José Domingo's Pen Purchase A Mathematical Solution

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In the world of mathematics, word problems often present intriguing scenarios that require careful analysis and logical deduction. This article delves into one such problem, a scenario involving José Domingo's first business venture and his subsequent purchase of pens. We will break down the problem step-by-step, employing mathematical principles to arrive at the solution. This kind of mathematical problem-solving is crucial for developing critical thinking skills and applying mathematical concepts in real-world situations. Understanding how to approach and solve these problems not only enhances our mathematical proficiency but also cultivates our ability to analyze information and make informed decisions.

Problem Statement

José Domingo, after a successful first business endeavor, earned 120 soles. With this money, he decided to purchase 20 pens. Some of these pens cost 9 soles each, while the others cost 3 soles each. The question we aim to answer is: How many pens costing 9 soles did José Domingo buy?

This problem presents a classic example of a system of equations, where we have two unknowns (the number of pens costing 9 soles and the number of pens costing 3 soles) and two pieces of information (the total number of pens and the total amount spent). To solve this, we will use algebraic techniques, setting up equations that represent the given information and then solving them simultaneously. This approach demonstrates the power of algebra in representing and solving real-world problems, highlighting the importance of algebraic thinking in mathematical reasoning and problem-solving strategies. The ability to translate word problems into mathematical equations is a fundamental skill that forms the basis for more advanced mathematical concepts and applications.

Setting Up the Equations

Let's define our variables:

  • Let x be the number of pens that cost 9 soles each.
  • Let y be the number of pens that cost 3 soles each.

We can form two equations based on the information given:

  1. Equation 1 (Total number of pens): x + y = 20
    This equation represents the fact that José Domingo bought a total of 20 pens. The sum of the number of 9-sole pens (x) and the number of 3-sole pens (y) must equal 20. This equation captures the constraint on the total quantity of pens purchased, a crucial piece of information for solving the problem. It illustrates how mathematical notation can concisely represent real-world constraints and relationships, a fundamental aspect of mathematical modeling. The equation also reinforces the idea of variable representation, where we use symbols to represent unknown quantities, allowing us to manipulate and solve for them using algebraic techniques.
  2. Equation 2 (Total cost of pens): 9x + 3y = 120 This equation represents the total amount José Domingo spent on the pens. The cost of the 9-sole pens (9x) plus the cost of the 3-sole pens (3y) must equal the total amount he spent, which is 120 soles. This equation incorporates the cost of each type of pen and the total expenditure, providing another critical constraint for the problem. It highlights the importance of linear equations in modeling real-world scenarios where quantities and costs are related linearly. The ability to formulate such equations is essential for mathematical problem-solving, as it allows us to translate verbal information into a symbolic form that can be analyzed and solved systematically.

Solving the System of Equations

Now we have a system of two equations with two variables:

  • x + y = 20
  • 9x + 3y = 120

There are several methods to solve such a system, including substitution, elimination, and graphing. We will use the substitution method here. First, we can solve the first equation for y:

  • y = 20 - x

Now, substitute this expression for y into the second equation:

  • 9x + 3(20 - x) = 120

Simplify and solve for x:

  • 9x + 60 - 3x = 120
  • 6x + 60 = 120
  • 6x = 60
  • x = 10

So, José Domingo bought 10 pens that cost 9 soles each. To find the number of pens that cost 3 soles each, we can substitute x = 10 back into the equation y = 20 - x:

  • y = 20 - 10
  • y = 10

Therefore, José Domingo also bought 10 pens that cost 3 soles each.

This process demonstrates the power of the substitution method in solving systems of equations. By expressing one variable in terms of the other, we can reduce the problem to a single equation with one variable, which is easier to solve. This technique is widely used in algebra and has applications in various fields, including physics, engineering, and economics. The ability to solve simultaneous equations is a fundamental skill in mathematics, as it allows us to tackle problems involving multiple unknowns and constraints. The systematic approach used here, involving isolating variables and substituting expressions, exemplifies the importance of logical reasoning and step-by-step problem-solving in mathematical contexts.

Verifying the Solution

To ensure our solution is correct, we can verify it by plugging the values of x and y back into the original equations:

  • Equation 1: 10 + 10 = 20 (Correct)
  • Equation 2: 9(10) + 3(10) = 90 + 30 = 120 (Correct)

Since both equations hold true, our solution is correct. José Domingo bought 10 pens that cost 9 soles each and 10 pens that cost 3 soles each. This step of verification is crucial in mathematical problem-solving. It ensures that the solution obtained satisfies all the conditions and constraints of the problem, preventing errors and promoting accuracy. Verifying solutions reinforces the importance of critical thinking and attention to detail in mathematics, as it requires us to revisit the original problem and confirm that our answer makes sense in the given context. This process also highlights the interconnectedness of mathematical concepts, as we use the values obtained from solving the equations to validate their correctness within the broader problem framework.

Answer

José Domingo bought 10 pens that cost 9 soles each.

This final answer provides a concrete resolution to the initial problem statement. It represents the culmination of our mathematical analysis and problem-solving efforts. The ability to arrive at a clear and concise answer is a key objective in mathematics, as it demonstrates a complete understanding of the problem and the application of appropriate techniques. This answer is not just a numerical value; it is the solution to a real-world scenario, highlighting the practical relevance of mathematics. The process of translating a word problem into mathematical equations, solving those equations, and interpreting the results in the context of the original problem is a fundamental skill that extends beyond the classroom, empowering us to tackle a wide range of challenges in various aspects of life.

Conclusion

This problem demonstrates the application of basic algebraic principles to solve a real-world scenario. By setting up a system of equations, we were able to determine the number of pens José Domingo bought at each price point. This exercise highlights the importance of mathematical literacy and problem-solving skills in everyday situations. The ability to translate word problems into mathematical models, solve those models, and interpret the results is a valuable skill that empowers individuals to make informed decisions and navigate complex situations. This particular problem, while seemingly simple, illustrates the power of algebra in representing and solving real-world challenges. The concepts and techniques used here, such as variable representation, equation formulation, and simultaneous equation solving, are foundational elements of mathematical thinking and have broad applications across various disciplines. Mastering these skills not only enhances our mathematical proficiency but also cultivates our ability to analyze information, think critically, and solve problems effectively.