Unraveling Profit Puzzle The Cost Price Of Item In Market Dealings

In the intricate world of commerce, understanding profit margins and comparative pricing is crucial. This article delves into a fascinating scenario involving Ram and Laxman, two individuals engaged in buying and selling the same item in the market. Through careful analysis of their transactions, we aim to unravel the profit puzzle and determine the original price of the item.

Decoding Ram's Transaction

Let's begin by dissecting Ram's transaction. Ram, an astute businessman, purchased an item from the market and sold it at a profit of 10%. To fully grasp the dynamics at play, we need to introduce a variable. Let's denote the original cost price of the item as 'X' rupees. This 'X' represents the amount Ram initially paid for the item before adding his profit margin.

Ram's profit margin is a crucial element in understanding the overall transaction. He sold the item at a 10% profit, meaning he added 10% of the cost price to the original price. Mathematically, this can be expressed as 10/100 * X, which simplifies to 0.1X. This 0.1X represents the profit Ram made on the sale.

To determine the selling price, we need to add the profit to the cost price. The selling price can be calculated as X + 0.1X, which equals 1.1X. This 1.1X is the amount Ram received when he sold the item in the market. It's the original cost price plus the 10% profit he added.

The profit Ram made from this transaction is the difference between the selling price and the cost price. We already know the selling price is 1.1X and the cost price is X. Therefore, Ram's profit can be calculated as 1.1X - X, which equals 0.1X. This 0.1X is the actual monetary gain Ram experienced from this particular transaction. Understanding this profit margin is essential for comparing it with Laxman's potential profit and unraveling the puzzle.

Analyzing Laxman's Potential Transaction

Now, let's shift our focus to Laxman, who presents an interesting twist in this market scenario. Laxman had the opportunity to purchase the same item, but he could have acquired it at a 10% lower price than Ram. This potential discount significantly alters the profit dynamics for Laxman. To fully analyze Laxman's potential transaction, we need to calculate the price he could have paid for the item.

The 10% lower price is a crucial factor in determining Laxman's potential profit. Since the original cost price of the item is 'X', a 10% reduction translates to 10/100 * X, which simplifies to 0.1X. This 0.1X represents the amount Laxman would have saved on the purchase compared to Ram. To find the price Laxman could have paid, we subtract this discount from the original cost price: X - 0.1X, which equals 0.9X. This 0.9X is the price Laxman could have potentially purchased the item for.

Laxman also had the opportunity to sell the item for 60 rupees more than Ram. This increased selling price directly impacts Laxman's potential profit. Recall that Ram sold the item for 1.1X. If Laxman sold it for 60 rupees more, his selling price would be 1.1X + 60. This higher selling price, combined with the lower purchase price, sets the stage for a potentially larger profit for Laxman.

The crux of the problem lies in understanding how much more profit Laxman could have made compared to Ram. The problem states that Laxman could have made 100 rupees more profit than Ram. This differential in profit is the key to solving the puzzle. We know Ram's profit was 0.1X. If Laxman made 100 rupees more, his profit would be 0.1X + 100. This information, coupled with the potential purchase and selling prices for Laxman, allows us to form an equation and solve for the unknown cost price, X.

Formulating the Profit Equation

To solve this intricate puzzle, we need to formulate an equation that accurately represents the profit dynamics between Ram and Laxman. This equation will serve as the key to unlocking the original cost price of the item. We know Laxman's profit is 100 rupees more than Ram's profit. We also have expressions for both Ram's profit (0.1X) and Laxman's potential profit based on his lower purchase price and higher selling price.

Laxman's profit can be calculated as the difference between his selling price and his cost price. We established that Laxman's potential selling price is 1.1X + 60, and his potential cost price is 0.9X. Therefore, Laxman's profit can be expressed as (1.1X + 60) - 0.9X. This expression represents the actual monetary gain Laxman would have experienced if he had purchased and sold the item under these conditions.

Now, we can set up the equation. We know Laxman's profit is equal to Ram's profit plus 100 rupees. This translates to the equation: (1.1X + 60) - 0.9X = 0.1X + 100. This equation encapsulates the entire scenario, linking the cost price, selling prices, and profit margins of both Ram and Laxman. Solving this equation will provide us with the value of X, the original cost price of the item.

Solving for the Unknown Cost Price (X)

With the profit equation firmly in place, we can now embark on the crucial step of solving for the unknown cost price, denoted by 'X'. This involves applying algebraic principles to isolate 'X' on one side of the equation, thereby revealing its value. The equation we derived is: (1.1X + 60) - 0.9X = 0.1X + 100. Let's break down the steps involved in solving this equation.

The first step is to simplify the equation by combining like terms. On the left side of the equation, we have 1.1X and -0.9X. Combining these terms gives us 0.2X. So, the equation now becomes: 0.2X + 60 = 0.1X + 100. This simplification makes the equation easier to manipulate and solve.

Next, we want to isolate the 'X' terms on one side of the equation. To do this, we can subtract 0.1X from both sides of the equation. This gives us: 0.2X - 0.1X + 60 = 0.1X - 0.1X + 100. Simplifying this, we get: 0.1X + 60 = 100. We've successfully moved all the 'X' terms to the left side of the equation.

Now, we need to isolate the 'X' term completely. To do this, we can subtract 60 from both sides of the equation: 0.1X + 60 - 60 = 100 - 60. This simplifies to: 0.1X = 40. We're getting closer to finding the value of 'X'.

Finally, to solve for 'X', we need to divide both sides of the equation by 0.1: (0.1X) / 0.1 = 40 / 0.1. This gives us: X = 400. Therefore, the original cost price of the item, X, is 400 rupees. This is the solution to the profit puzzle, revealing the initial investment Ram made in the item.

Verifying the Solution and Final Answer

Now that we've diligently solved for the original cost price (X = 400 rupees), it's essential to verify our solution. This crucial step ensures that our calculations are accurate and that the solution aligns with the conditions outlined in the problem. To verify, we'll substitute X = 400 back into the original problem and check if the profit difference between Ram and Laxman indeed equals 100 rupees.

First, let's revisit Ram's transaction. Ram bought the item for 400 rupees and sold it at a 10% profit. His profit, as we calculated earlier, is 0.1X. Substituting X = 400, Ram's profit is 0.1 * 400 = 40 rupees. This is the actual profit Ram made from the transaction.

Next, let's examine Laxman's potential transaction. Laxman could have bought the item for 10% less than Ram, meaning he could have purchased it for 0.9 * 400 = 360 rupees. Laxman could have sold the item for 60 rupees more than Ram, meaning his selling price would be 1.1 * 400 + 60 = 440 + 60 = 500 rupees. Laxman's profit would then be 500 - 360 = 140 rupees.

Now, let's compare the profits. Laxman's profit (140 rupees) is indeed 100 rupees more than Ram's profit (40 rupees). This confirms that our solution, X = 400 rupees, is correct. We have successfully unraveled the profit puzzle and determined the original cost price of the item.

Therefore, the original cost price of the item was 400 rupees. This detailed analysis of Ram and Laxman's market dealings highlights the importance of understanding profit margins, comparative pricing, and the power of algebraic equations in solving real-world problems.

Conclusion

This exploration of Ram and Laxman's market transactions provides valuable insights into the dynamics of profit, cost price, and selling price. By meticulously analyzing each individual's potential dealings and formulating a comprehensive equation, we successfully deciphered the unknown original cost price of the item. The solution, 400 rupees, underscores the significance of precise calculation and logical reasoning in navigating real-world financial scenarios. This problem serves as a testament to the power of mathematical principles in understanding and resolving complex commercial puzzles.