In the Bargain Bazar everyone purchases with a fair bargaining, so the traders markup the prices too much. A trader marked up an article at ₹ M ejected huge profit if it is .sold on the marked price. But a customer purchased it at `M//2` with his fine bargaining skills, so the expected profit of the trader diminished by `66.66%`. What is the percentage discount fetched by the customer through bargaining?

Options

A. a)`33.33%`
B. b)`50%`
C. c)`66.66%`
D. d)none of these

Answer

To solve the problem step by step, we will first define the variables and then calculate the required percentage discount.

### Step 1: Define the Variables
Let the marked price of the article be \( M \).

### Step 2: Calculate the Selling Price
The customer purchased the article at half the marked price, which can be expressed as:
\[ \text{Selling Price (SP)} = \frac{M}{2} \]

### Step 3: Understand the Profit Situation
Let’s denote the cost price (CP) of the article as \( CP \). The profit when sold at the marked price \( M \) is given by:
\[ \text{Profit at marked price} = M - CP \]

### Step 4: Calculate the Expected Profit Reduction
According to the problem, the expected profit diminishes by \( 66.66\% \) when sold at the selling price \( \frac{M}{2} \). This means the profit at the selling price is:
\[ \text{Profit at selling price} = \text{Profit at marked price} - 66.66\% \text{ of Profit at marked price} \]
This can be expressed as:
\[ \text{Profit at selling price} = (1 - 0.6666)(M - CP) = 0.3334(M - CP) \]

### Step 5: Calculate the Profit at Selling Price
The profit at the selling price can also be expressed as:
\[ \text{Profit at selling price} = \frac{M}{2} - CP \]

### Step 6: Set Up the Equation
Now we can set the two expressions for profit equal to each other:
\[ \frac{M}{2} - CP = 0.3334(M - CP) \]

### Step 7: Solve for CP
Expanding the right side:
\[ \frac{M}{2} - CP = 0.3334M - 0.3334CP \]
Rearranging gives:
\[ \frac{M}{2} - 0.3334M = CP - 0.3334CP \]
\[ \frac{M}{2} - 0.3334M = 0.6666CP \]
Calculating the left side:
\[ \frac{M}{2} - 0.3334M = \frac{0.5M - 0.3334M}{1} = \frac{0.1666M}{1} \]
So we have:
\[ 0.1666M = 0.6666CP \]
Now, solving for \( CP \):
\[ CP = \frac{0.1666M}{0.6666} = \frac{M}{4} \]

### Step 8: Calculate the Discount
Now, we can find the discount given by the customer:
\[ \text{Discount} = \text{Marked Price} - \text{Selling Price} = M - \frac{M}{2} = \frac{M}{2} \]

### Step 9: Calculate the Percentage Discount
The percentage discount can be calculated as:
\[ \text{Percentage Discount} = \left( \frac{\text{Discount}}{\text{Marked Price}} \right) \times 100 = \left( \frac{\frac{M}{2}}{M} \right) \times 100 = 50\% \]

### Final Answer
The percentage discount fetched by the customer through bargaining is \( 50\% \).

---