Bat Sold At A Gain: Find The Original Price!

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Hey guys! Let's break down this Brainly math question about a bat being sold. We need to figure out the original price of the bat before the seller made a profit. It's all about percentages and figuring out the cost price when we know the selling price and the profit percentage. So, buckle up, and let's solve this problem step-by-step. We'll use some basic math principles to get to the answer. Ready? Let’s dive in!

Understanding the Problem

Okay, so here's the core of the problem: A bat is sold for ₹25.56, and the seller makes a gain of 27 4/5%. The question asks us to find at what price the bat was originally bought. This means we need to determine the cost price (CP) of the bat. The selling price (SP) and the gain percentage are provided. To find the cost price, we'll use the formula that relates the selling price, cost price, and gain percentage.

First, let’s convert the mixed fraction gain percentage into a simple fraction or decimal. The gain percentage is given as 27 4/5%. To convert this, we first multiply the whole number (27) by the denominator (5), which gives us 135. Then, we add the numerator (4) to this result, which gives us 139. So, the gain percentage is 139/5%. Now, to make calculations easier, we can convert this fraction to a decimal by dividing 139 by 5, which equals 27.8%. So, the bat was sold at a 27.8% gain. This conversion simplifies our calculations moving forward.

Now that we have the gain percentage as a decimal, we can proceed to use the formula to find the cost price. The formula that relates the selling price (SP), cost price (CP), and gain percentage is:

SP = CP + (Gain % × CP)

We know that SP = ₹25.56 and Gain % = 27.8% (or 0.278 as a decimal). We need to rearrange the formula to solve for CP:

₹25. 56 = CP + (0.278 × CP)

₹26. 56 = CP × (1 + 0.278)

₹27. 56 = CP × 1.278

Now, we solve for CP by dividing both sides of the equation by 1.278:

CP = ₹28. 56 / 1.278

CP ≈ ₹20.00

So, the cost price of the bat is approximately ₹20.00. This is the price at which the bat was bought before it was sold at a gain. Therefore, the original price of the bat was about ₹20.00.

Step-by-Step Solution

To solve this problem, we will follow a structured, step-by-step approach:

  1. Convert the Gain Percentage: Convert the mixed fraction gain percentage (27 4/5%) into a decimal. This makes it easier to use in calculations. The conversion is as follows:

    27 4/5% = (27 + 4/5)% = (27 + 0.8)% = 27.8%

    So, the gain percentage is 27.8%, or 0.278 when expressed as a decimal.

  2. Use the Selling Price Formula: The formula that relates the selling price (SP), cost price (CP), and gain percentage is:

    SP = CP + (Gain % × CP)

    Where:

    • SP = ₹25.56 (the selling price of the bat)
    • Gain % = 27.8% or 0.278 (the gain percentage as a decimal)
    • CP = Cost Price (the original price we want to find)

  3. Rearrange the Formula to Solve for CP: We need to isolate CP in the formula. Start by substituting the known values:

    ₹29. 56 = CP + (0.278 × CP)

    Combine the terms involving CP:

    ₹30. 56 = CP × (1 + 0.278)

    ₹31. 56 = CP × 1.278

  4. Solve for CP: Divide both sides of the equation by 1.278 to find the cost price:

    CP = ₹32. 56 / 1.278

    CP ≈ ₹20.00

    Therefore, the cost price of the bat is approximately ₹20.00.

  5. State the Final Answer: The bat was bought for approximately ₹20.00.

By following these steps, we have successfully found the original price of the bat. This structured approach makes it easy to understand and solve similar problems involving cost price, selling price, and gain percentage.

Explanation of Key Concepts

To fully understand this problem, let's delve into the key concepts involved:

  • Cost Price (CP): The cost price is the original price at which an item is purchased before any profit or loss is applied. In this scenario, it's the price at which the bat was bought.

  • Selling Price (SP): The selling price is the price at which an item is sold after considering profit or loss. Here, the bat was sold for ₹25.56.

  • Gain Percentage: The gain percentage is the percentage of profit made on the cost price. It is calculated as:

    Gain % = ((Selling Price - Cost Price) / Cost Price) × 100

    In our case, the gain percentage was given as 27 4/5%, which we converted to 27.8%.

  • Relationship between SP, CP, and Gain %: The relationship between the selling price, cost price, and gain percentage is crucial for solving problems like this. The selling price can be expressed as:

    SP = CP + (Gain % × CP)

    This formula helps us understand how the selling price is determined based on the cost price and the profit margin.

Understanding these concepts thoroughly equips us with the knowledge to tackle a variety of similar problems. Knowing the definitions and formulas ensures accuracy and efficiency in solving such mathematical questions.

Alternative Method

Here’s another way to approach the problem, which can sometimes be more intuitive:

  1. Understand the Percentage Increase: A gain of 27 4/5% means the selling price is 127 4/5% of the cost price. Convert 127 4/5% to a decimal.

    127 4/5% = 127.8% = 1.278

  2. Set Up the Equation: Let CP be the cost price. Then:

    1. 278 × CP = ₹25.56

  3. Solve for CP: Divide both sides by 1.278:

    CP = ₹25.56 / 1.278

    CP ≈ ₹20.00

This method directly uses the fact that the selling price is a percentage of the cost price, making it a straightforward way to find the cost price when the gain percentage is known. It’s a useful alternative for those who prefer to think in terms of direct percentages of the original price.

Common Mistakes to Avoid

When solving problems like this, there are several common mistakes that students often make. Being aware of these pitfalls can help ensure accuracy and prevent errors:

  • Incorrectly Converting the Gain Percentage: A common mistake is mishandling the conversion of the mixed fraction gain percentage into a decimal. Ensure that you accurately convert 27 4/5% to 27.8% before using it in calculations. Miscalculating this value will lead to an incorrect final answer.
  • Using the Wrong Formula: It’s crucial to use the correct formula relating the selling price, cost price, and gain percentage. The formula SP = CP + (Gain % × CP) is essential. Confusing this with a different formula can lead to significant errors.
  • Incorrectly Rearranging the Formula: When solving for the cost price (CP), make sure you correctly rearrange the formula. The algebraic manipulation must be accurate to isolate CP properly. A mistake in this step will result in an incorrect value for the cost price.
  • Rounding Errors: Be cautious with rounding. If you round prematurely during the intermediate steps, it can affect the accuracy of the final answer. It’s best to keep as many decimal places as possible until the final step to minimize rounding errors.
  • Misunderstanding the Problem: Always ensure you fully understand the problem before attempting to solve it. Identify what is given (selling price, gain percentage) and what needs to be found (cost price). Misinterpreting the question can lead to using the wrong approach.

By avoiding these common mistakes, you can improve your accuracy and confidence in solving similar problems involving cost price, selling price, and gain percentage.

Real-World Applications

Understanding these concepts isn't just for solving textbook problems; they have numerous real-world applications. Here are a few examples:

  • Retail: Retail businesses use these calculations to determine the selling price of products. They need to consider the cost price, desired profit margin, and any additional expenses to set a competitive and profitable price.
  • Finance: In finance, understanding cost and selling prices is crucial for investment decisions. Investors need to know the original price they paid for an asset and the selling price to calculate their return on investment (ROI).
  • Real Estate: Real estate agents and investors use these concepts to evaluate property values. They consider the original purchase price, any improvements made, and the current market value to determine a fair selling price and potential profit.
  • Personal Finance: On a personal level, understanding these calculations can help you make informed purchasing decisions. For example, when buying a used car, you can estimate the original cost and depreciation to negotiate a fair price.
  • E-commerce: Online businesses use these formulas to optimize pricing strategies. They need to factor in costs such as manufacturing, shipping, and marketing to set prices that attract customers and generate profit.

By understanding these real-world applications, you can see how valuable these mathematical concepts are in various aspects of life. This knowledge empowers you to make informed decisions and understand the financial implications of different scenarios.

Summary

Alright, guys, let’s wrap things up! We tackled a Brainly question about a bat being sold for ₹25.56 at a gain of 27 4/5%. The goal was to find the original price (cost price) of the bat. We converted the gain percentage to a decimal, used the selling price formula, rearranged it to solve for the cost price, and found that the bat was originally bought for approximately ₹20.00.

We also explored key concepts like cost price, selling price, and gain percentage, and looked at an alternative method to solve the problem. Plus, we highlighted common mistakes to avoid and discussed real-world applications of these concepts.

Hopefully, this breakdown helps you understand how to solve similar problems and appreciate the practical uses of these calculations. Keep practicing, and you’ll become a pro at solving these types of questions!