Manu's Jacket Discount Puzzle Solving Profit And Percentage Problem

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In the bustling lanes of Sarojini market, Manu stumbled upon a stylish jacket priced at Rs. 1000. Inspired by its potential, Manu decided to mark up the price by P%. To attract customers, a discount of (0.4 x P)% was offered, yet Manu still managed to secure a profit percentage of (0.4 x P)%. This intriguing scenario presents a mathematical puzzle that requires a detailed analysis of cost price, marked price, discount, and profit to determine the amount of the discount offered. This comprehensive guide delves into the step-by-step solution, providing a clear understanding of the calculations involved.

Understanding the Problem

The core of this problem lies in understanding the relationships between cost price, marked price, discount, and profit. The cost price is the initial price Manu paid for the jacket, which is Rs. 1000. The marked price is the price after the markup, which is calculated by increasing the cost price by P%. The discount is the reduction offered on the marked price, calculated as (0.4 x P)%. The selling price is the final price after the discount, and the profit is the difference between the selling price and the cost price. The profit percentage is the profit expressed as a percentage of the cost price.

To solve this problem, we need to establish equations that represent these relationships. Let's break down the problem into smaller parts.

Defining the Variables

To make the problem easier to solve, let's define the variables:

  • Cost Price (CP) = Rs. 1000
  • Markup Percentage = P%
  • Discount Percentage = (0.4 x P)%
  • Profit Percentage = (0.4 x P)%
  • Marked Price (MP) = CP + (P% of CP)
  • Selling Price (SP) = MP - (Discount % of MP)

Calculating the Marked Price

The marked price is the price after the markup. It's calculated by adding the markup amount to the cost price. The markup amount is P% of the cost price.

MP = CP + (P/100) * CP MP = 1000 + (P/100) * 1000 MP = 1000 + 10P

Determining the Selling Price

The selling price is the price after the discount. It's calculated by subtracting the discount amount from the marked price. The discount amount is (0.4 x P)% of the marked price.

SP = MP - (0.4P/100) * MP SP = MP * (1 - 0.4P/100) SP = (1000 + 10P) * (1 - 0.4P/100)

Calculating the Profit

The profit is the difference between the selling price and the cost price.

Profit = SP - CP Profit = (1000 + 10P) * (1 - 0.4P/100) - 1000

Determining the Profit Percentage

The profit percentage is the profit expressed as a percentage of the cost price.

Profit % = (Profit / CP) * 100 (0.4P) = (((1000 + 10P) * (1 - 0.4P/100) - 1000) / 1000) * 100

Solving the Equation

Now we have an equation that relates P to the given information. Let's solve for P.

  1. 4P = (((1000 + 10P) * (1 - 0.4P/100) - 1000) / 1000) * 100
  2. 4P/100 = ((1000 + 10P) * (1 - 0.4P/100) - 1000) / 1000
  3. 4P/10000 = ((1000 + 10P) * (1 - 0.4P/100) - 1000) / 1000
  4. 4P = (1000 + 10P) * (1 - 0.4P/100) - 1000

Let's simplify the equation:

  1. 4P = (1000 + 10P) * (1 - 0.004P) - 1000
  2. 4P = 1000 - 4P + 10P - 0.04P^2 - 1000
  3. 4P = 6P - 0.04P^2
  4. 04P^2 - 1.6P = 0

Now, we can solve this quadratic equation. We can factor out P:

P(0.04P - 1.6) = 0

This gives us two possible solutions for P:

  • P = 0
  • 0.04P - 1.6 = 0 => 0.04P = 1.6 => P = 1.6 / 0.04 => P = 40

Since P cannot be 0 (as there would be no markup, discount, or profit), we have P = 40%.

Calculating the Discount Amount

Now that we have the value of P, we can calculate the discount amount. The discount percentage is (0.4 x P)%, which is (0.4 x 40)% = 16%.

First, we need to find the marked price:

MP = 1000 + 10P MP = 1000 + 10 * 40 MP = 1000 + 400 MP = 1400

Now, we can calculate the discount amount:

Discount Amount = (Discount Percentage / 100) * MP Discount Amount = (16 / 100) * 1400 Discount Amount = 0.16 * 1400 Discount Amount = 224

Therefore, the amount of discount is Rs. 224.

Final Answer

The amount of discount Manu offered on the jacket is Rs. 224. This detailed solution illustrates how to break down a complex problem into smaller, manageable steps. By understanding the relationships between cost price, marked price, discount, and profit, we can effectively solve such mathematical puzzles.

To effectively tackle this problem involving Manu's jacket purchase, a structured approach is necessary. We'll dissect each step, providing clarity and ensuring a comprehensive understanding of the solution.

1. Define the Variables and Understand the Relationships

The first crucial step is to define the variables and understand how they relate to each other. This provides a clear framework for the problem.

  • Cost Price (CP): The initial price Manu paid for the jacket (Rs. 1000).
  • Markup Percentage (P%): The percentage by which Manu increased the price.
  • Discount Percentage (0.4 x P)%: The percentage discount offered on the marked price.
  • Profit Percentage (0.4 x P)%: The percentage profit Manu made after the discount.
  • Marked Price (MP): The price after the markup, calculated as CP + (P% of CP).
  • Selling Price (SP): The final price after the discount, calculated as MP - (Discount % of MP).

Understanding these relationships is paramount to solving the problem. The marked price is the cost price plus the markup. The selling price is the marked price minus the discount. The profit is the selling price minus the cost price, and the profit percentage is the profit expressed as a percentage of the cost price.

2. Calculate the Marked Price (MP)

The marked price is the price after the initial markup. We calculate it by adding the markup amount to the cost price. The markup amount is P% of the cost price.

MP = CP + (P/100) * CP MP = 1000 + (P/100) * 1000 MP = 1000 + 10P

This equation represents the marked price in terms of P. For instance, if P is 20%, the marked price would be Rs. 1200.

3. Determine the Selling Price (SP)

The selling price is the price after the discount is applied. We calculate it by subtracting the discount amount from the marked price. The discount amount is (0.4 x P)% of the marked price.

SP = MP - (0.4P/100) * MP SP = MP * (1 - 0.4P/100) SP = (1000 + 10P) * (1 - 0.4P/100)

This equation gives us the selling price in terms of P. It considers the marked price and the discount percentage to arrive at the final selling price.

4. Calculate the Profit and Profit Percentage

The profit is the difference between the selling price and the cost price.

Profit = SP - CP Profit = (1000 + 10P) * (1 - 0.4P/100) - 1000

The profit percentage is the profit expressed as a percentage of the cost price. This is a crucial step as it connects the profit earned to the initial investment.

Profit % = (Profit / CP) * 100 (0.4P) = (((1000 + 10P) * (1 - 0.4P/100) - 1000) / 1000) * 100

This equation sets the profit percentage equal to (0.4 x P)%, as given in the problem statement. This is the key equation that allows us to solve for P.

5. Solve the Equation for P

Now we have an equation that relates P to the given information. Solving for P involves algebraic manipulation and simplification.

  1. 4P = (((1000 + 10P) * (1 - 0.4P/100) - 1000) / 1000) * 100
  2. 4P/100 = ((1000 + 10P) * (1 - 0.4P/100) - 1000) / 1000
  3. 4P/10000 = ((1000 + 10P) * (1 - 0.4P/100) - 1000) / 1000
  4. 4P = (1000 + 10P) * (1 - 0.4P/100) - 1000

Simplifying the equation further:

  1. 4P = (1000 + 10P) * (1 - 0.004P) - 1000
  2. 4P = 1000 - 4P + 10P - 0.04P^2 - 1000
  3. 4P = 6P - 0.04P^2
  4. 04P^2 - 1.6P = 0

This leads us to a quadratic equation. Factoring out P simplifies the solution:

P(0.04P - 1.6) = 0

This gives us two possible solutions for P:

  • P = 0 (This solution is not valid in this context as it implies no markup, discount, or profit.)
    1. 04P - 1.6 = 0 => 0.04P = 1.6 => P = 1.6 / 0.04 => P = 40

Therefore, the markup percentage P is 40%.

6. Calculate the Discount Amount

Now that we have the value of P, we can calculate the discount amount. The discount percentage is (0.4 x P)%, which is (0.4 x 40)% = 16%.

First, we calculate the marked price:

MP = 1000 + 10P MP = 1000 + 10 * 40 MP = 1000 + 400 MP = 1400

Now, we can calculate the discount amount:

Discount Amount = (Discount Percentage / 100) * MP Discount Amount = (16 / 100) * 1400 Discount Amount = 0.16 * 1400 Discount Amount = 224

Thus, the discount amount is Rs. 224.

7. Final Answer and Conclusion

The amount of discount Manu offered on the jacket is Rs. 224.

This step-by-step solution provides a clear and methodical approach to solving the problem. By breaking down the problem into smaller, manageable steps, we can understand the relationships between the variables and arrive at the correct solution. This approach is valuable for solving similar mathematical problems involving percentages, profit, and discount.

To fully grasp the solution to Manu's jacket problem, it's crucial to understand the underlying mathematical concepts. These concepts form the foundation for solving similar problems involving profit, discount, and percentages. Let's delve into the key concepts:

1. Cost Price (CP)

The cost price is the initial price at which an item is purchased. In this case, the cost price is the amount Manu paid for the jacket, which is Rs. 1000. Understanding the cost price is the first step in determining profit or loss.

The cost price serves as the base for all subsequent calculations, including markup, discount, and profit. It's the foundation upon which the entire transaction is analyzed.

2. Marked Price (MP)

The marked price is the price at which an item is listed for sale after a markup has been applied to the cost price. The markup is typically expressed as a percentage of the cost price. In Manu's case, the price was marked up by P%.

The marked price is calculated as:

MP = CP + (P/100) * CP

Understanding the marked price is essential because it's the price from which discounts are offered. It represents the seller's initial intention regarding the selling price.

The markup percentage reflects the seller's strategy in terms of desired profit margin and market competitiveness. A higher markup percentage indicates a higher profit expectation, while a lower percentage might be used to attract more customers.

3. Discount Percentage

A discount is a reduction in the marked price, offered to attract customers or clear out inventory. In this problem, Manu offered a discount of (0.4 x P)% on the marked price.

The discount percentage is applied to the marked price to determine the selling price. It's a critical factor in influencing sales volume and customer perception.

The discount percentage can be a strategic tool to increase sales, especially during seasonal promotions or to liquidate older stock. It can also be used to match competitors' pricing strategies.

4. Selling Price (SP)

The selling price is the final price at which an item is sold after applying any discounts to the marked price. It's the price the customer actually pays for the item.

The selling price is calculated as:

SP = MP - (Discount Percentage/100) * MP

The selling price is a key determinant of both revenue and profit. It needs to be carefully considered to ensure that the business achieves its financial goals.

The selling price directly impacts the customer's purchasing decision. A lower selling price can attract more customers, but it also reduces the profit margin. A higher selling price can increase profit per unit but might lead to lower sales volume.

5. Profit

Profit is the difference between the selling price and the cost price. It represents the financial gain from a transaction.

Profit = SP - CP

Profit is the ultimate goal of any business transaction. It's the reward for the risk taken and the effort invested.

Profit is a critical metric for assessing the financial performance of a business. It indicates whether the business is generating sufficient revenue to cover its costs and provide a return on investment.

6. Profit Percentage

The profit percentage is the profit expressed as a percentage of the cost price. It provides a standardized measure of profitability, allowing for comparisons across different transactions or businesses.

Profit % = (Profit / CP) * 100

The profit percentage is a more informative metric than the absolute profit amount because it takes into account the initial investment.

The profit percentage allows for a relative assessment of profitability. A higher profit percentage indicates a more efficient use of resources and a better return on investment.

7. Percentage Calculations

This problem heavily relies on percentage calculations, including finding a percentage of a number, increasing a number by a percentage, and decreasing a number by a percentage.

Understanding percentage calculations is fundamental to solving problems involving profit, discount, and markup.

Percentage calculations are essential tools for financial analysis, pricing strategies, and sales promotions. They provide a standardized way to express changes in value and compare different scenarios.

8. Algebraic Equations

Solving this problem involves setting up and solving algebraic equations. This requires the ability to translate word problems into mathematical expressions and manipulate equations to isolate the unknown variable.

Algebraic equations are essential for representing relationships between variables and solving for unknown quantities.

Algebraic skills are crucial for problem-solving in various fields, including finance, engineering, and science. They provide a framework for modeling real-world situations and finding solutions.

By understanding these key concepts, you can approach similar problems with confidence and clarity. The relationships between cost price, marked price, discount, selling price, profit, and profit percentage are fundamental to business and finance, and mastering these concepts will prove invaluable in various contexts.

Original Question: Manu bought a jacket from Sarojini market at Rs. 1000 and marked up its price by P%. He then gave a discount of (0.4 x P)% and still got a profit percentage of (0.4 x P)%. What is the amount of discount?

Rewritten Question: Manu purchased a jacket for Rs. 1000 and increased the price by P percent. After offering a discount of (0.4 multiplied by P) percent, he still made a profit of (0.4 multiplied by P) percent. Determine the discount amount in Rupees.