Calculating Discount Percentage: A Shopkeeper's Profit Puzzle

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Calculating Discount Percentage: A Shopkeeper's Profit Puzzle

Hey there, math enthusiasts! Let's dive into a classic percentage problem that often pops up in competitive exams and real-life scenarios. The core concept revolves around understanding how a shopkeeper marks up the price of an item, offers a discount, and still manages to make a profit. In this article, we'll break down the question: "A shopkeeper marks the price of an article 35% more than the cost price. After allowing a discount, he earns a 17% profit. The approximate discount percentage is:" Don't worry, we'll walk through it step-by-step, making sure you grasp the underlying principles and arrive at the correct solution. Let's get started!

Understanding the Problem and Setting Up the Variables

Alright, guys, before we jump into calculations, let's break down the problem statement. The question presents us with a common retail scenario. A shopkeeper wants to sell an item and make a profit. To do this, they first mark up the price above what it cost them to acquire the item (the cost price or CP). After that, they offer a discount to attract customers. The goal is to figure out the discount percentage they're giving while still making a profit. Here's a quick rundown of the key terms:

  • Cost Price (CP): The price the shopkeeper initially pays for the item.
  • Marked Price (MP): The price the shopkeeper labels the item for sale (often higher than the CP).
  • Discount: A reduction in the marked price to entice buyers.
  • Selling Price (SP): The price at which the item is actually sold (MP minus the discount).
  • Profit: The gain the shopkeeper makes (SP minus CP).

In our problem, we know the shopkeeper marks the price 35% above the cost price and makes a 17% profit after giving a discount. To solve this, it's easiest to assume a cost price. Let's suppose the cost price (CP) is 100. This makes calculating percentages a breeze. Since the marked price (MP) is 35% above the cost price, we calculate it as follows: MP = CP + 35% of CP = 100 + 35 = 135. So the marked price is 135. Now we know the shopkeeper makes a 17% profit. The profit is calculated on the cost price (CP). So profit = 17% of 100 = 17. The selling price (SP) is then CP + Profit = 100 + 17 = 117. Therefore, to make a profit of 17%, the item needs to be sold for 117. Now we have all the important values to calculate our final answer. The discount will be calculated on MP to SP.

Calculating the Discount Percentage: Step-by-Step

Now, let's calculate that all-important discount percentage! We've already laid the groundwork, so this part should be straightforward. We know the marked price (MP) is 135 and the selling price (SP) is 117. The discount is the difference between the marked price and the selling price. Let's calculate the discount amount first. Discount = MP - SP = 135 - 117 = 18. The discount is 18. This is the monetary value of the discount. To express this as a percentage, we need to calculate what percentage 18 is of the marked price (MP), because the discount is always given as a percentage of the marked price. Here is the formula:

Discount Percentage = (Discount / Marked Price) × 100

So, plugging in our values: Discount Percentage = (18 / 135) × 100. Let's do the math: (18 / 135) = 0.1333. Then, 0.1333 × 100 = 13.33%. Therefore, the approximate discount percentage is 13.33%. This means the shopkeeper offered a discount of approximately 13.33% to make a 17% profit. See, not so hard, right?

Alternative Approach and Important Considerations

Here’s another way to approach the problem, which can be useful for speed and accuracy, especially in time-constrained exams. This method uses a direct formula to find the discount percentage when the markup and profit percentages are known. The relationship between the cost price (CP), marked price (MP), selling price (SP), discount, and profit can be expressed as follows: If the shopkeeper marks up the price by X% and makes a profit of Y% after giving a discount, then the discount percentage can be calculated using the formula:

Discount Percentage = [(Markup % - Profit %) / (100 + Markup %)] * 100

Where Markup % is the percentage increase from the CP to MP, and Profit % is the percentage profit earned. In our problem, the markup is 35% and the profit is 17%. Plugging the values: Discount % = [(35 - 17) / (100 + 35)] * 100 = (18 / 135) * 100 = 13.33%. This formula gives the same result, but it helps speed things up. Remember, you should always double-check your calculations, especially during exams. Rounding errors can sometimes lead to slightly different answers, so be mindful of the level of precision required. If you're dealing with multiple-choice questions, it's often a good strategy to quickly scan the options and choose the closest answer to your calculation.

Important considerations: Remember that in real-world scenarios, shopkeepers might factor in other costs like overheads (rent, salaries, etc.) when setting prices and determining profit margins. Also, discounts can vary widely based on the type of product, the season, and the overall marketing strategy. Always read the question carefully and understand what's being asked. Be mindful of the units (percentages, amounts, etc.) and ensure your answer makes logical sense within the given context. Practice makes perfect, so solving similar problems will help build your confidence and improve your problem-solving skills.

Final Answer and Key Takeaways

So, guys, what's the deal? We've successfully calculated the approximate discount percentage! By following the step-by-step approach and using the formula, we determined that the shopkeeper offered a discount of approximately 13.33% on the marked price. The correct answer, therefore, is around 13%. Here are some key takeaways:

  • Understand the Terms: Clearly define cost price, marked price, selling price, discount, and profit. Knowing these definitions is fundamental to solving the problem.
  • Assume a CP: Assuming a CP of 100 simplifies the calculation of percentages. This is a common and effective problem-solving strategy.
  • Calculate Discount Amount: Find the monetary value of the discount by subtracting the selling price from the marked price.
  • Calculate Discount Percentage: Use the formula: (Discount / Marked Price) × 100. Always calculate the discount percentage concerning the marked price.
  • Use the Formula: To avoid complex calculations, you can use the direct formula: Discount Percentage = [(Markup % - Profit %) / (100 + Markup %)] * 100.

That's it, folks! I hope this explanation has been helpful. Keep practicing and exploring these concepts. You will find that these types of problems become easier and more manageable with practice. Keep up the excellent work! If you have any further questions or if you want to explore more problems, feel free to ask. Keep learning and growing your math skills. Happy calculating!