A dealer sells an article for Rs. 75 and gains as much per cent as the cost price of the article. Find the cost price of the article.

Explanation

To solve the problem step by step, let's denote the cost price of the article as \( x \).

### Step 1: Understand the Selling Price and Gain
The selling price (SP) of the article is given as Rs. 75. The gain percentage is equal to the cost price, which means if the cost price is \( x \), then the gain percentage is also \( x \% \).

### Step 2: Write the Formula for Selling Price
The formula for selling price in terms of cost price and gain percentage is:
\[ SP = CP \times \left(1 + \frac{Gain\%}{100}\right) \]
Substituting the known values:
\[ 75 = x \times \left(1 + \frac{x}{100}\right) \]

### Step 3: Simplify the Equation
Expanding the equation gives:
\[ 75 = x + \frac{x^2}{100} \]
To eliminate the fraction, multiply the entire equation by 100:
\[ 7500 = 100x + x^2 \]

### Step 4: Rearrange into Standard Quadratic Form
Rearranging the equation gives:
\[ x^2 + 100x - 7500 = 0 \]

### Step 5: Solve the Quadratic Equation
Now, we can solve the quadratic equation \( x^2 + 100x - 7500 = 0 \) using the middle term splitting method.

1. We need two numbers that multiply to \(-7500\) and add to \(100\). These numbers are \(150\) and \(-50\).
2. Rewrite the equation:
\[ x^2 + 150x - 50x - 7500 = 0 \]

### Step 6: Factor the Quadratic
Grouping the terms:
\[ x(x + 150) - 50(x + 150) = 0 \]
Factoring out the common term:
\[ (x + 150)(x - 50) = 0 \]

### Step 7: Find the Roots
Setting each factor to zero gives:
1. \( x + 150 = 0 \) → \( x = -150 \) (not applicable as cost price cannot be negative)
2. \( x - 50 = 0 \) → \( x = 50 \)

### Conclusion
The cost price of the article is Rs. 50.