Question

A shopkeeper can sell upto 20 units of both the books and stationery. If he makes a profit of \(\mathrm{Rs} 2\) on each book and \(\mathrm{Rs} 3\) on each unit of stationery, then the profit function is , if \(\mathrm{x}\) and \(\mathrm{y}\) denote the number of units of books and stationery sold. (1) \(p=2 x-3 y\) (2) \(p=2 x+3 y\) (3) \(\mathrm{p}=3 \mathrm{x}-2 \mathrm{y}\) (4) \(\mathrm{p}=3 \mathrm{x}+2 \mathrm{y}\)

Short answer

Answer: (2) \(p = 2x + 3y\)

Step-by-step solution

Identify the variables

In this problem, \(x\) denotes the number of units of books sold, while \(y\) represents the number of units of stationery sold.

Write down the profit for each unit sold

The profit for selling each unit of books is \(\mathrm{Rs} 2\). This can be written as \(2x\). The profit for selling each unit of stationery is \(\mathrm{Rs} 3\). This can be written as \(3y\).

Write down the profit function

Now, we need to write the profit function which calculates the total profit based on the number of units of books and stationery sold. It can be obtained by adding the profit from selling each unit of books to the profit from selling each unit of stationery. Mathematically: \(p = 2x + 3y\)

Match the profit function with the given options

We can see that the profit function derived in Step 3 matches option (2): \(p=2x+3y\) So, the correct answer is: (2) \(p = 2x + 3y\)

Key concepts

Understanding Linear Equations

Linear equations are mathematical expressions that equate two expressions by using a linear relationship between two variables, often labelled as \(x\) and \(y\). In the context of the profit function from the shopkeeper's exercise, the linear equation \(p = 2x + 3y\) represents how profit, \(p\), depends on the number of books and stationery sold. Here, \(x\) and \(y\) are the variables.
Linear equations have some characteristic features:

• Constant Coefficients: In the equation \(2x + 3y = p\), the numbers 2 and 3 are coefficients. They represent the constant amount of profit made per unit; \(2\) per book and \(3\) per piece of stationery.
• One Degree: Linear equations do not have variables raised to a power higher than 1.
• Graphical Representation: These equations are represented as straight lines when graphed on a coordinate plane.

Understanding linear equations helps in visualizing how different quantities like sales units affect total profits.

Profit Calculation Explained

Profit calculation involves determining the increase in revenue by selling products minus the costs. However, when simplifying for a function like this, it can simply be represented as the gains per unit sold. In our problem, this means considering how much profit is made per book and per stationery item.

• For Books: Each book sold yields a profit of Rs. 2. Thus, selling \(x\) books gives a profit of \(2x\).
• For Stationery: Similarly, each unit of stationery sold generates Rs. 3 in profit. Thus, selling \(y\) items gives a profit of \(3y\).

The overall profit is the sum of these two, resulting in the profit function \(p = 2x + 3y\). By calculating in this way, the shopkeeper can forecast the financial outcome based on projected sales.

The Role of Mathematical Modeling

Mathematical modeling is a method used to translate a real-world scenario into a mathematical format, making complex problems easier to analyze. For the shopkeeper's problem, we employ mathematical modeling to articulate the relation between sales and profit.

Creating the Model

• The variables \(x\) and \(y\) signify real-world components, specifically the number of books and stationery sold.
• Profits per unit (Rs 2 for books and Rs 3 for stationery) are essential in constructing the model as coefficients.

Analyzing and Solving

The model enables the user to compute total potential profit expediently. By adjusting the values of \(x\) and \(y\), the shopkeeper can project different profit scenarios. This process aids in making informed business decisions.
Mathematical modeling thus serves as a powerful tool, linking everyday business activities with strategic financial planning.