The marked price of a shirt is Rs. 10400. A shopkeeper allows a discount of 20% on the shirt and gains 30%. If no discount is given, what will be the profit per cent ?

Options

A. 62.5
B. 37.5
C. 44
D. 45

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Calculate the Selling Price (SP) after Discount
The marked price (MP) of the shirt is Rs. 10,400, and the discount allowed is 20%.

To find the selling price after the discount, we use the formula:
\[ \text{Selling Price (SP)} = \text{Marked Price (MP)} - \text{Discount} \]

First, we calculate the discount:
\[ \text{Discount} = \text{MP} \times \frac{\text{Discount Rate}}{100} \]
\[ \text{Discount} = 10,400 \times \frac{20}{100} = 10,400 \times 0.20 = 2,080 \]

Now, we can find the selling price:
\[ \text{SP} = 10,400 - 2,080 = 8,320 \]

### Step 2: Calculate the Cost Price (CP) using Gain Percentage
We know that the shopkeeper gains 30% on the cost price (CP). The gain can be calculated using the formula:
\[ \text{Gain} = \text{CP} \times \frac{\text{Gain Percentage}}{100} \]

The selling price can also be expressed as:
\[ \text{SP} = \text{CP} + \text{Gain} \]
Substituting the gain into the equation gives:
\[ \text{SP} = \text{CP} + \left(\text{CP} \times \frac{30}{100}\right) \]
\[ \text{SP} = \text{CP} \left(1 + \frac{30}{100}\right) = \text{CP} \times 1.30 \]

Now, we can express the cost price in terms of the selling price:
\[ \text{CP} = \frac{\text{SP}}{1.30} \]
Substituting the selling price we calculated:
\[ \text{CP} = \frac{8,320}{1.30} \approx 6,400 \]

### Step 3: Calculate Profit Percentage if No Discount is Given
If no discount is given, the selling price remains the marked price, which is Rs. 10,400.

Now we can calculate the profit:
\[ \text{Profit} = \text{SP} - \text{CP} \]
\[ \text{Profit} = 10,400 - 6,400 = 4,000 \]

Finally, we can calculate the profit percentage:
\[ \text{Profit Percentage} = \left(\frac{\text{Profit}}{\text{CP}} \times 100\right) \]
\[ \text{Profit Percentage} = \left(\frac{4,000}{6,400} \times 100\right) \approx 62.5\% \]

### Final Answer
The profit percentage if no discount is given is approximately **62.5%**.

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