Equation For Consecutive Numbers With Half Product Of 105

Hey guys! Today, we're diving into a fascinating mathematical puzzle. We're tasked with figuring out which equation helps us solve for n, the smaller of two consecutive numbers, given that half their product equals 105. Let's break this down step-by-step, making sure we understand the core concepts and how to translate word problems into algebraic equations. Math can be a wild ride, but with a clear roadmap, we can conquer any challenge!

Understanding the Problem: Consecutive Numbers and Their Product

First, let's clarify what consecutive numbers are. These are numbers that follow each other in order, each differing from the previous one by 1. Think of it like this: 5 and 6, 12 and 13, or even -3 and -2. They're a sequence of numbers marching right along. In our problem, we have two consecutive numbers, and we're calling the smaller one n. That means the next number in the sequence, the larger one, is simply n + 1. Got it?

Now, we need to consider the product of these numbers. In mathematics, the product refers to the result of multiplying two or more numbers together. So, the product of our consecutive numbers n and (n + 1) is n multiplied by (n + 1), which we can write as n(n + 1). This is a crucial piece of the puzzle, as it represents the total result before we apply the "half" condition.

But here's the twist: the problem states that half of this product equals 105. This means we need to take our product, n(n + 1), and divide it by 2. Mathematically, we represent this as [n(n + 1)] / 2. The key is to translate these words into mathematical expressions accurately. So far, so good, right? We're building our equation brick by brick.

Finally, we know that this expression, half the product of our consecutive numbers, is equal to 105. This gives us the foundation for our equation. We can now write: [n(n + 1)] / 2 = 105. This is the bridge between the word problem and the algebraic world. We're almost there!

Building the Equation: From Words to Algebra

Okay, guys, we've translated the problem into a basic equation: [n(n + 1)] / 2 = 105. But to match the answer choices provided, we need to do some algebraic maneuvering. This involves simplifying and rearranging the equation to get it into a recognizable form. Think of it like transforming raw ingredients into a delicious dish – we're using our algebraic skills as the recipe.

The first step in simplifying this equation is to get rid of the fraction. We can do this by multiplying both sides of the equation by 2. This is a fundamental rule of algebra: what you do to one side, you must do to the other to maintain the balance. So, multiplying both sides by 2 gives us: n(n + 1) = 210. See how we're clearing away the clutter and getting closer to our goal?

Next, we need to expand the left side of the equation. This means distributing the n across the parentheses. Remember the distributive property? It's a cornerstone of algebra! n multiplied by n is n², and n multiplied by 1 is simply n. So, our equation now becomes: n² + n = 210. We're transforming the equation step by step, like a mathematical makeover.

Now, to match the answer choices, we need to get all the terms on one side of the equation, leaving zero on the other side. This is a common strategy when dealing with quadratic equations (equations with a term like n²). To do this, we subtract 210 from both sides of the equation. This gives us: n² + n - 210 = 0. Bingo! We've arrived at one of the answer choices.

This equation, n² + n - 210 = 0, is a quadratic equation. It's in the standard form of a quadratic equation, which is ax² + bx + c = 0, where a, b, and c are constants. In our case, a = 1, b = 1, and c = -210. Recognizing this form is crucial because it allows us to use various methods to solve for n, such as factoring or the quadratic formula. But for this problem, we only needed to find the correct equation, and we've nailed it!

Comparing with the Answer Choices: Identifying the Correct Equation

Alright, we've successfully transformed the word problem into the equation n² + n - 210 = 0. Now, let's put on our detective hats and compare this equation with the answer choices provided. This is where we confirm our work and ensure we've landed on the right solution. It's like checking the map to make sure we're on the correct path.

The first answer choice is n² + n - 210 = 0. This is exactly the equation we derived! Hooray! This confirms that we've correctly translated the problem and manipulated the equation. It's a satisfying moment when all the pieces fall into place.

Let's quickly glance at the other answer choices to understand why they're incorrect. The second choice is n² + n - 105 = 0. This equation is similar, but it incorrectly uses 105 instead of 210. Remember, we multiplied both sides of the equation by 2 to eliminate the fraction, so 105 should have become 210. This highlights the importance of careful step-by-step manipulation.

The third choice is 2n² + 2n + 210 = 0. This equation seems to have multiplied the entire equation by 2 at some point, but it also changed the sign of the constant term. This is a common mistake – it's crucial to perform operations consistently on both sides of the equation and maintain the correct signs.

The fourth choice is 2n² + 2n + 105 = 0. This equation has a similar issue to the third choice, but it uses 105 instead of 210. It seems to have missed the step of multiplying both sides of the equation by 2 to eliminate the fraction. These incorrect choices serve as valuable reminders of the common pitfalls in algebraic manipulations.

By comparing our derived equation with the answer choices, we can confidently identify n² + n - 210 = 0 as the correct equation. We've not only solved the problem but also understood why the other options are incorrect. This deep understanding is what truly makes us mathematical masters!

Conclusion: Mastering the Art of Equation Building

So, there you have it, guys! We've successfully navigated the challenge of finding the equation to solve for n, the smaller of two consecutive numbers whose half-product is 105. We started by understanding the core concepts of consecutive numbers and products. Then, we meticulously translated the word problem into an algebraic equation, step by step. We simplified the equation, compared it with the answer choices, and emerged victorious!

This problem highlights the power of algebra in solving real-world puzzles. By translating words into symbols and manipulating equations, we can unlock hidden solutions. It's like having a secret code that allows us to decipher the mysteries of the universe. The key takeaways from this exercise are the importance of:

• Understanding the problem statement and identifying the key information.
• Translating words into mathematical expressions accurately.
• Using algebraic principles to simplify and rearrange equations.
• Comparing the derived equation with the answer choices to confirm the solution.

Remember, practice makes perfect. The more you engage with these kinds of problems, the more confident you'll become in your mathematical abilities. So, keep exploring, keep questioning, and keep building those equation-solving skills! You've got this!