Understanding the Article Transaction

The problem describes a common scenario involving buying and selling articles at different rates. Anil purchased a certain number of articles at one price and sold them at another price. We need to determine if this transaction resulted in a percentage gain or loss.

To figure this out, the easiest approach is to find the cost price (CP) and selling price (SP) of a single article or a convenient number of articles that is a multiple of both quantities purchased and sold (like the LCM of 6 and 10, which is 30). Let's calculate the CP and SP for one article first.

Calculating Cost Price (CP) and Selling Price (SP) per Article

Anil bought 6 articles for Rs. 8.

• Cost Price of 6 articles = Rs. 8
• Cost Price of 1 article (CP) = $\frac{\text{Total Cost}}{\text{Number of articles}} = \frac{8}{6}$ Rs.
• CP per article = $\frac{4}{3}$ Rs.

Anil sold 10 articles for Rs. 12.

• Selling Price of 10 articles = Rs. 12
• Selling Price of 1 article (SP) = $\frac{\text{Total Revenue}}{\text{Number of articles}} = \frac{12}{10}$ Rs.
• SP per article = $\frac{6}{5}$ Rs.

Determining Profit or Loss on Articles

Now we compare the CP and SP per article to find out if there was a profit or a loss. CP is $\frac{4}{3}$ Rs. and SP is $\frac{6}{5}$ Rs. To compare these fractions easily, we can find a common denominator. The least common multiple (LCM) of 3 and 5 is 15.

• CP per article = $\frac{4}{3} = \frac{4 \times 5}{3 \times 5} = \frac{20}{15}$ Rs.
• SP per article = $\frac{6}{5} = \frac{6 \times 3}{5 \times 3} = \frac{18}{15}$ Rs.

Comparing the values, we see that $\text{SP} \left(\frac{18}{15}\right)$ is less than $\text{CP} \left(\frac{20}{15}\right)$.

Since Selling Price (SP) < Cost Price (CP), there is a loss in the transaction.

Calculating Percentage Loss

The amount of loss per article is calculated as CP - SP.

• Loss per article = $\text{CP} - \text{SP} = \frac{4}{3} - \frac{6}{5}$
• Loss per article = $\frac{20}{15} - \frac{18}{15} = \frac{20 - 18}{15} = \frac{2}{15}$ Rs.

The percentage loss is calculated using the formula: $\text{Loss } \% = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100$.

• Loss % = $\left( \frac{2/15}{4/3} \right) \times 100$
• Loss % = $\left( \frac{2}{15} \times \frac{3}{4} \right) \times 100$ (Inverting the denominator fraction)
• Loss % = $\left( \frac{6}{60} \right) \times 100$
• Loss % = $\left( \frac{1}{10} \right) \times 100$
• Loss % = $10 \%$

Therefore, Anil incurred a 10% loss on the transaction.

Revision Table: Key Profit and Loss Formulas

Additional Information on Profit and Loss Calculations

When solving problems like this involving different quantities of items bought and sold, the most reliable method is to calculate the cost price and selling price per unit item. This standardizes the comparison.

Alternatively, one could find the cost price and selling price of the same total number of articles, such as the least common multiple of the number of articles bought and sold. In this case, the LCM of 6 and 10 is 30.

• Cost of 6 articles = Rs. 8
• Cost of 1 article = Rs. $\frac{8}{6}$
• Cost of 30 articles (CP) = $\frac{8}{6} \times 30 = 8 \times 5 = \text{Rs. } 40$
• Selling price of 10 articles = Rs. 12
• Selling price of 1 article = Rs. $\frac{12}{10}$
• Selling price of 30 articles (SP) = $\frac{12}{10} \times 30 = 12 \times 3 = \text{Rs. } 36$

Here, CP (Rs. 40) > SP (Rs. 36), confirming a loss.

• Total Loss = CP - SP = $40 - 36 = \text{Rs. } 4$
• Loss % = $\left( \frac{\text{Loss}}{\text{CP}} \right) \times 100 = \left( \frac{4}{40} \right) \times 100 = \left( \frac{1}{10} \right) \times 100 = 10 \%$

Both methods yield the same result: a 10% loss. Choosing the method that feels simpler for a given problem can save time during exams.