Expressing 2/(6x) + 2/8 As A Single Fraction In Lowest Terms A Step-by-Step Guide

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In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to represent complex equations in a more concise and manageable form. This article delves into the process of expressing the sum of two fractions, specifically 2/(6x) + 2/8, as a single fraction in its lowest terms. We will explore the underlying principles, step-by-step calculations, and the importance of simplification in mathematical problem-solving.

Understanding the Basics of Fraction Addition

Before we dive into the specific problem, let's revisit the fundamentals of adding fractions. To add fractions, they must have a common denominator. The common denominator is a multiple of both denominators in the original fractions. Once the fractions share a common denominator, we can add the numerators while keeping the denominator the same.

For instance, to add 1/3 and 1/4, we need to find a common denominator. The least common multiple (LCM) of 3 and 4 is 12. We then convert each fraction to an equivalent fraction with a denominator of 12:

1/3 = (1 * 4) / (3 * 4) = 4/12 1/4 = (1 * 3) / (4 * 3) = 3/12

Now, we can add the fractions:

4/12 + 3/12 = (4 + 3) / 12 = 7/12

The resulting fraction, 7/12, is in its simplest form because 7 and 12 share no common factors other than 1.

Finding the Least Common Denominator (LCD)

In our problem, we have 2/(6x) + 2/8. The denominators are 6x and 8. To find the least common denominator (LCD), we need to determine the least common multiple (LCM) of 6x and 8. This involves considering both the numerical coefficients and the variable term.

First, let's find the LCM of the numerical coefficients, 6 and 8. The multiples of 6 are 6, 12, 18, 24, 30, and so on. The multiples of 8 are 8, 16, 24, 32, and so on. The least common multiple of 6 and 8 is 24.

Next, we consider the variable term, x. Since 6x contains the variable x, the LCD must also include x. Therefore, the LCD of 6x and 8 is 24x.

Converting Fractions to Equivalent Fractions with the LCD

Now that we have the LCD, 24x, we need to convert each fraction in the original expression to an equivalent fraction with this denominator. To do this, we multiply the numerator and denominator of each fraction by the appropriate factor.

For the first fraction, 2/(6x), we need to multiply the denominator 6x by 4 to get 24x. So, we multiply both the numerator and denominator by 4:

2/(6x) = (2 * 4) / (6x * 4) = 8/(24x)

For the second fraction, 2/8, we need to multiply the denominator 8 by 3x to get 24x. So, we multiply both the numerator and denominator by 3x:

2/8 = (2 * 3x) / (8 * 3x) = 6x/(24x)

Now, our expression looks like this:

8/(24x) + 6x/(24x)

Adding the Fractions

With both fractions having the same denominator, 24x, we can add the numerators:

8/(24x) + 6x/(24x) = (8 + 6x) / (24x)

Simplifying the Fraction

The fraction (8 + 6x) / (24x) can be simplified further by finding the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest factor that divides both the numerator and denominator without leaving a remainder.

In the numerator, 8 + 6x, the GCF of 8 and 6 is 2. We can factor out 2 from the numerator:

8 + 6x = 2(4 + 3x)

In the denominator, 24x, the factors are 2, 3, 4, 6, 8, 12, 24, and x. The GCF of 2(4 + 3x) and 24x is 2. We can divide both the numerator and denominator by 2:

[2(4 + 3x)] / (24x) = (4 + 3x) / (12x)

Final Result

The simplified form of the expression 2/(6x) + 2/8 as a single fraction in its lowest terms is (4 + 3x) / (12x).

Importance of Simplifying Fractions

Simplifying fractions is crucial in mathematics for several reasons:

  1. Conciseness: Simplified fractions are easier to understand and work with.
  2. Accuracy: Simplifying reduces the risk of errors in subsequent calculations.
  3. Comparison: Simplified fractions make it easier to compare the relative sizes of different quantities.
  4. Generalization: Simplified expressions often reveal underlying patterns and relationships.

In conclusion, expressing 2/(6x) + 2/8 as a single fraction in its lowest terms involves finding the least common denominator, converting fractions to equivalent forms, adding the numerators, and simplifying the resulting fraction. The process highlights the importance of understanding fundamental mathematical principles and applying them systematically to solve problems. Simplifying fractions is not just a mechanical exercise; it's a crucial skill that enhances mathematical understanding and problem-solving abilities. By mastering these techniques, students can confidently tackle more complex mathematical challenges.

Step-by-Step Breakdown of the Solution

To further solidify the understanding, let's summarize the steps involved in solving this problem:

  1. Identify the fractions: We start with the expression 2/(6x) + 2/8.
  2. Find the Least Common Denominator (LCD): The LCD of 6x and 8 is 24x.
  3. Convert fractions to equivalent forms:
    • 2/(6x) becomes 8/(24x) by multiplying the numerator and denominator by 4.
    • 2/8 becomes 6x/(24x) by multiplying the numerator and denominator by 3x.
  4. Add the fractions: 8/(24x) + 6x/(24x) = (8 + 6x) / (24x).
  5. Simplify the fraction:
    • Factor out the GCF from the numerator: 8 + 6x = 2(4 + 3x).
    • Divide both the numerator and denominator by the GCF, 2: [2(4 + 3x)] / (24x) = (4 + 3x) / (12x).
  6. Final Result: The simplified fraction is (4 + 3x) / (12x).

By following these steps methodically, anyone can confidently express the sum of fractions as a single fraction in its lowest terms. The key lies in understanding the underlying principles and practicing the techniques until they become second nature. This skill is not only valuable for academic success but also for real-world problem-solving where simplifying expressions can lead to more efficient and accurate solutions.

Real-World Applications of Fraction Simplification

The ability to simplify fractions extends beyond the classroom and finds numerous applications in real-world scenarios. Here are a few examples:

  1. Cooking and Baking: Recipes often involve fractions, and simplifying them is crucial for scaling recipes up or down. For instance, if a recipe calls for 2/3 cup of flour and you want to double the recipe, you need to calculate 2/3 + 2/3 = 4/3, which simplifies to 1 1/3 cups.
  2. Construction and Engineering: Calculating dimensions, angles, and material quantities often involves fractions. Simplifying these fractions ensures accurate measurements and prevents costly errors. For example, if a beam needs to be 5/8 the length of another beam that is 16 feet long, you need to calculate (5/8) * 16 = 10 feet.
  3. Finance and Accounting: Financial calculations, such as interest rates, investment returns, and budget allocations, often involve fractions and percentages. Simplifying these fractions helps in making informed financial decisions. For instance, calculating a 15% discount on an item priced at $80 involves finding 15/100 of 80, which simplifies to (3/20) * 80 = $12.
  4. Science and Research: Scientific measurements and calculations frequently involve fractions, especially in fields like chemistry and physics. Simplifying fractions ensures accurate data analysis and interpretation. For example, calculating the concentration of a solution might involve simplifying a ratio of solute to solvent.
  5. Everyday Life: Even in everyday situations, we encounter fractions. Splitting a bill among friends, calculating the time remaining on a task, or understanding discounts at a store all involve fractions. Simplifying these fractions makes these calculations easier and more intuitive.

In each of these scenarios, the ability to simplify fractions is not just a mathematical skill but a practical tool that enhances efficiency and accuracy. By mastering the techniques of fraction simplification, individuals can navigate real-world challenges with greater confidence and competence. The principles learned in the classroom translate directly into tangible benefits in various aspects of life, highlighting the importance of mathematical literacy in today's world.

In summary, expressing 2/(6x) + 2/8 as a single fraction in its lowest terms demonstrates the power of mathematical simplification. By understanding the principles of fraction addition, finding the least common denominator, and simplifying the resulting fraction, we can transform a seemingly complex expression into a more manageable form. This skill is not only fundamental to mathematics but also essential for problem-solving in various real-world contexts. The ability to simplify fractions enhances our understanding of mathematical relationships and empowers us to tackle complex challenges with greater confidence and accuracy. As we continue our journey in mathematics, mastering these foundational skills will pave the way for more advanced concepts and applications.