Calculating Gain Percent in Profit and Loss Problems

This problem deals with the concept of profit and loss, specifically focusing on how to calculate the gain percent when the selling price of a certain number of items is equal to the cost price of a different number of items.

Understanding the Problem Statement

We are given that the selling price (SP) of 9 boxes is equal to the cost price (CP) of 15 boxes. We need to find the gain percent. It's also stated that the selling price of all boxes is the same, and the cost price of all boxes is the same. This simplifies things, as we can consider the price per box.

Setting up the Relationship between SP and CP

Let:

• The cost price of one box be \(C\).
• The selling price of one box be \(S\).

According to the problem statement, the total selling price of 9 boxes is equal to the total cost price of 15 boxes.

Mathematically, this can be written as:

\[ 9 \times S = 15 \times C \]

Solving for the Selling Price per Box

We can rearrange the equation to express the selling price (\(S\)) in terms of the cost price (\(C\)):

\[ S = \frac{15}{9} \times C \]\[ S = \frac{5}{3} \times C \]

This tells us that the selling price of one box is \( \frac{5}{3} \) times the cost price of one box. Since \(S > C\), there is a gain (profit) on each box.

Calculating the Gain per Box

The gain on one box is the difference between its selling price and its cost price:

\[ \text{Gain} = S - C \]\[ \text{Gain} = \frac{5}{3} C - C \]\[ \text{Gain} = \left(\frac{5}{3} - 1\right) C \]\[ \text{Gain} = \left(\frac{5 - 3}{3}\right) C \]\[ \text{Gain} = \frac{2}{3} C \]

The gain on one box is \( \frac{2}{3} \) of the cost price of one box.

Calculating the Gain Percent

The gain percent is calculated using the formula:

\[ \text{Gain Percent} = \left( \frac{\text{Gain}}{\text{Cost Price}} \right) \times 100\% \]

Using the gain per box and the cost price per box:

\[ \text{Gain Percent} = \left( \frac{\frac{2}{3} C}{C} \right) \times 100\% \]\[ \text{Gain Percent} = \left( \frac{2}{3} \right) \times 100\% \]\[ \text{Gain Percent} = \frac{200}{3}\% \]

Step-by-Step Solution Summary

Here are the steps followed to arrive at the gain percent:

1. Represent the cost price of one box as \(C\) and the selling price of one box as \(S\).
2. Formulate the equation based on the problem statement: \(9S = 15C\).
3. Solve the equation to find \(S\) in terms of \(C\): \(S = \frac{5}{3}C\).
4. Calculate the gain per box: Gain \( = S - C = \frac{5}{3}C - C = \frac{2}{3}C\).
5. Calculate the gain percent using the formula: Gain Percent \( = \left( \frac{\text{Gain}}{\text{CP}} \right) \times 100 \).
6. Substitute the values: Gain Percent \( = \left( \frac{\frac{2}{3}C}{C} \right) \times 100 = \frac{2}{3} \times 100 = \frac{200}{3}\%\).

Final Answer Analysis

The calculated gain percent is \( \frac{200}{3}\% \). This matches one of the provided options.

Revision Table: Profit and Loss Concepts

Additional Information on Profit Calculation

Understanding the relationship between the number of items bought and sold for the same cost or selling price is a common type of profit and loss problem in competitive exams. When the selling price of 'x' items equals the cost price of 'y' items, the gain or loss depends on the values of 'x' and 'y'.

• If \(y > x\), there is a gain. The gain is equivalent to the cost price of \((y - x)\) items.
• If \(y < x\), there is a loss. The loss is equivalent to the cost price of \((x - y)\) items.

In this specific problem, the SP of 9 boxes is equal to the CP of 15 boxes. Here, \(x = 9\) and \(y = 15\). Since \(y > x\) (\(15 > 9\)), there is a gain. The gain is equivalent to the cost price of \((15 - 9) = 6\) boxes.

So, the gain is equal to the CP of 6 boxes. The CP on which this gain is made is the CP of 9 boxes (since we are selling 9 boxes).

Gain % \( = \left(\frac{\text{Gain (CP of 6 boxes)}}{\text{CP of 9 boxes}}\right) \times 100 \)

Let CP of 1 box be \(C\). Then CP of 6 boxes is \(6C\) and CP of 9 boxes is \(9C\).

Gain % \( = \left(\frac{6C}{9C}\right) \times 100 = \left(\frac{6}{9}\right) \times 100 = \left(\frac{2}{3}\right) \times 100 = \frac{200}{3}\%\).

This alternative method yields the same result and provides another way to think about these types of problems.