Solving Consecutive Number Problems Finding The Right Equation

At the heart of many mathematical problems lies the challenge of translating words into equations. This is particularly true when dealing with word problems that involve consecutive numbers. These problems often require a blend of algebraic manipulation and logical reasoning to arrive at the correct solution. Today, we will delve into a specific problem of this nature, dissecting it step by step to not only find the answer but also understand the underlying mathematical principles. Our focus will be on a problem that involves the product of two consecutive numbers, where half of this product is given as 105. The goal is to identify the equation that can be used to solve for n, the smaller of these two numbers. This is a common type of problem in algebra, frequently encountered in middle school and high school mathematics curricula, and mastering it provides a solid foundation for more advanced mathematical concepts. By understanding how to approach such problems, students can improve their problem-solving skills and their ability to translate real-world scenarios into mathematical models.

Breaking Down the Problem The Essence of Consecutive Numbers

To tackle this problem effectively, let’s first break it down into its core components. The key concept here is that of consecutive numbers. Consecutive numbers are integers that follow each other in order, each differing from the previous one by 1. For example, 5 and 6 are consecutive numbers, as are -3 and -2. In our problem, we have two such numbers, and we’re calling the smaller one n. This means the next number in the sequence, the larger number, will be n + 1. Understanding this relationship is crucial because it allows us to express both numbers in terms of a single variable, n, which simplifies the process of forming an equation. The problem states that half of the product of these two consecutive numbers is 105. This is a critical piece of information that we will use to build our equation. The term “product” in mathematics refers to the result of multiplying two or more numbers. In this case, we need to multiply n and n + 1. The phrase “half of the product” tells us that after multiplying the two numbers, we need to divide the result by 2. Finally, the problem states that this result is equal to 105. By carefully dissecting the wording of the problem, we can translate it into a mathematical statement that forms the basis for our equation. This step-by-step approach is essential for solving any word problem, as it ensures that we understand all the components and their relationships before attempting to construct an equation. It’s not just about finding the right answer; it’s about developing a systematic approach to problem-solving that can be applied to a wide range of scenarios.

Forming the Equation From Words to Symbols

Now that we’ve dissected the problem and identified the key components, let’s translate it into a mathematical equation. This is a crucial step, as the equation will be the tool we use to solve for n. We know that the two consecutive numbers are n and n + 1. Their product is therefore n * ( n + 1). The problem states that half of this product is 105. Mathematically, “half of” implies division by 2, so we have (n * (n + 1)) / 2. Finally, the problem tells us that this expression is equal to 105. Putting it all together, we get the equation: (n * (n + 1)) / 2 = 105. This equation represents the core of our problem. It’s a concise mathematical statement that captures all the information given in the word problem. However, to solve for n, we need to manipulate this equation into a more standard form, typically a quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. Quadratic equations have a general form of ax² + bx + c = 0, where a, b, and c are constants. To transform our equation into this form, we need to eliminate the fraction and expand the product. Multiplying both sides of the equation by 2, we get n * (n + 1) = 210. Next, we expand the left side of the equation by distributing n: n² + n = 210. Finally, to get the equation into the standard quadratic form, we subtract 210 from both sides: n² + n - 210 = 0. This is the equation we will use to solve for n, the smaller of the two consecutive numbers. The process of translating a word problem into an equation is a fundamental skill in algebra, and it requires careful attention to detail and a solid understanding of mathematical notation.

Analyzing the Options Finding the Right Fit

With the equation n² + n - 210 = 0 in hand, we can now turn our attention to the given options and see which one matches our derived equation. This is a critical step in solving multiple-choice problems, as it allows us to verify our work and ensure that we haven’t made any errors in our calculations. The options provided are:

• n² + n - 210 = 0
• n² + n - 105 = 0
• 2n² + 2n + 210 = 0
• 2n² + 2n + 105 = 0

By comparing each option to our derived equation, it becomes clear that the first option, n² + n - 210 = 0, is an exact match. This confirms that our algebraic manipulations were correct and that we have successfully translated the word problem into the appropriate mathematical equation. The other options can be eliminated because they do not accurately represent the relationship described in the problem. The second option, n² + n - 105 = 0, is incorrect because it doesn’t account for the multiplication by 2 when we removed the fraction in our original equation. The third and fourth options, 2n² + 2n + 210 = 0 and 2n² + 2n + 105 = 0, respectively, are also incorrect. These options seem to have multiplied the entire equation by 2 at some point, but they also have the wrong sign for the constant term. In the third option the sign is wrong, and in the fourth option both the sign and the constant are incorrect. Analyzing the options carefully and comparing them to our derived equation is an essential part of the problem-solving process. It not only helps us identify the correct answer but also reinforces our understanding of the mathematical concepts involved.

Solving the Quadratic Equation (Optional) Deeper into the Problem

While the original question asks only for the equation, let’s take it a step further and solve the quadratic equation we derived: n² + n - 210 = 0. This will give us the actual values of n that satisfy the problem's conditions. Solving a quadratic equation involves finding the roots, or the values of the variable that make the equation true. There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is a straightforward approach. Factoring involves expressing the quadratic equation as a product of two binomials. We need to find two numbers that multiply to -210 and add up to 1 (the coefficient of the n term). These numbers are 15 and -14. Therefore, we can factor the quadratic equation as: (n + 15) * (n - 14) = 0. For this product to be zero, at least one of the factors must be zero. This gives us two possible solutions: n + 15 = 0 or n - 14 = 0. Solving these equations for n, we get n = -15 or n = 14. These are the two possible values for the smaller of the two consecutive numbers. If n = -15, then the next consecutive number is -14. If n = 14, then the next consecutive number is 15. In both cases, half of the product of the two consecutive numbers is indeed 105: ((-15) * (-14)) / 2 = 105 and (14 * 15) / 2 = 105. This demonstrates that both solutions are valid in the context of the original problem. Solving the quadratic equation provides a deeper understanding of the problem and reinforces our algebraic skills. It also highlights the fact that quadratic equations can have multiple solutions, and it's important to consider all possibilities when interpreting the results.

Conclusion Mastering Mathematical Problem-Solving

In conclusion, the equation that can be used to solve for n, the smaller of the two consecutive numbers, when half of their product is 105, is n² + n - 210 = 0. This problem demonstrates the importance of carefully translating word problems into mathematical equations. By breaking down the problem into its core components, understanding the concept of consecutive numbers, and applying algebraic manipulation techniques, we were able to derive the correct equation. Furthermore, we explored the process of solving the quadratic equation, which provided a deeper understanding of the problem and reinforced our algebraic skills. Mastering mathematical problem-solving is a crucial skill that extends far beyond the classroom. It enhances our ability to think critically, analyze information, and solve complex problems in various aspects of life. By practicing and refining these skills, we can become more confident and effective problem-solvers in all areas.

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