Mohan sold an article for RS.56 which coste him RS.x. He finds that he has gained x% on his outlay. Find x.

Answer

To solve the problem step by step, we start by understanding the information given:

1. **Given Information**:
- Selling Price (SP) = ₹56
- Cost Price (CP) = ₹x
- Gain = x% of CP

2. **Understanding Gain**:
The gain can be expressed as:
\[ \text{Gain} = \text{Selling Price} - \text{Cost Price} = 56 - x \]
Since the gain is also given as x% of the cost price, we can write:
\[ \text{Gain} = \frac{x}{100} \times x = \frac{x^2}{100} \]

3. **Setting Up the Equation**:
Now we can equate the two expressions for gain:
\[ 56 - x = \frac{x^2}{100} \]

4. **Clearing the Fraction**:
To eliminate the fraction, multiply the entire equation by 100:
\[ 100(56 - x) = x^2 \]
This simplifies to:
\[ 5600 - 100x = x^2 \]

5. **Rearranging the Equation**:
Rearranging gives us a standard quadratic equation:
\[ x^2 + 100x - 5600 = 0 \]

6. **Factoring the Quadratic**:
We need to find two numbers that multiply to -5600 and add to 100. The numbers are 140 and -40. Thus, we can factor the equation:
\[ (x + 140)(x - 40) = 0 \]

7. **Finding the Roots**:
Setting each factor to zero gives:
\[ x + 140 = 0 \quad \Rightarrow \quad x = -140 \quad (\text{not valid since cost cannot be negative}) \]
\[ x - 40 = 0 \quad \Rightarrow \quad x = 40 \]

8. **Conclusion**:
Therefore, the cost price (CP) is ₹40.

### Final Answer:
The value of \( x \) is 40.

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