Solving Consecutive Number Problems Finding The Equation For N

In the realm of mathematics, problem-solving often involves translating word problems into algebraic equations. This article delves into a specific problem concerning consecutive numbers and their product. We aim to dissect the problem, understand the underlying concepts, and arrive at the correct equation to solve for the unknown. The problem states: "Half of the product of two consecutive numbers is 105. Which equation can be used to solve for n, the smaller of the two numbers?" This seemingly simple question opens up a fascinating journey into the world of quadratic equations and algebraic manipulation. Let's embark on this journey together, unraveling the complexities and arriving at a clear, concise solution.

Understanding Consecutive Numbers

To begin, let's define what we mean by "consecutive numbers." Consecutive numbers are integers that follow each other in order, each differing from the previous number by 1. For example, 5 and 6 are consecutive numbers, as are -3 and -2. In algebraic terms, if we represent the smaller number as n, the next consecutive number would be n + 1. This fundamental understanding is crucial for translating the word problem into a mathematical equation. The beauty of mathematics lies in its ability to represent real-world scenarios using abstract symbols and equations. By grasping the concept of consecutive numbers, we've taken the first step towards solving our problem. It's like having the right key to unlock a door; now, we need to find the right lock.

Translating the Word Problem into an Equation

Now that we understand consecutive numbers, let's tackle the core of the problem: "Half of the product of two consecutive numbers is 105." This sentence is packed with information, and our task is to unpack it and express it mathematically. First, let's identify the key components. We have two consecutive numbers, which we know can be represented as n and n + 1. The word "product" implies multiplication, so we need to multiply these two expressions: n(n + 1). Next, we encounter the phrase "half of the product," which means we need to divide the product by 2: [n(n + 1)] / 2. Finally, we're told that this result is equal to 105. Putting it all together, we get the equation: [n(n + 1)] / 2 = 105. This equation is the bridge between the word problem and the algebraic solution. It's like having a map that guides us from the starting point to the destination. Now, we need to simplify this equation to match the given options.

Simplifying the Equation

The equation we've derived, [n(n + 1)] / 2 = 105, is a good starting point, but it's not in the standard form of a quadratic equation. To simplify it, we need to get rid of the fraction and rearrange the terms. The first step is to multiply both sides of the equation by 2. This eliminates the denominator and gives us: n(n + 1) = 210. Next, we need to expand the left side of the equation by distributing n: n² + n = 210. Now, we have a quadratic expression, but it's not yet in the standard form, which is ax² + bx + c = 0. To achieve this, we need to subtract 210 from both sides of the equation: n² + n - 210 = 0. This is the simplified form of the equation, and it matches one of the options provided in the problem. The process of simplification is like refining a rough diamond; we take the raw equation and polish it until it shines in its clearest form.

Identifying the Correct Equation

After simplifying the equation, we arrived at n² + n - 210 = 0. Now, we need to compare this equation with the given options and identify the correct one. The options provided were:

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

By direct comparison, it's clear that the first option, n² + n - 210 = 0, matches our simplified equation. This is the equation that can be used to solve for n, the smaller of the two consecutive numbers. The other options either have incorrect constants or have been multiplied by 2 without adjusting the constant term. Identifying the correct equation is like finding the missing piece of a puzzle; it completes the solution and brings clarity to the problem.

Why Other Options Are Incorrect

To solidify our understanding, let's briefly discuss why the other options are incorrect. The second option, n² + n - 105 = 0, is incorrect because it doesn't account for the multiplication by 2 when we eliminated the denominator in the original equation. The third option, 2n² + 2n + 210 = 0, has the correct constant term (210), but the sign is incorrect, and the entire equation has been multiplied by 2. The fourth option, 2n² + 2n + 105 = 0, suffers from both the sign error and the incorrect constant term. Understanding why the incorrect options are wrong is just as important as knowing why the correct option is right. It deepens our grasp of the concepts and helps us avoid similar errors in the future.

Solving the Quadratic Equation

While the problem only asked for the equation, let's take it a step further and solve the quadratic equation n² + n - 210 = 0. To do this, we can use factoring, the quadratic formula, or completing the square. In this case, factoring is the most straightforward approach. We need to find two numbers that multiply to -210 and add up to 1. These numbers are 15 and -14. Therefore, we can factor the equation as (n + 15)(n - 14) = 0. Setting each factor equal to zero, we get two possible solutions: n = -15 and n = 14. Since the problem asks for the smaller of the two numbers, both solutions are valid in the mathematical sense. However, if the problem specified positive integers, we would choose n = 14. Solving the equation is like adding the final brushstrokes to a painting; it completes the picture and gives us a full understanding of the solution.

Practical Applications and Importance

Understanding how to translate word problems into algebraic equations is a fundamental skill in mathematics. It has practical applications in various fields, including physics, engineering, economics, and computer science. For instance, engineers might use quadratic equations to model the trajectory of a projectile, while economists might use them to analyze supply and demand curves. The ability to solve these types of problems is not just an academic exercise; it's a valuable tool for problem-solving in the real world. The importance of this skill cannot be overstated; it's like having a universal translator that allows us to decipher the language of the universe.

Conclusion

In conclusion, we've successfully dissected the problem of finding the equation to solve for two consecutive numbers whose product, when halved, equals 105. We identified the correct equation as n² + n - 210 = 0. Along the way, we explored the concept of consecutive numbers, translated the word problem into an algebraic equation, simplified the equation, and discussed why the other options were incorrect. We even went a step further and solved the quadratic equation to find the possible values of n. This journey through the world of algebra has not only provided us with a solution to a specific problem but has also reinforced the importance of mathematical reasoning and problem-solving skills. The solution is not just an answer; it's a testament to the power of mathematical thinking.

Solving Consecutive Number Problems Finding the Equation for n

• Solving for consecutive numbers
• Half of the product of two consecutive numbers is 105
• Which equation can be used to solve for $n$, the smaller of the two numbers?