Finding Three Consecutive Even Numbers Summing To 144 A Step By Step Solution

Introduction: The Puzzle of Consecutive Even Numbers

In the realm of mathematics, consecutive even numbers present a fascinating topic, offering a blend of simplicity and complexity. These numbers, following each other in sequence with a difference of 2, hold a unique charm. In this article, we delve into a specific problem: finding three consecutive even whole numbers that sum up to 144. This exploration will not only solve the problem at hand but also illuminate the underlying principles of algebra and number patterns. We'll embark on a journey of mathematical discovery, unraveling the mystery behind these numbers and understanding the elegance of mathematical solutions. Let’s dive into the world of equations and numbers, where we aim to demystify the process of finding these elusive consecutive even numbers. We will dissect the problem, understand the nature of even numbers, and apply algebraic techniques to arrive at a solution. This exercise is not just about finding an answer; it’s about appreciating the beauty and logic of mathematics in action. Through this exploration, we hope to enhance your problem-solving skills and ignite a passion for mathematical challenges.

Understanding Consecutive Even Numbers: Setting the Stage

Before we jump into solving the equation, it’s crucial to understand what consecutive even numbers are. Even numbers are integers that are exactly divisible by 2, such as 2, 4, 6, and so on. Consecutive even numbers are a sequence of even numbers that follow each other in order, each differing by 2. For example, 10, 12, and 14 are consecutive even numbers. The concept of consecutiveness in even numbers is essential to grasp, as it forms the foundation for our algebraic representation. In our specific problem, we're looking for three such numbers. Let’s denote the first even number as 'x'. Since the next even number will be 2 greater than the previous one, we can represent the second number as 'x + 2', and the third as 'x + 4'. This algebraic representation is key to translating the word problem into a solvable equation. By understanding this sequential relationship, we can effectively model the problem mathematically. This approach not only simplifies the problem but also allows us to apply algebraic principles to find a solution. The beauty of algebra lies in its ability to represent abstract concepts with symbols and equations, making complex problems more manageable. In the next section, we will use these representations to form an equation that captures the essence of the problem.

Formulating the Equation: The Heart of the Solution

Now that we understand what consecutive even numbers are and how to represent them algebraically, we can proceed to formulate the equation. The problem states that the sum of three consecutive even numbers is 144. Using our representations, the first number is 'x', the second is 'x + 2', and the third is 'x + 4'. The sum of these three numbers can be written as x + (x + 2) + (x + 4). According to the problem, this sum equals 144. Therefore, our equation is: x + (x + 2) + (x + 4) = 144. This equation is the heart of our solution. It encapsulates the problem in a concise and manageable form. The next step is to simplify and solve this equation. By combining like terms and isolating the variable, we can find the value of 'x', which represents our first even number. The ability to translate a word problem into an algebraic equation is a fundamental skill in mathematics. It requires a clear understanding of the problem's conditions and the ability to represent these conditions using mathematical symbols. Our equation now provides a roadmap for solving the problem. We'll move forward by simplifying the equation and solving for 'x', which will lead us to the three consecutive even numbers we're seeking.

Solving the Equation: Unraveling the Mystery

With the equation x + (x + 2) + (x + 4) = 144, we are now poised to solve for 'x'. The first step is to simplify the equation by combining like terms. Adding the 'x' terms together, we have 3x. Adding the constants 2 and 4, we get 6. Thus, the simplified equation is 3x + 6 = 144. The next step is to isolate the term with 'x'. To do this, we subtract 6 from both sides of the equation, which gives us 3x = 138. Finally, to solve for 'x', we divide both sides of the equation by 3. This yields x = 46. This value of 'x' is the first of our three consecutive even numbers. Now that we have found 'x', we can easily find the other two numbers by adding 2 and 4 to 'x', respectively. This process of solving the equation demonstrates the power of algebra in simplifying complex problems. By following a systematic approach of simplifying, isolating the variable, and performing inverse operations, we can unravel the mystery and find the solution. The value of 'x' we've found is a crucial stepping stone, leading us to the final answer. In the next section, we will use this value to determine the three consecutive even numbers that satisfy the conditions of the problem.

Finding the Three Numbers: Completing the Puzzle

Now that we have determined that x = 46, we can find the three consecutive even numbers. The first number is x, which is 46. The second number is x + 2, which is 46 + 2 = 48. The third number is x + 4, which is 46 + 4 = 50. Therefore, the three consecutive even numbers are 46, 48, and 50. To verify our solution, we can add these numbers together: 46 + 48 + 50 = 144. This confirms that our solution is correct and satisfies the condition given in the problem. Finding these numbers completes our puzzle. We started with a word problem, translated it into an algebraic equation, solved the equation, and finally, identified the three consecutive even numbers. This process highlights the interconnectedness of different mathematical concepts and the power of a systematic approach to problem-solving. Our journey through this problem has not only provided us with an answer but also reinforced our understanding of algebraic principles and problem-solving strategies. In the next section, we will summarize our findings and reflect on the process we undertook.

Conclusion: Reflecting on the Mathematical Journey

In conclusion, we successfully solved the problem of finding three consecutive even numbers that sum up to 144. The numbers we found are 46, 48, and 50. This problem-solving journey has been a testament to the power and elegance of mathematics. We began by understanding the concept of consecutive even numbers and representing them algebraically. We then formulated an equation that captured the essence of the problem, solved the equation using algebraic techniques, and finally, identified the numbers that satisfied the given conditions. This exercise underscores the importance of a systematic approach to problem-solving. By breaking down a complex problem into smaller, manageable steps, we can effectively navigate through the solution. Furthermore, this exploration has highlighted the beauty of algebra in translating real-world scenarios into mathematical models. The ability to represent problems algebraically is a fundamental skill that empowers us to tackle a wide range of mathematical challenges. As we conclude this exploration, we hope that you have gained not only a solution to this specific problem but also a deeper appreciation for the power and beauty of mathematics. The skills and strategies we've employed here can be applied to a variety of mathematical problems, making you a more confident and capable problem solver.