Solving For X In The Equation 4/x = 5/10 A Step-by-Step Guide

by ADMIN 62 views

Introduction

In this article, we will delve into the process of solving for the variable x in the equation 4/x = 5/10. This is a fundamental algebraic problem that involves manipulating fractions and applying the principles of cross-multiplication or equivalent fractions. Understanding how to solve such equations is crucial for various mathematical and real-world applications. Whether you're a student learning algebra or someone looking to refresh your math skills, this step-by-step guide will provide a clear and concise explanation of the solution. We will explore the underlying concepts, demonstrate the algebraic manipulations required, and arrive at the final answer. By the end of this article, you will have a solid understanding of how to tackle similar problems involving fractions and variables.

Understanding the Equation

Before we dive into the solution, let's first understand the equation we are dealing with: 4/x = 5/10. This equation states that the fraction 4 divided by x is equal to the fraction 5 divided by 10. Our goal is to isolate x on one side of the equation to find its value. To achieve this, we will employ techniques such as cross-multiplication and simplification of fractions. Recognizing the structure of the equation and the relationship between the terms is the first step toward finding the solution. In this case, we have a proportion, which is an equation stating that two ratios (or fractions) are equal. Proportions are a common occurrence in mathematics and have various applications, making it essential to understand how to solve them effectively. The ability to manipulate fractions and solve for unknowns is a cornerstone of algebra and essential for more advanced mathematical concepts.

Methods to Solve for x

There are primarily two methods to solve for x in the equation 4/x = 5/10. The first method involves cross-multiplication, and the second method involves recognizing equivalent fractions. Both methods are equally valid and will lead to the same solution. Understanding both approaches can provide a more comprehensive understanding of the problem and allow you to choose the method that you find most intuitive or efficient. Cross-multiplication is a standard technique used to solve proportions, while recognizing equivalent fractions can sometimes simplify the problem and make it easier to solve. Let's explore each method in detail:

Method 1: Cross-Multiplication

Cross-multiplication is a widely used technique for solving proportions. The basic principle behind cross-multiplication is that if a/b = c/d, then ad = bc. In other words, we multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction. Applying this to our equation, 4/x = 5/10, we multiply 4 by 10 and set it equal to 5 multiplied by x. This gives us the equation 4 * 10 = 5 * x, which simplifies to 40 = 5x. Now, to isolate x, we divide both sides of the equation by 5. This yields x = 40/5, which simplifies to x = 8. Therefore, the value of x that satisfies the equation 4/x = 5/10 is 8. Cross-multiplication is a reliable method for solving proportions and is particularly useful when the fractions are not easily simplified or when the relationship between the numerators and denominators is not immediately apparent.

Method 2: Equivalent Fractions

Another way to solve the equation 4/x = 5/10 is by recognizing equivalent fractions. First, we can simplify the fraction 5/10 by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This simplifies 5/10 to 1/2. Now our equation becomes 4/x = 1/2. To solve for x, we need to find a value that, when plugged into the denominator, makes the fraction 4/x equivalent to 1/2. We can think of this as finding a fraction equivalent to 1/2 that has a numerator of 4. To achieve this, we multiply both the numerator and the denominator of 1/2 by 4. This gives us (1 * 4) / (2 * 4) = 4/8. Therefore, 4/x = 4/8. By comparing the two fractions, we can see that x must be equal to 8. This method highlights the concept of equivalent fractions and provides an alternative approach to solving the equation. Recognizing equivalent fractions can sometimes lead to a quicker and more intuitive solution, especially when the fractions involved can be easily simplified.

Step-by-Step Solution

Let's consolidate the steps to solve the equation 4/x = 5/10, incorporating the method of cross-multiplication:

  1. Cross-multiply: Multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction. This gives us 4 * 10 = 5 * x.
  2. Simplify: Perform the multiplication on both sides of the equation. 4 * 10 = 40, so the equation becomes 40 = 5x.
  3. Isolate x: To isolate x, divide both sides of the equation by the coefficient of x, which is 5. This gives us x = 40/5.
  4. Solve for x: Perform the division to find the value of x. 40/5 = 8, so x = 8.

Therefore, the solution to the equation 4/x = 5/10 is x = 8. This step-by-step approach provides a clear and concise pathway to solving the equation, emphasizing the key algebraic manipulations involved. Each step is logically connected to the previous one, ensuring a smooth and understandable solution process. By following these steps, you can confidently solve similar equations involving fractions and variables.

Verification

To ensure that our solution is correct, it is always a good practice to verify it by substituting the value of x back into the original equation. In our case, we found that x = 8. Let's substitute this value into the equation 4/x = 5/10:

4/8 = 5/10

Now, we simplify both fractions. 4/8 simplifies to 1/2 by dividing both the numerator and the denominator by their greatest common divisor, which is 4. Similarly, 5/10 simplifies to 1/2 by dividing both the numerator and the denominator by their greatest common divisor, which is 5. So, we have:

1/2 = 1/2

Since both sides of the equation are equal, our solution x = 8 is correct. This verification step provides assurance that we have accurately solved the equation and that our answer is valid. By substituting the solution back into the original equation, we can confirm that it satisfies the equality and that there are no errors in our calculations.

Conclusion

In conclusion, we have successfully solved the equation 4/x = 5/10 using both cross-multiplication and the concept of equivalent fractions. The solution we found is x = 8. We also verified this solution by substituting it back into the original equation and confirming that it satisfies the equality. This exercise demonstrates the fundamental principles of solving proportions and manipulating fractions, which are essential skills in algebra and various other mathematical contexts. Understanding how to solve equations like this is crucial for problem-solving in both academic and real-world scenarios. By mastering these techniques, you will be well-equipped to tackle more complex algebraic problems and apply mathematical reasoning to a wide range of situations. Remember to always verify your solutions to ensure accuracy and to build confidence in your mathematical abilities.