Calculating Profit or Loss Percentage

This problem involves a classic scenario where the cost price of a certain number of items is equal to the selling price of a different number of items. We need to determine if this results in a profit or a loss and calculate the percentage.

Understanding the Given Information

We are given that:

• The purchase price (Cost Price or CP) of 30 items is the same as
• The selling price (SP) of 40 items.

Let's denote the cost price per item as $CP_{item}$ and the selling price per item as $SP_{item}$.

Setting Up the Relationship

According to the problem statement, the total cost price of 30 items is equal to the total selling price of 40 items. We can write this as an equation:

$$30 \times CP_{item} = 40 \times SP_{item}$$

Finding the Ratio of CP to SP

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

$$\frac{CP_{item}}{SP_{item}} = \frac{40}{30}$$

Simplifying the fraction, we get:

$$\frac{CP_{item}}{SP_{item}} = \frac{4}{3}$$

This ratio tells us that for every 4 units of cost price, there are 3 units of selling price.

Determining Profit or Loss

Since the ratio $\frac{CP_{item}}{SP_{item}} = \frac{4}{3}$ is greater than 1 (because the numerator 4 is greater than the denominator 3), it implies that the cost price per item ($CP_{item}$) is greater than the selling price per item ($SP_{item}$).

When the selling price of an item is less than its cost price, there is a loss.

Loss = Cost Price - Selling Price

Calculating the Percentage Loss

To calculate the loss percentage, we use the formula:

$$\text{Loss Percentage} = \frac{\text{Loss}}{\text{Cost Price}} \times 100$$

From the ratio $\frac{CP_{item}}{SP_{item}} = \frac{4}{3}$, we can assume $CP_{item} = 4x$ and $SP_{item} = 3x$ for some value $x$ (where $x > 0$).

Loss per item = $CP_{item} - SP_{item} = 4x - 3x = x$

Now, substitute these values into the loss percentage formula:

$$\text{Loss Percentage} = \frac{x}{4x} \times 100$$

Cancel out $x$ from the numerator and denominator:

$$\text{Loss Percentage} = \frac{1}{4} \times 100$$

$$\text{Loss Percentage} = 25\%$$

Therefore, there is a 25% loss in sales.

Summary of Calculation

The calculated result is a 25% loss.

Revision Table: Profit & Loss Basics

Additional Information: Relating CP and SP Ratio to Profit/Loss

When you have a relationship like \(N_1 \times \text{CP} = N_2 \times \text{SP}\), where \(N_1\) and \(N_2\) are the number of items:

• The ratio \(\frac{\text{CP}}{\text{SP}} = \frac{N_2}{N_1}\).
• If $N_2 > N_1$, then $\frac{N_2}{N_1} > 1$, meaning CP > SP, which results in a loss.
• If $N_2 < N_1$, then $\frac{N_2}{N_1} < 1$, meaning CP < SP, which results in a profit.

In this specific problem, $N_1 = 30$ and $N_2 = 40$. Since $N_2 (40) > N_1 (30)$, we know there must be a loss.

The percentage loss (or profit) can be quickly calculated from the ratio \(\frac{CP}{SP} = \frac{N_2}{N_1}\). If it's a loss, the loss fraction is \(\frac{CP-SP}{CP} = \frac{N_2/N_1 \times SP - SP}{N_2/N_1 \times SP} = \frac{SP(N_2/N_1 - 1)}{SP(N_2/N_1)} = \frac{(N_2 - N_1)/N_1}{N_2/N_1} = \frac{N_2 - N_1}{N_2}\). However, calculating loss percentage is always on CP. From the ratio CP/SP = 4/3, loss is 4-3=1 unit, and CP is 4 units. Loss % = (1/4) * 100 = 25%.

Another way to think about it from \(N_1 \times \text{CP} = N_2 \times \text{SP}\) is that the effective CP for the transaction involving 40 items (which are sold) is the CP of 40 items, which is equivalent to the CP of 30 items according to the question. So, the SP of 40 items is equal to the CP of 30 items. Since 40 items are being sold for the cost price of only 30 items, there is a loss.

The loss is equivalent to the cost price of $40 - 30 = 10$ items.

The loss percentage is calculated on the original cost price of the items sold. In this case, 40 items were sold. The effective cost price of these 40 items, in terms of their individual CP, is $40 \times CP_{item}$. However, the question relates SP of 40 items to CP of 30 items. Let's go back to CP/SP = 4/3.

Alternative quick method: Assume CP of 1 item is 4 and SP of 1 item is 3. Selling 40 items gives total revenue $40 \times 3 = 120$. The cost of these 40 items would be $40 \times 4 = 160$. Loss = 160 - 120 = 40. Loss % = (Loss / Original CP) * 100 = (40 / 160) * 100 = (1/4) * 100 = 25%. This confirms the 25% loss.