A dealer sells an article for ₹ 24 and gains as much percent as the price of the article, then the cost price of the article is :

Question

Accepted Answer

To determine the cost price of the article, we need to understand the relationship between the selling price, cost price, and the gain percentage. The problem states that the dealer sells an article for ₹24 and gains as much percent as the cost price of the article.

Let's denote the cost price of the article as ₹x. According to the problem, the gain percentage is equal to the cost price. Therefore, the gain percentage is x%.

The formula for gain percentage is:

$$ \text{Gain Percentage} = \left( \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \right) \times 100 $$

Substituting the given values:

$$ x = \left( \frac{24 - x}{x} \right) \times 100 $$

Simplifying the equation:

$$ x = \frac{24 - x}{x} \times 100 $$

$$ x^2 = 2400 - 100x $$

$$ x^2 + 100x - 2400 = 0 $$

This is a quadratic equation in the form of $ ax^2 + bx + c = 0 $. Solving this using the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Where $ a = 1 $, $ b = 100 $, and $ c = -2400 $.

$$ x = \frac{-100 \pm \sqrt{100^2 - 4 \times 1 \times (-2400)}}{2 \times 1} $$

$$ x = \frac{-100 \pm \sqrt{10000 + 9600}}{2} $$

$$ x = \frac{-100 \pm \sqrt{19600}}{2} $$

$$ x = \frac{-100 \pm 140}{2} $$

The two possible solutions are:

$$ x = \frac{40}{2} = 20 $$

$$ x = \frac{-240}{2} = -120 $$

Since the cost price cannot be negative, the only feasible solution is $ x = 20 $.

Therefore, the cost price of the article is ₹20, which corresponds to Option 4.

Explanation for why other options are incorrect:

1. **Option 1 (₹15):** If the cost price were ₹15, the gain percentage would be calculated as $\left(\frac{24 - 15}{15}\right) \times 100 = 60\%$. This does not match the condition that the gain percentage equals the cost price.

2. **Option 2 (₹17):** If the cost price were ₹17, the gain percentage would be $\left(\frac{24 - 17}{17}\right) \times 100 \approx 41.18\%$. This does not satisfy the condition.

3. **Option 3 (₹18):** If the cost price were ₹18, the gain percentage would be $\left(\frac{24 - 18}{18}\right) \times 100 = 33.33\%$. This also does not meet the condition.

Thus, Option 4 (₹20) is the only correct answer that satisfies the condition given in the problem.