Find The Number: A Challenging Math Equation

Have you ever come across a word problem that seemed to twist your brain into knots? You're not alone! Math word problems can be tricky, especially when they involve fractions and algebraic expressions. Today, we're going to dive deep into a particularly interesting problem that asks us to find a specific number based on a set of conditions. The core of this problem lies in translating a verbal description into a mathematical equation. It's like learning a new language, but instead of speaking to people, you're speaking to numbers! We'll break down the sentence, identify the key phrases, and build the equation step-by-step. This process isn't just about solving a puzzle; it's about strengthening your analytical skills and building a solid foundation in algebra. By the end of this exploration, you'll not only understand how to solve this particular problem but also gain a clearer perspective on how to approach similar algebraic challenges in the future. So, grab your thinking cap, and let's embark on this mathematical adventure together!

Decoding the Word Problem: Turning Words into Math

Let's start by dissecting the problem statement: "The difference between one-half of a number and one-sixth of the number is equal to ten more than one-eighth of that number." Our goal is to find the number. To do this, we first need to represent the unknown number with a variable. Let's use 'n' to represent the number we are trying to find. Now, let's break down each part of the sentence and translate it into algebraic terms. "One-half of a number" can be written as $\frac{1}{2}n$. Similarly, "one-sixth of the number" is $\frac{1}{6}n$. The phrase "the difference between one-half of a number and one-sixth of the number" implies subtraction. So, this part translates to $\frac{1}{2}n - \frac{1}{6}n$. Next, we have "one-eighth of that number," which is $\frac{1}{8}n$. Finally, "ten more than one-eighth of that number" means we add 10 to $\frac{1}{8}n$, resulting in $\frac{1}{8}n + 10$. The word "is equal to" signifies the equals sign (=). Therefore, putting it all together, the equation becomes $\frac{1}{2}n - \frac{1}{6}n = \frac{1}{8}n + 10$. This equation accurately represents the conditions given in the word problem. It's crucial to pay close attention to every word, as subtle differences in phrasing can lead to different mathematical expressions. For instance, "ten less than" would mean subtraction, while "ten times" would imply multiplication. In this case, "difference between" and "ten more than" are the key operational phrases that guide our translation. We have successfully transformed a verbal description into a solvable algebraic equation, which is a significant step in finding the unknown number.

Building the Equation: A Step-by-Step Approach

Now that we've broken down the sentence, let's systematically build the equation. We've already established that 'n' represents our unknown number. The first part of the statement is "The difference between one-half of a number and one-sixth of the number." As we identified, "one-half of a number" is $\frac{1}{2}n$, and "one-sixth of the number" is $\frac{1}{6}n$. The "difference between" them is $\frac{1}{2}n - \frac{1}{6}n$. This is the left side of our equation. The next part of the statement is "is equal to ten more than one-eighth of that number." "One-eighth of that number" is $\frac{1}{8}n$. "Ten more than" means we add 10. So, this part translates to $\frac{1}{8}n + 10$. This is the right side of our equation. Finally, we equate the two parts: $\frac{1}{2}n - \frac{1}{6}n = \frac{1}{8}n + 10$. This equation precisely mirrors the problem's conditions. It's important to double-check our translation to ensure accuracy. We've correctly interpreted "difference between" as subtraction and "ten more than" as addition. The placement of the 'n' with each fraction is also correct, indicating that the fraction is of the number 'n'. This structured approach ensures that no part of the original problem is overlooked, leading to a correct mathematical representation. This methodical process is fundamental to solving any word problem, as a faulty equation will inevitably lead to an incorrect solution, no matter how skillfully you perform the subsequent calculations. Therefore, investing time in accurately translating the words into symbols is paramount for success in algebraic problem-solving. We have now successfully constructed the equation that will help us find the value of 'n'.

Solving for 'n': Uncovering the Number

With our equation $\frac{1}{2}n - \frac{1}{6}n = \frac{1}{8}n + 10$ in hand, we can now proceed to solve for 'n'. The first step in solving this equation is to simplify both sides. On the left side, we have $\frac{1}{2}n - \frac{1}{6}n$. To subtract these fractions, we need a common denominator. The least common multiple of 2 and 6 is 6. So, we convert $\frac{1}{2}n$ to $\frac{3}{6}n$. Now, the left side becomes $\frac{3}{6}n - \frac{1}{6}n = \frac{2}{6}n$, which can be simplified to $\frac{1}{3}n$. So, our equation is now $\frac{1}{3}n = \frac{1}{8}n + 10$. The next step is to gather all the terms involving 'n' on one side of the equation. We can do this by subtracting $\frac{1}{8}n$ from both sides: $\frac{1}{3}n - \frac{1}{8}n = 10$. To subtract $\frac{1}{3}n$ and $\frac{1}{8}n$, we need a common denominator for 3 and 8, which is 24. We convert $\frac{1}{3}n$ to $\frac{8}{24}n$ and $\frac{1}{8}n$ to $\frac{3}{24}n$. So, the equation becomes $\frac{8}{24}n - \frac{3}{24}n = 10$, which simplifies to $\frac{5}{24}n = 10$. To isolate 'n', we need to multiply both sides by the reciprocal of $\frac{5}{24}$, which is $\frac{24}{5}$. So, $n = 10 \times \frac{24}{5}$. Now, we can perform the multiplication: $n = \frac{10 \times 24}{5} = \frac{240}{5}$. Finally, we divide 240 by 5: $n = 48$. Therefore, the number is 48. This step-by-step process of combining like terms and isolating the variable is fundamental to solving linear equations. It's always a good idea to check your answer by substituting the value of 'n' back into the original equation to ensure it holds true. If $\frac{1}{2}(48) - \frac{1}{6}(48) = \frac{1}{8}(48) + 10$, then $24 - 8 = 6 + 10$, which means $16 = 16$. The equation holds true, confirming our solution.

Understanding the Options: Why Other Equations Might Be Incorrect

When presented with multiple-choice options for an equation, it's vital to understand why the correct one is right and why the others are wrong. Let's consider the options provided and analyze them in relation to our problem statement: "The difference between one-half of a number and one-sixth of the number is equal to ten more than one-eighth of that number." Our derived correct equation is $\frac{1}{2}n - \frac{1}{6}n = \frac{1}{8}n + 10$. Now, let's examine potential incorrect options, such as A: $\frac{1}{2}-\frac{1}{6} n=10-\frac{1}{8} n$ and B: $\frac{1}{2} n-\frac{1}{8} n=10+\frac{1}{6} n$. Option A is incorrect because it misinterprets "one-half of a number" as simply $\frac{1}{2}$ instead of $\frac{1}{2}n$, and similarly for other terms. It also incorrectly places the 'n' on some terms and not others, and it reverses the operations on the right side. The phrase "difference between" typically means the first quantity minus the second, but option A seems to misunderstand the structure entirely. Option B, $\frac{1}{2} n-\frac{1}{8} n=10+\frac{1}{6} n$, is also incorrect. While it correctly represents "one-half of a number" as $\frac{1}{2}n$, it incorrectly states the "difference between one-half of a number and one-sixth of the number" as $\frac{1}{2}n - \frac{1}{8}n$. The problem specifies the difference between $\frac{1}{2}n$ and $\frac{1}{6}n$, not $\frac{1}{8}n$. Furthermore, it incorrectly equates this to $10 + \frac{1}{6}n$, where the positions of $\frac{1}{6}n$ and 10 are swapped from the problem's description of "ten more than one-eighth of that number." The key takeaway here is that each part of the word problem must be translated precisely. Small errors in representing fractions, operations (addition, subtraction, multiplication, division), or the relationship between terms can lead to an entirely different equation. It's like building with LEGOs; if one brick is out of place, the whole structure might be unstable or incorrect. By carefully comparing each option against the original wording, we can confidently identify the correct equation and rule out the distractors.

Conclusion: Mastering the Art of Mathematical Translation

We've journeyed through the process of solving a complex word problem, transforming a verbal description into a concrete algebraic equation. The problem, "The difference between one-half of a number and one-sixth of the number is equal to ten more than one-eighth of that number," required us to carefully dissect each phrase and translate it into mathematical symbols. By assigning the variable 'n' to the unknown number, we were able to represent "one-half of a number" as $\frac{1}{2}n$, "one-sixth of the number" as $\frac{1}{6}n$, and "one-eighth of that number" as $\frac{1}{8}n$. The phrases "difference between" and "ten more than" guided us to use subtraction and addition, respectively. Ultimately, we arrived at the correct equation: $\frac{1}{2}n - \frac{1}{6}n = \frac{1}{8}n + 10$. Solving this equation involved finding common denominators to combine fractions, isolating the variable 'n' by performing operations on both sides of the equation, and finally calculating the value of 'n', which we found to be 48. This exercise underscores the importance of precise mathematical translation and systematic problem-solving. Each step, from decoding the words to performing algebraic manipulations, is critical for reaching the correct solution. Mastering these skills not only helps in solving specific problems but also builds a strong foundation for tackling more advanced mathematical concepts. Remember, practice is key! The more word problems you encounter and solve, the more comfortable and proficient you will become in translating them into equations. Don't be afraid to break down complex sentences into smaller, manageable parts. For further practice and exploration of algebraic concepts, you can visit Khan Academy for excellent resources and tutorials on various math topics.