Rs 7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received Rs 100 more. Find the original number of children.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the variables
Let the original number of children be \( x \).

### Step 2: Determine the amount each child receives
If Rs 7500 is divided equally among \( x \) children, then each child receives:
\[ \text{Amount per child} = \frac{7500}{x} \]

### Step 3: Set up the scenario with 20 fewer children
If there were 20 fewer children, the number of children would be \( x - 20 \). In this case, each child would receive:
\[ \text{New amount per child} = \frac{7500}{x - 20} \]

### Step 4: Establish the relationship between the two scenarios
According to the problem, if there were 20 fewer children, each child would receive Rs 100 more than before. Therefore, we can set up the equation:
\[ \frac{7500}{x - 20} = \frac{7500}{x} + 100 \]

### Step 5: Clear the fractions
To eliminate the fractions, we can multiply through by \( x(x - 20) \):
\[ 7500x = 7500(x - 20) + 100x(x - 20) \]

### Step 6: Expand and simplify the equation
Expanding both sides gives:
\[ 7500x = 7500x - 150000 + 100x^2 - 2000x \]
Now, subtract \( 7500x \) from both sides:
\[ 0 = -150000 + 100x^2 - 2000x \]
Rearranging gives:
\[ 100x^2 - 2000x + 150000 = 0 \]

### Step 7: Simplify the quadratic equation
Dividing the entire equation by 100 simplifies it:
\[ x^2 - 20x + 1500 = 0 \]

### Step 8: Factor the quadratic equation
We need to factor the quadratic equation:
\[ x^2 - 50x + 30x + 1500 = 0 \]
This factors to:
\[ (x - 50)(x + 30) = 0 \]

### Step 9: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 50 = 0 \quad \text{or} \quad x + 30 = 0 \]
Thus, we find:
\[ x = 50 \quad \text{or} \quad x = -30 \]

### Step 10: Determine the valid solution
Since the number of children cannot be negative, we reject \( x = -30 \) and accept:
\[ x = 50 \]

### Final Answer
The original number of children is \( \boxed{50} \).

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