Understanding the Profit and Loss Problem

This problem deals with the concepts of cost price, selling price, percentage loss, and percentage gain. We are given the loss percentage when an article is sold at a certain price, and the gain percentage if it were sold for a higher price. The difference in the selling prices is also provided. Our goal is to find the original cost price of the article.

Key Concepts Involved

• Cost Price (CP): The price at which an article is bought.
• Selling Price (SP): The price at which an article is sold.
• Loss: Occurs when SP < CP. Loss = CP - SP.
• Gain (Profit): Occurs when SP > CP. Gain = SP - SP.
• Loss Percentage: $\text{Loss} \% = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100$.
• Gain Percentage: $\text{Gain} \% = \left( \frac{\text{Gain}}{\text{CP}} \right) \times 100$.

Setting up the Equations

Let the cost price of the article be $\text{CP}$.

Initially, the article was sold at a loss of 24%. The selling price in this case (let's call it $\text{SP}_1$) can be calculated as:

$\text{SP}_1 = \text{CP} - 24\% \text{ of } \text{CP}$

$\text{SP}_1 = \text{CP} - \left( \frac{24}{100} \times \text{CP} \right)$

$\text{SP}_1 = \text{CP} \left( 1 - \frac{24}{100} \right) = \text{CP} \left( \frac{100 - 24}{100} \right) = \text{CP} \times \frac{76}{100} = 0.76 \times \text{CP}$

If the article were sold for Rs. 1,596 more, there would have been a gain of 18%. The new selling price (let's call it $\text{SP}_2$) can be calculated as:

$\text{SP}_2 = \text{CP} + 18\% \text{ of } \text{CP}$

$\text{SP}_2 = \text{CP} + \left( \frac{18}{100} \times \text{CP} \right)$

$\text{SP}_2 = \text{CP} \left( 1 + \frac{18}{100} \right) = \text{CP} \left( \frac{100 + 18}{100} \right) = \text{CP} \times \frac{118}{100} = 1.18 \times \text{CP}$

We are given that the difference between the new selling price ($\text{SP}_2$) and the initial selling price ($\text{SP}_1$) is Rs. 1,596.

$\text{SP}_2 - \text{SP}_1 = 1596$

Solving for the Cost Price

Now, substitute the expressions for $\text{SP}_1$ and $\text{SP}_2$ in terms of $\text{CP}$ into the difference equation:

$(1.18 \times \text{CP}) - (0.76 \times \text{CP}) = 1596$

Combine the terms involving $\text{CP}$:

$(1.18 - 0.76) \times \text{CP} = 1596$

$0.42 \times \text{CP} = 1596$

To find the cost price ($\text{CP}$), divide both sides of the equation by 0.42:

$\text{CP} = \frac{1596}{0.42}$

To make the division easier, we can remove the decimal from the denominator by multiplying both the numerator and denominator by 100:

$\text{CP} = \frac{1596 \times 100}{0.42 \times 100} = \frac{159600}{42}$

Now, perform the division:

$\frac{159600}{42} = \frac{79800}{21} = \frac{26600}{7} = 3800$

So, the cost price of the article is Rs. 3800.

Let's verify this:

• Initial loss of 24%: $\text{SP}_1 = 3800 \times (1 - 0.24) = 3800 \times 0.76 = 2888$
• Gain of 18%: $\text{SP}_2 = 3800 \times (1 + 0.18) = 3800 \times 1.18 = 4484$
• Difference: $\text{SP}_2 - \text{SP}_1 = 4484 - 2888 = 1596$

The calculated difference matches the given difference, confirming the cost price is correct.

Conclusion on Cost Price Calculation

By setting up equations based on the given loss and gain percentages and using the difference in selling prices, we were able to calculate the cost price of the article. The cost price is Rs. 3800.

Revision Table: Profit and Loss Concepts

Additional Information on Profit and Loss Calculations

In profit and loss problems, the cost price (CP) is usually considered the base for calculating percentage gain or loss. Understanding the relationship between CP, SP, and the percentage change is crucial.

When a percentage change shifts from a loss to a gain (or vice versa), the total percentage difference covering the change in selling price is the sum of the absolute values of the percentage loss and the percentage gain.

In this problem, the initial state was a 24% loss, and the final state was an 18% gain. The total "jump" in percentage terms, relative to the cost price, is $24\% (\text{to reach CP from the loss SP}) + 18\% (\text{to reach the gain SP from CP}) = 42\%$.

This 42% of the cost price corresponds to the Rs. 1,596 difference in the selling prices.

So, $42\%$ of $\text{CP} = 1596$.

$\frac{42}{100} \times \text{CP} = 1596$

$\text{CP} = \frac{1596 \times 100}{42} = \frac{159600}{42} = 3800$

This alternative approach yields the same result and confirms our previous calculation method. Both methods rely on the fact that the Rs. 1,596 difference in selling price accounts for bridging the gap from a 24% loss scenario to an 18% gain scenario, relative to the cost price.