Calculating Selling Price With Loss A Step-by-Step Guide

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In this intricate mathematical problem, we delve into the complexities of cost price, selling price, and loss percentage. We'll dissect the scenario step by step to determine the crucial selling price of sugar per kilogram. Let's embark on this journey of calculations and logical deductions.

Understanding the Problem

In order to solve this problem, it's very important to understand the core components first. Here we have someone who invested in 100 kg of sugar, this is our starting point. The cost price is Rs. 2770, meaning that's how much was spent to acquire the sugar. Now, the twist is that the sugar wasn't sold at a profit; there was a loss involved. This loss is quantified in a rather peculiar way: it's 37.5% more than the amount received from selling 28 kg of sugar. Our ultimate goal is to find the selling price per kilogram that resulted in this loss scenario. To begin, we will need to break down each piece of information and see how they fit together, turning them into equations that reflect the real world situation being described. This process will help simplify the complexity of the problem into manageable calculations, bringing clarity and paving the way for an accurate solution.

Step-by-Step Solution

1. Define Variables

Let's start by defining the variables to represent the unknowns in our problem. This will help us translate the word problem into mathematical expressions. We'll use:

  • SP to denote the selling price of the sugar per kg (in Rs.). This is the key value we're trying to find.

2. Calculate the Loss in Terms of SP

Here's where the problem gets interesting. The loss is tied to the selling price of a specific quantity of sugar. The problem states that the total loss is 37.5% more than the amount obtained by selling 28 kg of sugar. Let's break this down:

  • The amount obtained by selling 28 kg of sugar is simply 28 * SP. This is because each kilogram is sold at a price SP. Understanding this part is crucial, as it directly links the unknown selling price to a tangible revenue from a subset of the total sale.

  • 37.5% is equivalent to 3/8, which is a common fractional representation to remember for percentage calculations. When we say "37.5% more than," we mean adding 37.5% of the original amount to itself. In mathematical terms, this means the loss is 3/8 more than the amount received from selling 28 kg of sugar. This can be written as:

    Loss = (28 * SP) + (3/8) * (28 * SP)

3. Simplify the Loss Expression

Now, let's simplify the expression for the loss. Simplifying expressions makes them easier to work with in the subsequent steps. We start with:

Loss = (28 * SP) + (3/8) * (28 * SP)

To simplify, we can factor out the common term 28 * SP:

Loss = (28 * SP) * (1 + 3/8)

Next, we add the terms inside the parentheses. To do this, we need a common denominator, which in this case is 8. So, we rewrite 1 as 8/8:

Loss = (28 * SP) * (8/8 + 3/8)

Now we can add the fractions:

Loss = (28 * SP) * (11/8)

Finally, we multiply to get the simplified expression for the loss:

Loss = (28 * 11 * SP) / 8

This can be further simplified by dividing 28 by 4 to get 7, and dividing 8 by 4 to get 2:

Loss = (7 * 11 * SP) / 2

Which gives us:

Loss = 77 * SP / 2

This simplified expression, Loss = 77 * SP / 2, is much easier to work with. It tells us that the total loss is directly proportional to the selling price SP. Specifically, the loss is 77/2 times the selling price. This simplification makes it more straightforward to use this expression in further calculations, especially when we bring in the concept of the cost price and the fundamental equation that links cost price, selling price, and loss.

4. Express Loss in Terms of CP and SP

The fundamental relationship we need to use here is:

Loss = Cost Price - Selling Price

This is a basic principle in commerce and mathematics, where the loss is defined as the difference between what you originally paid for an item (the cost price) and what you sold it for (the selling price). The cost price represents the initial investment, while the selling price represents the revenue generated from the sale. The difference between these two is the profit or loss.

In our problem, we know the cost price (CP) is Rs. 2770. The selling price (SP) is what we're trying to find, and we've just derived an expression for the loss in terms of SP. We need to apply this relationship to the entire quantity of sugar, which is 100 kg. The total selling price for 100 kg of sugar would be 100 times the selling price per kg, or 100 * SP. Therefore, we can express the total loss as:

Loss = 2770 - (100 * SP)

Here, we've translated the verbal description of the loss into an equation, using the known cost price and the unknown selling price. This equation represents the total financial loss incurred from selling the sugar, which is the cost price minus the total revenue from sales.

5. Equate the Two Loss Expressions

Now, we have two different ways to express the loss:

  • From Step 3, we have Loss = 77 * SP / 2
  • From Step 4, we have Loss = 2770 - (100 * SP)

Since both expressions represent the same loss, we can set them equal to each other. This is a crucial step because it allows us to create an equation with only one unknown variable (SP), which we can then solve. The act of equating the two expressions mathematically represents the connection between the proportional loss (derived from the partial sale) and the overall loss (calculated from the cost and total selling prices). By setting these two expressions equal, we're essentially saying that the loss calculated based on the 28 kg sale scenario is the same as the overall loss experienced from selling all 100 kg.

Setting these equal gives us the equation:

77 * SP / 2 = 2770 - (100 * SP)

This equation is the bridge that connects the initial conditions of the problem to its solution. Solving it will give us the selling price of the sugar per kg, which is the goal of the problem.

6. Solve for SP

We now have the equation:

77 * SP / 2 = 2770 - (100 * SP)

To solve for SP, we need to isolate it on one side of the equation. Our first step will be to eliminate the fraction by multiplying every term in the equation by 2. This will clear the denominator and make the equation easier to work with. When we multiply each term by 2, we get:

2 * (77 * SP / 2) = 2 * 2770 - 2 * (100 * SP)

This simplifies to:

77 * SP = 5540 - 200 * SP

Now, we need to get all the SP terms on one side of the equation. To do this, we'll add 200 * SP to both sides of the equation. This will move the -200 * SP term from the right side to the left side:

77 * SP + 200 * SP = 5540 - 200 * SP + 200 * SP

This simplifies to:

277 * SP = 5540

Finally, to isolate SP, we'll divide both sides of the equation by 277. This will give us the value of SP:

277 * SP / 277 = 5540 / 277

This simplifies to:

SP = 5540 / 277

Now we perform the division. 5540 divided by 277 is exactly 20:

SP = 20

So, the selling price of the sugar per kg is Rs. 20. This is the value that satisfies the conditions of the problem, where the loss is 37.5% more than the amount obtained by selling 28 kg of sugar. This solution is the result of carefully translating the word problem into mathematical expressions, setting up an equation, and solving for the unknown. It's a process that combines algebraic manipulation with a solid understanding of the problem's context.

7. State the Answer

The selling price of sugar is Rs. 20 per kg.

Conclusion

Through careful analysis and step-by-step calculations, we've successfully determined the selling price of the sugar. This problem highlights the importance of understanding the relationships between cost price, selling price, and loss, as well as the ability to translate word problems into mathematical equations. By breaking down the problem into smaller, manageable steps, we can solve even the most complex mathematical puzzles. This approach not only helps in finding the solution but also enhances our problem-solving skills, which are valuable in various aspects of life.