Solve: Pencil & Pen Cost Problem [Ex 3.1], Step-by-Step Guide

Destin Kuphal 22 May 2025

Ever found yourself meticulously calculating the price of individual stationery items, pondering the seemingly simple yet intricate dance of numbers? The quest to determine the cost of a single pencil and pen, given a set of combined prices, is a classic mathematical puzzle that sharpens our analytical skills and reveals the power of linear equations.

Consider this scenario: five pencils and seven pens collectively amount to Rs. 50, while a separate purchase of seven pencils and five pens totals Rs. 46. The challenge lies in dissecting these combined costs to isolate the individual price of each pencil and each pen. This isn't merely an academic exercise; it's a microcosm of real-world problem-solving, demanding precision and a keen understanding of algebraic principles. The problem, often presented as an exercise (Ex 3.1, 1), is a gateway to understanding the practical applications of forming and solving pairs of linear equations, and its a staple in introductory algebra curricula.

Category	Details
Problem Type	Pair of Linear Equations in Two Variables
Given Information	5 pencils + 7 pens = Rs. 50 7 pencils + 5 pens = Rs. 46
Objective	Determine the cost of one pencil and one pen.
Variables	x = cost of one pencil (in Rs.) y = cost of one pen (in Rs.)
Equations	5x + 7y = 50 7x + 5y = 46
Solution Methods	Graphical Method, Substitution Method, Elimination Method
Reference	Example Educational Website

The algebraic formulation is straightforward. If we denote the cost of one pencil as 'x' and the cost of one pen as 'y', we can represent the given information as two linear equations: 5x + 7y = 50 and 7x + 5y = 46. These equations encapsulate the core of the problem, transforming a wordy scenario into a precise mathematical statement. The next step involves choosing a solution method, with graphical representation being one option explicitly mentioned in the initial instructions. However, algebraic methods such as substitution or elimination often provide a more accurate and efficient route to the answer.

Initially, one might misinterpret a similar problem as involving a bulk purchase, perhaps envisioning a pack of five pencils. However, the problem clearly distinguishes between individual pencils and pens, emphasizing the need to calculate the unit price of each item. Moreover, anecdotal experiences with stationery purchases, such as receiving multiple packages of pencils with varying contents (e.g., one package containing four pencils, another containing five), highlight the importance of precise accounting and attention to detail. Such real-world nuances underscore the relevance of this mathematical exercise to everyday financial literacy.

Beyond the core algebraic challenge, the context surrounding such problems can introduce additional layers of complexity. Imagine a scenario where the pencils in question are not uniform in quality or color. Perhaps a batch contains a mix of blue and orange pencils, with varying lead quality. For instance, a collection might include 9 orange and 15 blue pencils, but only a fraction of them possess usable leads. Such variables, while not directly impacting the initial cost calculation, speak to the broader considerations that often accompany real-world purchasing decisions.

The practical implications extend beyond the classroom. Consider a shopkeeper managing their inventory. They might earn a profit of Re. 1 by selling a pen but incur a loss of 40 paise per pencil from an old stock. If, in a particular month, the shopkeeper experiences a net loss of Rs. 5 despite selling 45 pens, the problem becomes one of determining how many pencils were sold. This is a direct application of the same linear equation principles, but framed within the context of business and financial management.

To solve this type of problem, let's revisit the initial scenario: 5 pencils and 7 pens cost Rs. 50, and 7 pencils and 5 pens cost Rs. 46. We established the equations 5x + 7y = 50 and 7x + 5y = 46. The goal is to find the values of 'x' and 'y' that satisfy both equations simultaneously. One common method is elimination. Multiplying the first equation by 7 and the second equation by 5, we obtain: 35x + 49y = 350 and 35x + 25y = 230. Subtracting the second modified equation from the first eliminates 'x', resulting in 24y = 120. Dividing both sides by 24 gives us y = 5. This means the cost of one pen is Rs. 5.

Now, substitute the value of 'y' back into one of the original equations to find 'x'. Using the first equation, 5x + 7(5) = 50, which simplifies to 5x + 35 = 50. Subtracting 35 from both sides gives 5x = 15, and dividing by 5 yields x = 3. Therefore, the cost of one pencil is Rs. 3.

Another scenario involves determining the number of boys and girls who participated in a quiz, given that the number of girls is 4 more than the number of boys. This problem, while distinct in its context, still relies on the same foundational principles of forming and solving linear equations. If we let 'b' represent the number of boys and 'g' represent the number of girls, we can express the given information as g = b + 4. If additional information is provided, such as a relationship between their scores or a total number of participants, a second equation can be formed, allowing us to solve for both 'b' and 'g'.

Consider a slightly different variation of the initial problem. Suppose we're told that the combined cost of 5 pencils and 7 pens is Rs. 195, represented by the equation 5x + 7y = 195. And the combined cost of 7 pencils and 5 pens is Rs. 153, represented by the equation 7x + 5y = 153. To solve this system, we can use the elimination method again. Multiplying the first equation by 7 and the second equation by 5 yields 35x + 49y = 1365 and 35x + 25y = 765. Subtracting the second equation from the first, we get 24y = 600, which means y = 25. Substituting y = 25 back into the first equation, we have 5x + 7(25) = 195, which simplifies to 5x + 175 = 195. Thus, 5x = 20, and x = 4. In this case, the cost of one pencil is Rs. 4, and the cost of one pen is Rs. 25.

It is important to note that the method of solving such problems is not limited to algebraic manipulation. The problem statement often explicitly encourages graphical solutions. This involves plotting the two linear equations on a graph and identifying the point of intersection, which represents the solution to the system. While graphical methods can provide a visual understanding of the problem, they may not always yield precise results, particularly if the solution involves non-integer values. As such, algebraic methods are often preferred for their accuracy and efficiency.

The underlying concept of these exercises is to illustrate the power of mathematical modeling. By translating real-world scenarios into algebraic equations, we can leverage the tools of mathematics to solve problems and gain insights that might not be immediately apparent. Whether it's calculating the cost of stationery items or determining the number of participants in a quiz, the ability to formulate and solve linear equations is a valuable skill that transcends the boundaries of the classroom.

In another scenario, if a man bought 6 pencils for Rs. 5, the cost price of one pencil would be Rs. 5 divided by 6, which is approximately Rs. 0.833. If he then sold 5 pencils for Rs. 6, the selling price of one pencil would be Rs. 6 divided by 5, which is Rs. 1.20. This simple example illustrates how basic arithmetic principles can be used to analyze profitability and make informed business decisions.

Furthermore, consider the scenario where 5 pens and 6 pencils together cost Rs. 9, and 3 pens and 2 pencils cost Rs. 5. Let 'p' be the cost of one pen and 'c' be the cost of one pencil. The equations are 5p + 6c = 9 and 3p + 2c = 5. Multiplying the second equation by 3 gives us 9p + 6c = 15. Subtracting the first equation from this new equation, we get 4p = 6, so p = 1.5. Substituting p = 1.5 into the first equation, we have 5(1.5) + 6c = 9, which simplifies to 7.5 + 6c = 9. Thus, 6c = 1.5, and c = 0.25. In this case, the cost of one pen is Rs. 1.50, and the cost of one pencil is Rs. 0.25.

The recurring theme across these various examples is the application of mathematical principles to solve real-world problems. Whether it's determining the individual prices of pencils and pens, calculating profits and losses, or analyzing the composition of a group, the ability to translate information into equations and solve them is a valuable skill that is applicable across a wide range of disciplines. The problem of finding the cost of one pencil and one pen serves as a fundamental building block for more complex mathematical concepts and real-world applications. It reinforces the idea that mathematics is not just an abstract subject, but a powerful tool for understanding and navigating the world around us.

One might even consider the impact of discounts and promotions on the final cost. For example, a "Doms 100 color erasers jar + 100 c3 pearly pencil combo pack" might be offered at a discounted price, such as Rs. 699 with an 18% discount. Calculating the actual cost per pencil and eraser in this scenario would involve understanding percentages and applying them to the total price. This adds another layer of complexity to the problem, requiring not only algebraic skills but also a practical understanding of financial calculations.

Furthermore, the quality of the pencils and pens can also influence their perceived value. A pencil with smooth, break-resistant lead might be considered more valuable than a pencil with brittle lead. Similarly, a pen with consistent ink flow and a comfortable grip might be preferred over a pen with inconsistent ink and an uncomfortable design. These qualitative factors, while not directly captured in the mathematical equations, can influence the willingness of consumers to pay a certain price for these items.

In conclusion, while the problem of finding the cost of one pencil and one pen may seem simple on the surface, it serves as a gateway to understanding a wide range of mathematical and real-world concepts. From formulating and solving linear equations to analyzing profits and losses, these exercises provide valuable insights into the power of mathematical modeling and its relevance to everyday life. They encourage critical thinking, problem-solving skills, and a deeper appreciation for the role of mathematics in shaping our understanding of the world.