Solving Fraction And Equation Problems A Step-by-Step Guide
H2 Introduction
In this comprehensive guide, we will delve into solving mathematical problems involving fractions and equations. We will explore two specific questions that highlight fundamental concepts in algebra and arithmetic. These questions require us to translate word problems into mathematical expressions and then solve for the unknown variables. Mastering these techniques is crucial for success in various mathematical disciplines and real-world applications. Our journey will begin with a detailed analysis of the first question, where we will learn how to set up an equation involving fractions and solve for the unknown number. Then, we will move on to the second question, which introduces the concept of reciprocals and mixed fractions. By the end of this guide, you will have a solid understanding of how to approach and solve similar mathematical problems. Solving mathematical problems is a skill that can be honed through practice and a clear understanding of the underlying principles. This guide will provide you with the necessary tools and techniques to tackle these types of challenges with confidence. We will break down each problem step-by-step, ensuring that you grasp the logic behind each operation. Whether you are a student preparing for an exam or simply someone who enjoys the challenge of mathematical puzzles, this guide will offer valuable insights and strategies. The key to success in mathematics lies in the ability to translate real-world scenarios into mathematical models, and this is precisely what we aim to achieve in this article. By the time you reach the conclusion, you will have not only solved the two presented questions but also gained a broader perspective on how to approach a wide range of mathematical problems.
H2 Question 69 Finding the Unknown Number
Let's tackle the first problem: "If $ rac1}{3}$ of a number is added to $ rac{1}{5}$ of the same number, the result is 8. Find the number." This question is a classic example of an algebraic word problem that requires us to translate the given information into a mathematical equation. Finding the unknown number starts with representing the unknown number with a variable. Let's denote the number by x. According to the problem statement, we have two fractions of this number3}$ of x and $ rac{1}{5}$ of x. These fractions are then added together, and the result is 8. Mathematically, this can be expressed as3}x + rac{1}{5}x = 8$. Now, our goal is to solve this equation for x. To do this, we need to combine the terms on the left side of the equation. Since we are adding fractions, we need to find a common denominator. The least common multiple of 3 and 5 is 15. So, we rewrite the fractions with a common denominator of 1515}x + rac{3}{15}x = 8$. Now that the fractions have the same denominator, we can add them15} = 8$, which simplifies to $ rac{8x}{15} = 8$. To isolate x, we need to get rid of the fraction on the left side. We can do this by multiplying both sides of the equation by 1515} = 8 imes 15$. This simplifies to $8x = 120$. Finally, to solve for x, we divide both sides of the equation by 88} = rac{120}{8}$. This gives us $x = 15$. Therefore, the number we were looking for is 15. We can check our answer by plugging it back into the original equation{3}(15) + rac{1}{5}(15) = 5 + 3 = 8$, which confirms our solution.
H2 Step-by-Step Solution for Question 69
To further clarify the solution process for question 69, let's break it down into distinct steps. This step-by-step approach will help reinforce the method and make it easier to apply to similar problems. The first step is to represent the unknown. We begin by assigning a variable to the unknown number. In this case, we let x represent the number we are trying to find. This is a fundamental step in algebra, as it allows us to translate the word problem into a mathematical equation. The second step involves formulating the equation. We need to translate the information given in the problem into a mathematical equation. The problem states that $ rac1}{3}$ of the number added to $ rac{1}{5}$ of the same number equals 8. This translates directly to the equation $ rac{1}{3}x + rac{1}{5}x = 8$. It's crucial to accurately capture the relationships described in the problem statement. The third step is to find a common denominator. To add fractions, they must have the same denominator. The least common multiple (LCM) of 3 and 5 is 15. We rewrite the fractions with the common denominator15}x + rac{3}{15}x = 8$. This step ensures that we can combine the fractions correctly. The fourth step is to combine the fractions. Now that the fractions have the same denominator, we can add them together15} = 8$, which simplifies to $ rac{8x}{15} = 8$. This step reduces the equation to a simpler form. The fifth step is to isolate the variable. To solve for x, we need to isolate it on one side of the equation. We start by multiplying both sides of the equation by 1515} = 8 imes 15$, which simplifies to $8x = 120$. This eliminates the fraction and moves us closer to solving for x. The sixth and final step is to solve for x. We divide both sides of the equation by 8{8} = rac{120}{8}$, which gives us $x = 15$. This is the solution to the problem. By following these steps, we have successfully found the unknown number. To ensure our solution is correct, we can always check it by substituting the value of x back into the original equation, as we did earlier. This step-by-step approach not only helps in solving the problem but also provides a clear and organized way to tackle similar mathematical challenges.
H2 Question 70 Finding the Value of x
Now, let's move on to the second problem: "If $ rac1}{x}=1 rac{1}{2}$, find $x$." This question involves reciprocals and mixed fractions. Finding the value of x requires us to first convert the mixed fraction into an improper fraction. The mixed fraction $1 rac{1}{2}$ can be converted to an improper fraction by multiplying the whole number (1) by the denominator (2) and adding the numerator (1), then placing the result over the original denominator. So, $1 rac{1}{2} = rac{(1 imes 2) + 1}{2} = rac{3}{2}$. Now our equation looks like thisx} = rac{3}{2}$. To solve for x, we need to understand the concept of reciprocals. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. If $ rac{1}{x} = rac{3}{2}$, then x is the reciprocal of $ rac{1}{x}$, and $ rac{2}{3}$ is the reciprocal of $ rac{3}{2}$. Therefore, taking the reciprocal of both sides of the equation, we get3}$. Thus, the value of x is $ rac{2}{3}$. This problem demonstrates the importance of understanding fractions, mixed fractions, and reciprocals in solving algebraic equations. To further reinforce this concept, let's consider another example. Suppose we have $ rac{1}{y} = 2 rac{1}{4}$. First, we convert the mixed fraction to an improper fraction{4} = rac{(2 imes 4) + 1}{4} = rac{9}{4}$. So, $ rac{1}{y} = rac{9}{4}$. Taking the reciprocal of both sides, we get $y = rac{4}{9}$. These types of problems are common in algebra and can be solved efficiently by following these steps. Understanding reciprocals and how to manipulate fractions is a key skill in mathematics. It allows us to solve equations and simplify expressions with ease. In summary, to solve the equation $ rac{1}{x} = 1 rac{1}{2}$, we first converted the mixed fraction to an improper fraction, then took the reciprocal of both sides to find the value of x. This approach is straightforward and can be applied to similar problems involving reciprocals and fractions.
H2 Step-by-Step Solution for Question 70
To provide a clearer understanding of the solution for question 70, we will break it down into a step-by-step process. This will help solidify the concepts and provide a repeatable method for solving similar problems. The first step is to convert the mixed fraction to an improper fraction. The given equation is $ rac1}{x}=1 rac{1}{2}$. The mixed fraction $1 rac{1}{2}$ needs to be converted into an improper fraction. To do this, we multiply the whole number (1) by the denominator (2) and add the numerator (1), placing the result over the original denominator2} = rac{(1 imes 2) + 1}{2} = rac{3}{2}$. So, the equation becomes $ rac{1}{x} = rac{3}{2}$. This step is crucial for simplifying the equation and making it easier to solve. The second step is to take the reciprocal of both sides. To solve for x, we need to isolate it. Since x is in the denominator, we can take the reciprocal of both sides of the equation. The reciprocal of $ rac{1}{x}$ is x, and the reciprocal of $ rac{3}{2}$ is $ rac{2}{3}$. Therefore, we have3}$. Taking the reciprocal is a common technique in algebra for solving equations where the variable is in the denominator. The third step, although not always necessary, is to verify the solution. To ensure our solution is correct, we can substitute the value of x back into the original equationx} = rac{1}{ rac{2}{3}}$. Dividing by a fraction is the same as multiplying by its reciprocal, so{ rac{2}{3}} = 1 imes rac{3}{2} = rac{3}{2}$. Since $ rac{3}{2}$ is equal to the original mixed fraction $1 rac{1}{2}$, our solution is correct. This verification step is a good practice to ensure accuracy. By following these steps, we have successfully found the value of x in the equation $ rac{1}{x}=1 rac{1}{2}$. This step-by-step approach not only helps in solving this specific problem but also provides a clear method for tackling other equations involving reciprocals and fractions. Understanding these techniques is essential for success in algebra and other areas of mathematics. In summary, we converted the mixed fraction to an improper fraction, took the reciprocal of both sides, and verified the solution. This methodical approach ensures that we arrive at the correct answer with confidence.
H2 Conclusion
In conclusion, we have successfully solved two mathematical problems involving fractions and equations. These problems highlight the importance of understanding fundamental concepts such as translating word problems into mathematical expressions, working with fractions, finding common denominators, and using reciprocals. Solving these mathematical problems not only requires a solid grasp of algebraic principles but also the ability to apply these principles in a logical and step-by-step manner. In the first problem, we learned how to set up and solve an equation involving fractions to find an unknown number. We represented the unknown number with a variable, formulated an equation based on the given information, found a common denominator to combine fractions, and then isolated the variable to solve for its value. This process is a cornerstone of algebraic problem-solving and can be applied to a wide range of similar problems. In the second problem, we tackled an equation involving a reciprocal and a mixed fraction. We converted the mixed fraction to an improper fraction and then used the concept of reciprocals to solve for the unknown variable. This problem emphasized the importance of understanding different types of fractions and how to manipulate them effectively. By breaking down each problem into distinct steps, we have demonstrated a clear and organized approach to problem-solving. This method not only helps in finding the correct answer but also promotes a deeper understanding of the underlying mathematical concepts. The key to success in mathematics is practice and a methodical approach. By consistently applying these techniques, you can build your confidence and improve your problem-solving skills. Whether you are a student preparing for an exam or simply someone who enjoys the challenge of mathematical puzzles, the principles and methods discussed in this guide will be invaluable. We encourage you to continue practicing and exploring different types of mathematical problems. The more you engage with these concepts, the more proficient you will become. Remember, mathematics is not just about memorizing formulas; it's about understanding the logic and applying it creatively. With a solid foundation and a systematic approach, you can overcome any mathematical challenge.
