times R \times T}{100} - \frac{P \times R \times T}{100} \]
This means:
\[ \frac{3P}{100} - \frac{3P}{100} = 2.25 \]
This indicates that we need to find the correct relationship.

### Step 8: Final Calculation
Using the correct relationship:
\[ \text{Difference} = \frac{P \times R^2 \times T^2}{100^2} \]
Where \( R = 6 \) and \( T = \frac{1}{2} \):
\[ 2.25 = \frac{P \times 6^2 \times \left(\frac{1}{2}\right)^2}{100^2} \]
\[ 2.25 = \frac{P \times 36 \times \frac{1}{4}}{10000} \]
\[ 2.25 = \frac{9P}{10000} \]
Now, multiplying both sides by \( 10000 \):
\[ 22500 = 9P \]
Dividing by 9:
\[ P = 2500 \]

### Final Answer
The sum is \( 2500 \).