Question

A uader sells two articles, one at a loss of \(10 \%\) and another at a profit of \(15 \%\) but finally there is no loss or gain. If the total sale price of these two articles is Rs. 30,000 , find the difference between their cost prices: (a) Rs. 5000 (b) Rs. 6000 (c) Rs. 7500 (d) none of these

Short answer

Answer: (d) none of these. The difference between the cost prices of the two articles is ₹1500.

Step-by-step solution

Define the variables

Let the cost price of the first article be \(x\) and that of the second article be \(y\).

Calculate the selling prices

According to the problem, the first article is sold at a loss of 10%, i.e., the selling price is 90% of the cost price. Therefore, the selling price of the first article = \(\frac{90}{100}\cdot x\). Similarly, the second article is sold at a profit of 15%, so the selling price of the second article = \(\frac{115}{100}\cdot y\).

Write the equation for total selling prices

The total selling prices of both articles add up to ₹30,000. So, we have the equation: \(\frac{90}{100}\cdot x + \frac{115}{100}\cdot y = 30000\)

Write the equation for overall loss or gain

There is no overall loss or gain, which means that the total cost prices of the articles are equal to the total selling prices: \(x + y\) = \(\frac{90}{100}\cdot x + \frac{115}{100}\cdot y\)

Solve the linear system of equations

First, let’s simplify the equation by multiplying both sides by 100: 10x + 15y = 3000000 Solve the equation mentioned in step 4 for x: x = \(\frac{25}{100}\cdot y\) Substitute the value of x from step 5 into the equation from step 6: \(\frac{90}{100}\cdot\frac{25}{100}\cdot y + \frac{115}{100}\cdot y = 30000\) Solve the resulting equation for y: \(15y = 30000\) Divide both sides by 15: \(y = 2000\) Now, substitute the value of y into the equation from step 5: x = \(\frac{25}{100}\cdot 2000\) x = 500

Calculate the difference between cost prices

The cost prices of the two articles are ₹500 and ₹2000 respectively. The difference between the cost prices is: 2000 - 500 = ₹1500 Therefore, the answer is not among the given options (a), (b), or (c). The correct answer is: (d) none of these

Key concepts

Linear Equations

Linear equations are mathematical expressions that define a straight line when graphed. They involve variables and constants and are found in the format of one or more algebraic expressions being set equal to each other. In problems involving profit and loss, such as this one, linear equations are instrumental in linking various elements like cost price and selling price.
For instance, in the exercise, we have the equation \(\frac{90}{100}\cdot x + \frac{115}{100}\cdot y = 30000\). This equation showcases the relationship between the selling prices of two articles. Linear equations allow us to isolate variables and find unknown values by balancing both sides of the equation. It's crucial to understand the rules of algebra to effectively manipulate and solve these equations, especially when handling multiple unknowns as seen here.

Cost Price

The cost price of an article is the original price at which it was purchased. Understanding cost price is vital for calculating profit and loss. It's the baseline from which we determine how well an item performed financially.
In the given problem, we let the cost price of the first article be \(x\) and the second be \(y\). These act as our variables in the linear equations. Knowing the cost price helps us calculate the selling price and determine the profit or loss. The challenge often lies in expressing the cost price in terms of the selling price and understanding how they relate through profit and loss percentages. Using these variables, we are eventually able to solve for their exact cost prices if given additional relationships, like selling price totals.

Selling Price

Selling price is the price at which an article is sold to a customer. It is crucial in determining profit or loss. It is calculated from the cost price by either adding a profit margin or reducing a loss percentage.
In this exercise, the selling price of the first article is determined by applying a 10% loss to its cost price, represented as \(\frac{90}{100}\cdot x\). For the second article, a 15% profit is applied, represented as \(\frac{115}{100}\cdot y\). These transformations involve basic percentage calculations, which are vital for developing equations linking the cost price and selling price, as we see with their summed equation totaling ₹30,000.

Percentage Calculation

Percentage calculations are essential in determining how much profit or loss was made on an article. A percentage expresses a number as a fraction of 100. Many profit and loss calculations depend on understanding and applying percentages correctly.
For this problem, we calculate the adjusted selling prices using percentages: 10% loss means retaining 90% of the cost price, and a 15% gain results in 115% of the cost price. To set up equations that solve for unknown values, understanding what these percentages represent is crucial. Percentage calculations enable us to bridge the gap between cost and selling prices in straightforward terms, simplifying the formation of useful linear equations.