Retailer Profit and Loss Calculation Explained

This problem involves calculating the cost price of an item given its selling price and profit percentage, and then determining the profit or loss percentage if the item is sold at a different price. We are given the initial selling price, the profit made at that price, and a new selling price. We need to find the profit or loss percentage at the new selling price.

Step 1: Calculate the Cost Price (CP)

We know the retailer sells the item for Rs. 486 and makes an 8% profit. The selling price (SP) when there is a profit is related to the cost price (CP) by the formula:

SP = CP + Profit

Profit is a percentage of the cost price, so:

\( \text{Profit} = \left( \frac{\text{Profit Percentage}}{100} \right) \times \text{CP} \)

Substituting this into the SP formula:

\( \text{SP} = \text{CP} + \left( \frac{\text{Profit Percentage}}{100} \right) \times \text{CP} \)

\( \text{SP} = \text{CP} \left( 1 + \frac{\text{Profit Percentage}}{100} \right) \)

We are given SP = Rs. 486 and Profit Percentage = 8%. Let's plug these values into the formula:

\( 486 = \text{CP} \left( 1 + \frac{8}{100} \right) \)

\( 486 = \text{CP} \left( 1 + 0.08 \right) \)

\( 486 = \text{CP} \times 1.08 \)

Now, we can solve for CP:

\( \text{CP} = \frac{486}{1.08} \)

\( \text{CP} = \frac{48600}{108} \)

Performing the division:

\( \text{CP} = 450 \)

So, the cost price of the item is Rs. 450.

Step 2: Calculate Profit or Loss at the New Selling Price

The retailer plans to sell the item for Rs. 414. This is our new selling price (SP2 = Rs. 414). We compare this new selling price with the cost price (CP = Rs. 450) we just calculated.

Since the new selling price (Rs. 414) is less than the cost price (Rs. 450), the retailer would make a loss.

\( \text{SP2} < \text{CP} \)

\( 414 < 450 \)

The amount of loss is calculated as:

\( \text{Loss} = \text{CP} - \text{SP2} \)

\( \text{Loss} = 450 - 414 \)

\( \text{Loss} = 36 \)

The loss is Rs. 36.

Step 3: Calculate the Loss Percentage

The loss percentage is calculated based on the cost price using the formula:

\( \text{Loss Percentage} = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100 \)

Plugging in the values for Loss and CP:

\( \text{Loss Percentage} = \left( \frac{36}{450} \right) \times 100 \)

\( \text{Loss Percentage} = \frac{36 \times 100}{450} \)

\( \text{Loss Percentage} = \frac{3600}{450} \)

\( \text{Loss Percentage} = \frac{360}{45} \)

Simplifying the fraction:

\( \text{Loss Percentage} = 8 \)

So, if the retailer sells the item for Rs. 414, he would make an 8% loss.

The final result indicates an 8% loss.

Revision Table: Key Concepts in Profit and Loss

Additional Information: Understanding Profit vs. Loss

In business, profit and loss are fundamental concepts used to determine the financial performance of selling goods or services. They are calculated by comparing the selling price (the price at which an item is sold) to the cost price (the price at which the item was bought or produced).

• Cost Price (CP): This is the original price at which an item is purchased or the cost incurred to produce it.
• Selling Price (SP): This is the price at which an item is sold to the customer.

When SP is greater than CP, the result is a profit. The amount of profit is the difference between SP and CP. When CP is greater than SP, the result is a loss. The amount of loss is the difference between CP and SP.

Profit and loss are often expressed as a percentage of the cost price. This allows for easier comparison across different items or businesses, regardless of the absolute monetary values involved.

Calculating profit or loss percentage is crucial for retailers to understand their margin on products and make informed decisions about pricing and inventory.