The simple interest and the true discount on a certain sum for a given time and at a given rate are Rs. 25 and Rs. 20 respectively. The sum is-

Options

A. Rs. 500
B. Rs. 200
C. Rs. 250
D. Rs. 100

Answer

To find the sum based on the given simple interest (SI) and true discount (TD), we can follow these steps:

### Step 1: Define Variables
Let the sum (principal) be denoted as \( X \) and the rate of interest be \( R \) percent.

### Step 2: Set Up the Simple Interest Equation
The formula for simple interest is given by:
\[ SI = \frac{X \cdot R \cdot T}{100} \]
Since the time \( T \) is not specified, we will assume it to be 1 year for simplicity. Given that the simple interest is Rs. 25, we can write:
\[ \frac{X \cdot R}{100} = 25 \]
Multiplying both sides by 100, we get:
\[ X \cdot R = 2500 \quad \text{(Equation 1)} \]

### Step 3: Set Up the True Discount Equation
The true discount is given by the formula:
\[ TD = \frac{(X - TD) \cdot R \cdot T}{100} \]
Again, assuming \( T = 1 \), and given that the true discount is Rs. 20, we can write:
\[ \frac{(X - 20) \cdot R}{100} = 20 \]
Multiplying both sides by 100, we get:
\[ (X - 20) \cdot R = 2000 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( X \cdot R = 2500 \)
2. \( (X - 20) \cdot R = 2000 \)

From Equation 1, we can express \( R \) in terms of \( X \):
\[ R = \frac{2500}{X} \]

Substituting this value of \( R \) into Equation 2:
\[ (X - 20) \cdot \frac{2500}{X} = 2000 \]
Multiplying through by \( X \) to eliminate the fraction:
\[ (X - 20) \cdot 2500 = 2000X \]
Expanding and rearranging gives:
\[ 2500X - 50000 = 2000X \]
\[ 2500X - 2000X = 50000 \]
\[ 500X = 50000 \]
Dividing both sides by 500:
\[ X = 100 \]

### Conclusion
The sum is Rs. 100.