A shopkeeper sold an item for Rs 1,510 after giving a discount of `24(1)/(2)%` and thereby incurred a loss of 10%. Had he sold the item without discount, his net profit would have been

Options

A. Rs 641
B. Rs `322(1)/(9)`
C. Rs. `422(2)/(9)`
D. Rs. `322(2)/(9)`

Answer

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

### Step 1: Identify the Selling Price (SP) and Discount
The selling price after discount is given as Rs 1,510. The discount is given as \(24 \frac{1}{2}\% = \frac{49}{2}\%\).

### Step 2: Calculate the Market Price (MP)
The formula for selling price after discount is:
\[ SP = MP - \text{Discount} \]
Where the discount can be expressed as:
\[ \text{Discount} = \frac{MP \times \text{Discount Rate}}{100} \]

Substituting the values, we have:
\[ 1510 = MP - \frac{MP \times \frac{49}{2}}{100} \]

This can be rewritten as:
\[ 1510 = MP \left(1 - \frac{49}{200}\right) \]
\[ 1510 = MP \left(\frac{200 - 49}{200}\right) \]
\[ 1510 = MP \left(\frac{151}{200}\right) \]

Now, solving for MP:
\[ MP = 1510 \times \frac{200}{151} \]
\[ MP = 2000 \]

### Step 3: Calculate the Cost Price (CP)
We know the selling price and that there is a loss of 10%. The formula for selling price in terms of cost price and loss is:
\[ SP = CP - \text{Loss} \]
Where the loss can be expressed as:
\[ \text{Loss} = \frac{10}{100} \times CP = 0.1 \times CP \]

So we can write:
\[ 1510 = CP - 0.1 \times CP \]
\[ 1510 = 0.9 \times CP \]

Now, solving for CP:
\[ CP = \frac{1510}{0.9} \]
\[ CP = 1677.78 \text{ (approximately)} \]

### Step 4: Calculate Profit if Sold at Market Price
If the item were sold at the market price (without discount), the profit would be:
\[ \text{Profit} = MP - CP \]
Substituting the values:
\[ \text{Profit} = 2000 - 1677.78 \]
\[ \text{Profit} = 322.22 \text{ (approximately)} \]

### Final Answer
The net profit had he sold the item without discount would be approximately Rs 322.22.

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