Solving Profit and Loss Problems in Lemon Sales: A Mathematical Approach

Understanding how to calculate profit and loss in business scenarios can be quite beneficial. In this article, we delve into a typical problem involving the sale of lemons. We explore the step-by-step process to solve a real-life problem and provide a comprehensive solution.

Problem Statement

A man is selling 20 lemons for Rs 40 and incurs a loss of 20%. We need to determine how many lemons he should sell for Rs 24 to gain 20% on the transaction.

Approach to Solving the Problem

Let's break down the problem into manageable steps:

Step 1: Determine the Cost Price (CP) of 20 Lemons

To solve this problem, our first step is to find the cost price of 20 lemons. Given that the man sells 20 lemons for Rs 40 and incurs a loss of 20%, we can use the following formula:

SP CP × (1 - Loss%)

Given:

Selling Price (SP) for 20 lemons Rs 40 Loss 20%

Substituting the given values into the formula:

40 CP × (1 - 0.20)

40 CP × 0.80

Solving for CP:

CP 40 / 0.80 50

So, the cost price of 20 lemons is Rs 50.

Step 2: Determine the Cost Price (CP) of Each Lemon

Now that we know the cost price of 20 lemons, we can find the cost price of each lemon:

CP of 1 lemon 50 / 20 2.5 Rs

Step 3: Calculate the Selling Price for a 20% Gain

To gain a 20% profit, the selling price (SP) should be 120% of the cost price (CP).

SP CP × (1 Gain%)

For 1 lemon:

SP of 1 lemon 2.5 × 1.20 3 Rs

Step 4: Determine the Number of Lemons to Sell for Rs 24 for a 20% Gain

We need to find how many lemons need to be sold for Rs 24 to achieve the 20% gain. Let x be the number of lemons sold. The equation can be set up as follows:

3x 24

Solving for x:

x 24 / 3 8

Therefore, the man should sell 8 lemons for Rs 24 to gain a 20% profit.

Additional Insights

In another approach, a similar problem was solved in a more simplified manner:

For 45 Rs, the man incurs a loss of 20, so the selling price (SP) should be 80% of the cost price (CP).

For 24 Rs, we need to gain 20%, so the SP should be 120% of the CP. Using a combined approach:

SP CP × (1 Gain%) CP × 1.20

The solution provided was:

CP (45 × 24 × 80) / (4120) 18 pieces

Conclusion

This problem illustrates the application of basic mathematical concepts in solving real-life business scenarios. By understanding the relationship between cost price, selling price, and profit or loss percentages, it is possible to make informed business decisions. Whether you are a seller, buyer, or business owner, these principles can help you optimize your sales strategy.