Understanding the Profit and Loss Problem

This question asks us to determine the approximate loss or gain percentage when the selling price (SP) of a certain number of articles is equal to the cost price (CP) of a different number of articles. Specifically, we are told that the selling price of 75 articles is equal to the cost price of 60 articles.

Setting up the Relationship between Selling Price and Cost Price

Let SP be the selling price of one article and CP be the cost price of one article. According to the problem statement, the total selling price of 75 articles is equal to the total cost price of 60 articles.

We can write this relationship as an equation:

\( 75 \times SP \text{ (of one article)} = 60 \times CP \text{ (of one article)} \)

We can rearrange this equation to find the ratio of the selling price to the cost price:

\( \frac{SP}{CP} = \frac{60}{75} \)

Now, we can simplify the fraction \(\frac{60}{75}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 15:

\( \frac{SP}{CP} = \frac{60 \div 15}{75 \div 15} = \frac{4}{5} \)

So, the ratio of the selling price to the cost price of one article is 4/5.

Determining Whether There is a Loss or Gain

From the ratio \(\frac{SP}{CP} = \frac{4}{5}\), we can see that the selling price (SP) is \(\frac{4}{5}\) times the cost price (CP). Since \(\frac{4}{5}\) is less than 1, the selling price is less than the cost price (SP < CP).

When the selling price of an article is less than its cost price, the seller incurs a loss.

Calculating the Loss Percentage

The loss is calculated as the difference between the cost price and the selling price: Loss = CP - SP.

From the ratio \(\frac{SP}{CP} = \frac{4}{5}\), we can assume, for calculation purposes, that the cost price of one article is 5 units and the selling price is 4 units. (Any multiple of this ratio, like CP=10 and SP=8, would also work).

• Let CP = 5 units
• Let SP = 4 units

Loss = CP - SP = 5 units - 4 units = 1 unit.

The loss percentage is calculated with respect to the cost price using the formula:

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

Substituting the values we have:

\( \text{Loss Percentage} = \left( \frac{1 \text{ unit}}{5 \text{ units}} \right) \times 100 \)

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

\( \text{Loss Percentage} = 0.20 \times 100 \)

\( \text{Loss Percentage} = 20\% \)

Therefore, there is a loss of 20%.

Summary of Steps to Calculate Loss/Gain Percent

1. Write the given condition relating the total selling price of one quantity of articles to the total cost price of another quantity of articles.
2. Form an equation using SP and CP of a single article.
3. Find the ratio of SP to CP.
4. If SP/CP < 1, it's a loss. If SP/CP > 1, it's a profit. If SP/CP = 1, it's no profit/no loss.
5. Calculate the loss or profit amount using the ratio.
6. Use the appropriate formula \(\left( \frac{\text{Loss/Profit}}{CP} \right) \times 100\) to find the percentage.

Revision Table: Profit and Loss Concepts

Additional Information on Profit and Loss Percent

Profit and loss percentages are always calculated with respect to the cost price (CP), unless explicitly stated otherwise (like on selling price). Understanding the relationship between SP and CP is crucial in these types of problems.

In this specific problem, the core idea is that to sell 75 articles, you spent money equivalent to the cost of only 60 articles. This immediately suggests you are selling more articles for the same cost money. However, the problem statement says "selling price of 75 articles is equal to cost price of 60 articles", which means the total revenue from selling 75 items is the same as the initial expense for 60 items. Since you sold 75 items but only recovered the cost of 60 items, there must be a loss on each item sold.

Another way to think about the ratio \(\frac{SP}{CP} = \frac{4}{5}\) is that for every 5 rupees of cost, you only get back 4 rupees in selling price. The difference, 1 rupee, is a loss. This loss of 1 rupee is on the original cost of 5 rupees. So, the loss percentage is \(\frac{1}{5} \times 100 = 20\%\).