Understanding the Profit and Loss Problem

This problem involves calculating the loss percentage when articles are bought at one rate and sold at another. To find the loss percentage, we first need to determine the cost price (CP) and selling price (SP) of a common number of articles or per article.

Calculating the Cost Price (CP)

Gaurav bought articles at the rate of 5 for Rs. 6. This means:

• Cost of 5 articles = Rs. 6

To find the cost price of one article, we divide the total cost by the number of articles:

\( \text{CP of 1 article} = \frac{\text{Total Cost}}{\text{Number of articles}} = \frac{6}{5} \text{ Rs.} \)

Calculating the Selling Price (SP)

Gaurav sold the articles at the rate of 10 for Rs. 11. This means:

• Selling Price of 10 articles = Rs. 11

To find the selling price of one article, we divide the total selling price by the number of articles:

\( \text{SP of 1 article} = \frac{\text{Total Selling Price}}{\text{Number of articles}} = \frac{11}{10} \text{ Rs.} \)

Determining Profit or Loss

Now we compare the CP and SP of one article:

• CP of 1 article = \( \frac{6}{5} \) Rs.
• SP of 1 article = \( \frac{11}{10} \) Rs.

To easily compare, we can find a common denominator:

• \( \frac{6}{5} = \frac{6 \times 2}{5 \times 2} = \frac{12}{10} \)
• \( \frac{11}{10} \) remains \( \frac{11}{10} \)

Comparing \( \frac{12}{10} \) and \( \frac{11}{10} \), we see that \( \text{CP} > \text{SP} \). This indicates a loss.

The loss per article is calculated as:

\( \text{Loss} = \text{CP} - \text{SP} = \frac{12}{10} - \frac{11}{10} = \frac{12 - 11}{10} = \frac{1}{10} \text{ Rs.} \)

Calculating the Loss Percentage

The loss percentage is calculated using the formula:

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

Using the values we found:

\( \text{Loss Percentage} = \frac{\frac{1}{10}}{\frac{6}{5}} \times 100 \)

Simplify the expression:

\( \text{Loss Percentage} = \frac{1}{10} \times \frac{5}{6} \times 100 \)

\( \text{Loss Percentage} = \frac{5}{60} \times 100 \)

\( \text{Loss Percentage} = \frac{1}{12} \times 100 \)

\( \text{Loss Percentage} = \frac{100}{12} \)

Divide 100 by 12:

\( \frac{100}{12} = \frac{50}{6} = \frac{25}{3} \)

Convert the improper fraction to a mixed number:

\( \frac{25}{3} = 8 \frac{1}{3} \)

So, the loss percentage is \( 8\frac{1}{3}\% \).

Revision Table: Profit and Loss Concepts

Additional Information on Profit and Loss Calculations

Alternatively, we can find the cost and selling price of a common number of articles. The number of articles bought (5) and sold (10) have a Least Common Multiple (LCM) of 10. Let's calculate the CP and SP of 10 articles.

• Cost of 5 articles = Rs. 6
• Cost of 10 articles = \( \frac{6}{5} \times 10 = 6 \times 2 = 12 \) Rs. (This is the CP of 10 articles)

• Selling Price of 10 articles = Rs. 11 (This is the SP of 10 articles)

Now compare the CP and SP of 10 articles:

• CP of 10 articles = Rs. 12
• SP of 10 articles = Rs. 11

Since CP > SP, there is a loss.

\( \text{Loss} = \text{CP} - \text{SP} = 12 - 11 = 1 \text{ Rs. (Loss on 10 articles)} \)

The loss percentage is calculated based on the CP:

\( \text{Loss Percentage} = \frac{\text{Loss}}{\text{CP}} \times 100 = \frac{1}{12} \times 100 = \frac{100}{12} = \frac{25}{3} = 8\frac{1}{3}\% \)

Both methods yield the same result, confirming the loss percentage is \( 8\frac{1}{3}\% \).