₹ 250 were divided equally among a certain number of children. If there were 25 more children, each would have received 50 paise less. Find the number of children.

Answer

To solve the problem, we need to find the number of children (let's denote this as \( x \)) who are equally dividing ₹ 250. The problem states that if there were 25 more children, each child would receive 50 paise (or ₹ 0.50) less.

Let's break down the solution step by step:

### Step 1: Set Up the Initial Equation
Let \( x \) be the number of children. If ₹ 250 is divided equally among \( x \) children, then each child receives:
\[ y = \frac{250}{x} \]

### Step 2: Set Up the New Scenario
If there are 25 more children, the new number of children becomes \( x + 25 \). The amount each child would receive in this case is:
\[ y - 0.5 = \frac{250}{x + 25} \]

### Step 3: Write the Equation Based on the New Scenario
From the new scenario, we can write the equation:
\[ \frac{250}{x + 25} = \frac{250}{x} - 0.5 \]

### Step 4: Clear the Fractions
To eliminate the fractions, we can multiply through by \( x(x + 25) \):
\[ 250x = 250(x + 25) - 0.5x(x + 25) \]

### Step 5: Expand and Rearrange the Equation
Expanding the right-hand side:
\[ 250x = 250x + 6250 - 0.5x^2 - 12.5x \]
Now, simplify the equation:
\[ 0 = 6250 - 0.5x^2 - 12.5x \]
Rearranging gives:
\[ 0.5x^2 + 12.5x - 6250 = 0 \]

### Step 6: Multiply Through by 2 to Eliminate Decimal
To make calculations easier, multiply the entire equation by 2:
\[ x^2 + 25x - 12500 = 0 \]

### Step 7: Factor the Quadratic Equation
Now, we need to factor the quadratic equation. We are looking for two numbers that multiply to \(-12500\) and add to \(25\). The factors are \(125\) and \(-100\):
\[ (x + 125)(x - 100) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives:
\[ x + 125 = 0 \quad \Rightarrow \quad x = -125 \quad (\text{not valid, as } x \text{ must be positive}) \]
\[ x - 100 = 0 \quad \Rightarrow \quad x = 100 \]

### Conclusion
The number of children is:
\[ \boxed{100} \]