Solving For Pedro's Newspaper Delivery Earnings A Mathematical Exploration
Solve for the additional amount earned per newspaper (*x*) given Pedro's earnings for delivering 5 and 20 newspapers.
Introduction
In this article, we will delve into a mathematical problem involving Pedro, a diligent newspaper deliverer. Pedro's earnings are determined by a base pay plus an additional amount for each newspaper he successfully delivers. We are presented with two scenarios: last week, Pedro delivered 5 newspapers and earned $37.50, and this week, he delivered 20 newspapers and earned $75. Our objective is to unravel the underlying mathematical relationship between the number of newspapers delivered and Pedro's total earnings. By employing algebraic techniques, we will formulate equations, solve for unknown variables, and gain a comprehensive understanding of Pedro's compensation structure. This exploration will not only enhance our mathematical problem-solving skills but also provide insights into real-world scenarios where linear relationships play a crucial role.
Problem Statement
The core of our mathematical problem lies in deciphering Pedro's earnings. We know that his pay consists of two components: a fixed base pay and a variable amount that depends on the number of newspapers he delivers. To represent this mathematically, let's use 'x' to denote the additional amount Pedro earns per newspaper delivered. Our goal is to determine the values of both the base pay and the per-newspaper delivery amount (x). The information provided gives us two distinct data points: (1) 5 newspapers delivered result in $37.50 earnings, and (2) 20 newspapers delivered lead to $75 earnings. These data points will serve as the foundation for constructing a system of equations, which we will then solve to find the unknown values. By carefully analyzing these equations, we can unlock the secrets of Pedro's compensation structure and gain a deeper appreciation for the power of mathematical modeling in real-life situations. This mathematical problem is a practical application of linear equations, a fundamental concept in algebra.
Setting up the Equations
To solve this mathematical problem, we need to translate the given information into mathematical equations. Let's represent Pedro's base pay as 'b'. From the problem statement, we know that Pedro's total earnings are the sum of his base pay and the amount he earns from delivering newspapers. We can express this relationship as:
Total Earnings = Base Pay + (Amount per Newspaper * Number of Newspapers)
Using this general equation, we can formulate two specific equations based on the information provided:
For last week:
$37.50 = b + 5x
For this week:
$75 = b + 20x
We now have a system of two linear equations with two unknowns (b and x). This system can be solved using various methods, such as substitution, elimination, or matrix methods. The key is to manipulate these equations in a way that allows us to isolate one variable and then solve for the other. This process is a cornerstone of algebraic problem-solving and will enable us to determine Pedro's base pay and the amount he earns per newspaper. This step in the mathematical problem is crucial for understanding how to translate word problems into a solvable mathematical framework.
Solving the Equations
With our equations set up, the next step in this mathematical problem is to solve for the unknowns 'b' (base pay) and 'x' (amount per newspaper). We have two equations:
- $37.50 = b + 5x
- $75 = b + 20x
Let's use the elimination method to solve this system. We can subtract equation 1 from equation 2 to eliminate 'b':
($75 - $37.50) = (b + 20x) - (b + 5x)
$37.50 = 15x
Now, we can solve for 'x' by dividing both sides by 15:
x = $37.50 / 15
x = $2.50
Now that we have the value of 'x', we can substitute it back into either equation 1 or equation 2 to solve for 'b'. Let's use equation 1:
$37.50 = b + 5($2.50)
$37.50 = b + $12.50
Subtract $12.50 from both sides to isolate 'b':
b = $37.50 - $12.50
b = $25
Therefore, Pedro's base pay (b) is $25, and he earns $2.50 per newspaper delivered (x). This solution demonstrates the power of algebraic techniques in solving real-world mathematical problems.
Interpretation of the Solution
Having solved the equations, we've arrived at a crucial juncture in this mathematical problem: interpreting the solution within the context of the original problem. We found that Pedro's base pay (b) is $25, and he earns $2.50 for each newspaper delivered (x). This means that regardless of how many newspapers Pedro delivers, he receives a guaranteed $25. On top of that, for every newspaper he successfully delivers, he earns an additional $2.50. This breakdown of Pedro's earnings provides a clear understanding of his compensation structure.
Let's consider a few scenarios to further illustrate this: If Pedro delivers zero newspapers, he still earns his base pay of $25. If he delivers 10 newspapers, he earns $25 (base pay) + 10 * $2.50 (delivery pay) = $50. This linear relationship between the number of newspapers delivered and his total earnings highlights the practical application of linear equations in everyday situations. By understanding this relationship, Pedro can predict his earnings based on the number of newspapers he delivers, and we can appreciate how mathematical problem-solving skills are valuable in understanding real-world financial scenarios.
Verification of the Solution
In any mathematical problem, it's essential to verify the solution to ensure its accuracy and validity. We found that Pedro's base pay is $25 and he earns $2.50 per newspaper delivered. To verify our solution, we can substitute these values back into the original equations and check if they hold true.
Recall our equations:
- $37.50 = b + 5x
- $75 = b + 20x
Substituting b = $25 and x = $2.50 into equation 1:
$37.50 = $25 + 5($2.50)
$37.50 = $25 + $12.50
$37.50 = $37.50 (True)
Substituting b = $25 and x = $2.50 into equation 2:
$75 = $25 + 20($2.50)
$75 = $25 + $50
$75 = $75 (True)
Since our values satisfy both equations, we can confidently conclude that our solution is correct. This verification step reinforces the importance of rigorousness in mathematical problem-solving and ensures that our conclusions are well-founded.
Real-World Applications
This mathematical problem, though seemingly simple, exemplifies a common real-world scenario: linear compensation models. Many jobs, such as sales positions, delivery services, and freelance work, involve a base pay plus a commission or per-unit payment. Understanding linear equations allows individuals to predict their earnings, set financial goals, and make informed decisions about their work. Furthermore, businesses use linear models to calculate employee compensation, forecast expenses, and analyze profitability.
The principles we've applied in solving Pedro's earnings can be extended to various other situations. For instance, consider a taxi fare that consists of a base charge plus a per-mile rate. Or, think about the cost of renting a car, which might include a daily fee plus a per-mile charge. These scenarios, like Pedro's earnings, can be modeled using linear equations, making the concepts we've explored highly relevant and applicable. This highlights the broader significance of mathematical problem-solving skills in navigating the complexities of the modern world.
Conclusion
In conclusion, we have successfully dissected a mathematical problem involving Pedro's newspaper delivery earnings. By translating the word problem into a system of linear equations, we were able to solve for the unknowns and determine Pedro's base pay and the amount he earns per newspaper delivered. We then interpreted our solution within the context of the problem, verified its accuracy, and explored its real-world applications. This exercise demonstrates the power of algebraic techniques in modeling and solving practical problems.
Furthermore, this exploration underscores the importance of mathematical problem-solving skills in various aspects of life, from personal finance to business management. The ability to analyze situations, formulate equations, and interpret solutions is a valuable asset in today's world. By mastering these skills, we can gain a deeper understanding of the world around us and make more informed decisions. The case of Pedro's earnings serves as a compelling example of how mathematics can be used to illuminate real-world scenarios and empower us to solve problems effectively.
Keywords
Solve for x given the following scenario: Pedro works as a newspaper deliverer. He receives a base pay plus an additional amount per newspaper delivered. Last week, Pedro delivered 5 newspapers and earned $37.50. This week, he delivered 20 newspapers and earned $75. Let x equal what?