Rs 480 is dividend equally among 'x' children. If the number of children were 20 more than each would have got Rs 12 less. Find 'x'.

Answer

To solve the problem step by step, we will follow the reasoning used in the video transcript.

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

### Step 2: Calculate the amount each child receives
The total amount of Rs 480 is divided equally among \( x \) children. Therefore, the amount each child receives is given by:
\[ \text{Amount per child} = \frac{480}{x} \]

### Step 3: Consider the scenario with 20 more children
If the number of children increases by 20, the new number of children will be \( x + 20 \). The amount each child would receive in this case is:
\[ \text{New amount per child} = \frac{480}{x + 20} \]

### Step 4: Set up the equation based on the problem statement
According to the problem, if the number of children were 20 more, each child would receive Rs 12 less. Therefore, we can set up the equation:
\[ \frac{480}{x} - \frac{480}{x + 20} = 12 \]

### Step 5: Simplify the equation
To solve this equation, we can first find a common denominator:
\[ \frac{480(x + 20) - 480x}{x(x + 20)} = 12 \]
This simplifies to:
\[ \frac{480 \cdot 20}{x(x + 20)} = 12 \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 480 \cdot 20 = 12 \cdot x(x + 20) \]
This simplifies to:
\[ 9600 = 12x^2 + 240x \]

### Step 7: Rearrange the equation into standard quadratic form
Rearranging the equation gives:
\[ 12x^2 + 240x - 9600 = 0 \]

### Step 8: Simplify the quadratic equation
Dividing the entire equation by 12 to simplify:
\[ x^2 + 20x - 800 = 0 \]

### Step 9: Factor the quadratic equation
Now we need to factor the quadratic equation. We look for two numbers that multiply to -800 and add to 20. The numbers are 40 and -20. Thus, we can write:
\[ (x + 40)(x - 20) = 0 \]

### Step 10: Solve for \( x \)
Setting each factor to zero gives:
1. \( x + 40 = 0 \) → \( x = -40 \) (not valid, as number of children cannot be negative)
2. \( x - 20 = 0 \) → \( x = 20 \)

### Conclusion
The total number of children is \( x = 20 \).