A person sells an article for Rs. 11 and gains as much percent as the cost price of the article in rupees. The cost price of the article is

Options

A. (a) 10
B. (b) 9
C. (c) 8
D. (d) 7

Answer

To find the cost price (CP) of the article, we can follow these steps:

### Step 1: Define the Variables
Let the cost price of the article be \( x \) rupees.

### Step 2: Understand the Selling Price and Profit
The selling price (SP) of the article is given as Rs. 11. The profit percentage is equal to the cost price in rupees, which means the profit percentage is \( x \% \).

### Step 3: Write the Profit Formula
The formula for selling price in terms of cost price and profit percentage is:
\[ SP = CP + \text{Profit} \]
Where profit can be expressed as:
\[ \text{Profit} = \frac{\text{Profit Percentage}}{100} \times CP \]
Thus, we can rewrite the selling price as:
\[ SP = CP + \frac{x}{100} \times CP \]
This simplifies to:
\[ SP = CP \left(1 + \frac{x}{100}\right) \]

### Step 4: Substitute the Known Values
Substituting the known values into the equation:
\[ 11 = x \left(1 + \frac{x}{100}\right) \]

### Step 5: Rearrange the Equation
Expanding the equation gives:
\[ 11 = x + \frac{x^2}{100} \]
Multiplying through by 100 to eliminate the fraction:
\[ 1100 = 100x + x^2 \]
Rearranging this gives us a standard quadratic equation:
\[ x^2 + 100x - 1100 = 0 \]

### Step 6: Solve the Quadratic Equation
We can solve this quadratic equation using the factorization method. We need to find two numbers that multiply to \(-1100\) and add to \(100\). The numbers are \(110\) and \(-10\):
\[ (x + 110)(x - 10) = 0 \]

### Step 7: Find the Roots
Setting each factor to zero gives:
\[ x + 110 = 0 \quad \Rightarrow \quad x = -110 \quad \text{(not valid since cost price cannot be negative)} \]
\[ x - 10 = 0 \quad \Rightarrow \quad x = 10 \]

### Conclusion
Thus, the cost price of the article is \( \text{Rs. } 10 \).