A shopkeeper mark up the price of a article `(3)/(2)th` time the cost price. He gave some discount to a customer and earn a profit such that, percentage of profit he earn and percentage of discount he gives is same. Find the discount percent.

Options

A. A)`30%`
B. B)`20%`
C. C)`40%`
D. D)`25%`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between Cost Price (CP), Marked Price (MP), and Selling Price (SP)

Given that the shopkeeper marks up the price of an article to \( \frac{3}{2} \) times the cost price, we can express this mathematically:
- Let the Cost Price (CP) be \( 2 \) units.
- Then, the Marked Price (MP) will be:
\[ MP = \frac{3}{2} \times CP = \frac{3}{2} \times 2 = 3 \text{ units} \]

**Hint:** Remember that the Marked Price is a multiple of the Cost Price based on the markup given.

### Step 2: Calculate the Markup Percentage

The markup percentage can be calculated as follows:
\[ \text{Markup Percentage} = \left( \frac{MP - CP}{CP} \right) \times 100 \]
Substituting the values:
\[ \text{Markup Percentage} = \left( \frac{3 - 2}{2} \right) \times 100 = \left( \frac{1}{2} \right) \times 100 = 50\% \]

**Hint:** The markup percentage is the increase in price relative to the cost price.

### Step 3: Set up the Profit and Discount Relationship

Let the discount percentage be \( K \). According to the problem, the profit percentage is equal to the discount percentage:
\[ \text{Profit Percentage} = K \]
Using the formula for profit percentage:
\[ \text{Profit Percentage} = \text{Markup Percentage} - \text{Discount Percentage} - \left( \frac{\text{Markup Percentage} \times \text{Discount Percentage}}{100} \right) \]
Substituting the known values:
\[ K = 50 - K - \left( \frac{50 \times K}{100} \right) \]

**Hint:** The profit percentage is derived from the markup and discount percentages.

### Step 4: Simplify the Equation

Rearranging the equation:
\[ K + K + \frac{50K}{100} = 50 \]
This simplifies to:
\[ 2K + \frac{1}{2}K = 50 \]
Combining the terms:
\[ \frac{5}{2}K = 50 \]

**Hint:** Combine like terms to simplify the equation.

### Step 5: Solve for \( K \)

To isolate \( K \):
\[ K = 50 \times \frac{2}{5} = 20 \]

**Hint:** When isolating a variable, ensure to perform the same operation on both sides of the equation.

### Conclusion

The discount percentage \( K \) is \( 20\% \).

**Final Answer:** The discount percentage is \( 20\% \).