Solving Number Puzzle Finding A Number Where One Tenth Minus 30 Equals One Fifth

In the realm of mathematics, problem-solving often involves translating word problems into algebraic equations, the problem presents a classic scenario where the difference between a fraction of a number and a constant is equal to another fraction of the same number. This article delves into the step-by-step process of solving this problem, emphasizing the core concepts of algebraic manipulation and equation solving. Our focus keyword is on finding a specific number by translating the word problem into an algebraic equation. We aim to provide a comprehensive explanation that caters to both students and math enthusiasts alike, making the solution accessible and understandable. Understanding the fundamental principles of algebra is crucial for tackling such problems, and we'll be highlighting these principles throughout our explanation. The problem at hand requires us to identify a number that satisfies a particular condition involving fractions and differences. Let's begin by dissecting the problem statement and converting it into a mathematical equation.

The heart of solving mathematical word problems lies in the ability to translate the given information into a concise and accurate equation. This involves identifying the unknowns, assigning variables, and representing the relationships between these variables using mathematical symbols. In this specific problem, the unknown is the number we're trying to find, which we'll denote as 'x'. The phrase "the difference of $\frac{1}{10}$ of a number and 30" can be translated to $\frac{x}{10} - 30$, where $\frac{x}{10}$ represents one-tenth of the number x, and the subtraction of 30 represents the difference. Similarly, "$\frac{1}{5}$ of the number" is expressed as $\frac{x}{5}$. The problem states that these two expressions are equal, which leads us to the equation $\frac{x}{10} - 30 = \frac{x}{5}$. This equation forms the backbone of our solution, and solving it will lead us to the answer. The process of translating words into equations is a fundamental skill in algebra and is essential for problem-solving. By carefully dissecting the problem statement and identifying the key components, we can construct an equation that accurately represents the given information. This step is crucial for success in solving mathematical word problems. In the following sections, we will demonstrate how to solve this equation using various algebraic techniques.

Now that we have our equation, $\frac{x}{10} - 30 = \frac{x}{5}$, we can embark on the journey of solving it. The primary goal here is to isolate the variable 'x' on one side of the equation. To achieve this, we'll employ a series of algebraic manipulations. The first step involves eliminating the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is 10. Multiplying both sides by 10 gives us: $10 * (\frac{x}{10} - 30) = 10 * (\frac{x}{5})$. This simplifies to $x - 300 = 2x$. Next, we want to gather all the terms involving 'x' on one side of the equation. We can subtract 'x' from both sides, resulting in $-300 = x$. Thus, we find that the number we're looking for is -300. This step-by-step approach highlights the importance of maintaining balance in an equation. Any operation performed on one side must also be performed on the other side to preserve equality. This principle is the cornerstone of equation solving. By carefully applying algebraic techniques, we can systematically isolate the variable and arrive at the solution. In the next section, we'll verify this solution to ensure its correctness.

After obtaining a solution, it's crucial to verify its accuracy. This involves substituting the solution back into the original equation and checking if it satisfies the equation. In our case, we found that x = -300. Let's substitute this value into the original equation: $\frac{x}{10} - 30 = \frac{x}{5}$. Substituting x = -300, we get $\frac{-300}{10} - 30 = \frac{-300}{5}$. Simplifying the left side, we have -30 - 30 = -60. Simplifying the right side, we have -60. Since both sides are equal (-60 = -60), our solution x = -300 is correct. This verification process is an essential step in problem-solving. It helps us catch any errors we might have made during the solution process and ensures that our final answer is accurate. By substituting the solution back into the original equation, we can confirm that it satisfies the given conditions. This practice reinforces the understanding of the problem and builds confidence in the solution. In the subsequent sections, we'll summarize the entire solution process and discuss the key takeaways from this problem.

In this article, we tackled a mathematical word problem that involved finding a number based on fractional relationships and differences. We successfully translated the problem into an algebraic equation, solved the equation using algebraic manipulations, and verified the solution for accuracy. The key takeaways from this exercise are the importance of translating word problems into mathematical equations, the systematic application of algebraic techniques to solve equations, and the necessity of verifying solutions to ensure correctness. Translating word problems into equations requires careful reading and understanding of the problem statement. Identifying the unknowns, assigning variables, and representing relationships using mathematical symbols are crucial steps in this process. Algebraic manipulations, such as multiplying both sides of an equation by a constant or adding/subtracting terms from both sides, are essential tools for isolating the variable and solving the equation. Verification of the solution is a critical step that should not be overlooked. It helps identify and correct errors, ensuring the accuracy of the final answer. By mastering these skills, one can confidently approach and solve a wide range of mathematical problems. In conclusion, this article demonstrates the power of algebra in solving real-world problems and highlights the importance of a systematic and methodical approach to problem-solving. We trust that this detailed explanation has provided valuable insights and enhanced your understanding of algebraic problem-solving techniques.

The journey of solving this mathematical puzzle underscores the significance of a structured approach to problem-solving. From translating the word problem into an algebraic equation to verifying the solution, each step plays a crucial role in arriving at the correct answer. The ability to translate words into equations is a fundamental skill in mathematics, and it forms the bedrock of solving word problems. The systematic application of algebraic techniques, such as isolating variables and simplifying equations, is essential for arriving at the solution. Verification of the solution is a non-negotiable step that ensures the accuracy of the final answer. By adopting these practices, individuals can enhance their problem-solving prowess and approach mathematical challenges with confidence. This article serves as a testament to the power of algebra in unraveling numerical mysteries and highlights the importance of perseverance and attention to detail in the realm of mathematics. We encourage readers to apply these principles to other problems and continue to explore the fascinating world of mathematics. Understanding algebraic concepts can be beneficial for academic success and real-world scenarios. The process of translating word problems into algebraic equations and subsequently solving them not only aids in enhancing mathematical skills but also fosters critical thinking, analytical reasoning, and problem-solving abilities. These skills are valuable assets in various aspects of life, ranging from personal financial planning to professional decision-making. As you continue to hone your mathematical expertise, keep in mind that problem-solving is an iterative process that involves learning from mistakes and refining strategies. Embrace the challenges, celebrate the victories, and never stop exploring the captivating realm of mathematics.