Which Of The Following Pairs Of Numbers Contains Like Fractions

Which of the Following Pairs of Numbers Contains Like Fractions? A Deep Dive into Fraction Equivalence

Understanding fractions is fundamental to mathematics, and a crucial stepping stone to more advanced concepts. One of the earliest and most important things to grasp is the idea of like fractions. But what exactly are like fractions, and how can we identify them? This comprehensive guide will explore this concept, providing you with the tools and knowledge to confidently distinguish like fractions from unlike fractions. We'll delve into the definition, explore examples, and even tackle some practice problems.

What are Like Fractions?

Like fractions, also known as similar fractions, are fractions that share the same denominator. The denominator, you'll recall, is the bottom number in a fraction; it represents the total number of equal parts into which a whole is divided. The numerator, the top number, represents the number of those parts being considered.

In simpler terms: If two or more fractions have the same number on the bottom (the denominator), they are like fractions.

Examples of Like Fractions

Let's look at some examples to solidify this understanding:

• 1/5 and 3/5: Both fractions have a denominator of 5. Therefore, they are like fractions.
• 2/7 and 5/7: The denominator in both is 7, making them like fractions.
• 11/12 and 7/12: Again, the denominator is the same (12), confirming that these are like fractions.
• -3/8 and 5/8: Even with a negative sign on one of the numerators, as long as the denominators are the same (8), these are like fractions.

Examples of Unlike Fractions

Conversely, unlike fractions have different denominators:

• 1/2 and 1/3: These have denominators of 2 and 3, respectively. They are unlike fractions.
• 3/4 and 2/5: The denominators (4 and 5) are different, thus these are unlike fractions.
• 5/6 and 7/8: Again, different denominators (6 and 8) mean these are unlike fractions.

Why are Like Fractions Important?

The importance of like fractions stems from their ease of addition, subtraction, and comparison. You cannot directly add or subtract unlike fractions; you must first find a common denominator. However, with like fractions, this step is unnecessary.

Adding and Subtracting Like Fractions

Adding or subtracting like fractions involves simply adding or subtracting the numerators while keeping the denominator the same.

Example:

1/5 + 3/5 = (1+3)/5 = 4/5

2/7 - 5/7 = (2-5)/7 = -3/7

Comparing Like Fractions

Comparing like fractions is also straightforward. The fraction with the larger numerator is the larger fraction.

Example:

Which is larger, 3/5 or 1/5?

Since 3 > 1, then 3/5 > 1/5.

Identifying Like Fractions: A Step-by-Step Approach

When presented with a set of fractions, follow these steps to identify the like fractions:

1. Examine the denominators: Focus solely on the bottom numbers of each fraction.
2. Identify matching denominators: Look for fractions that share the same denominator.
3. Group like fractions: Separate the fractions with matching denominators into groups. These groups represent sets of like fractions.

Practice Problems: Putting Your Knowledge to the Test

Let's put your newfound knowledge to the test with a series of practice problems. For each pair, determine whether the fractions are like or unlike.

Problem Set 1:

1. 2/9 and 5/9
2. 1/3 and 1/4
3. 7/10 and -3/10
4. 5/8 and 11/16
5. 1/6 and 5/6

Problem Set 2 (Slightly More Challenging):

1. Determine which pairs of fractions from the following set are like fractions: {1/2, 3/4, 5/8, 1/2, 2/4, 3/6}
2. Consider the fractions {2/5, 4/10, 6/15, 8/20}. Are these like fractions? Explain your answer and simplify the fractions.
3. A recipe calls for 2/3 cup of flour and 1/3 cup of sugar. Are these like or unlike fractions? How many cups of ingredients are needed in total?

Answers:

(Problem Set 1):

1. Like
2. Unlike
3. Like
4. Unlike
5. Like

(Problem Set 2):

1. Like fractions: {1/2, 1/2}, {3/4, 2/4} (Note: 2/4 simplifies to 1/2), {3/6} (Note: 3/6 simplifies to 1/2). This problem highlights the importance of simplifying fractions to identify like fractions.
2. While the numerators differ, all four fractions can be simplified to 2/5. Therefore, they are like fractions. This emphasizes that like fractions may appear differently but reduce to the same simplified form.
3. Like fractions. Total ingredients needed: 2/3 + 1/3 = 3/3 = 1 cup.

Beyond the Basics: Working with Unlike Fractions

While this guide focuses on like fractions, it's crucial to understand how to work with unlike fractions as well. To add, subtract, or compare unlike fractions, you must find a common denominator. This is the smallest multiple that both denominators share.

Finding a Common Denominator

To find a common denominator, consider the multiples of each denominator until you find a common value. For example, to find a common denominator for 1/3 and 1/4:

• Multiples of 3: 3, 6, 9, 12, 15...
• Multiples of 4: 4, 8, 12, 16...

The least common multiple (LCM) is 12. This becomes the common denominator. You then convert each fraction to have this denominator.

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

Now you can easily add or compare the fractions.

Conclusion: Mastering the Fundamentals of Like Fractions

Understanding like fractions is a cornerstone of fractional arithmetic. The ability to quickly and accurately identify like fractions simplifies various mathematical operations. By mastering this concept and practicing regularly, you'll build a strong foundation for tackling more complex fraction problems in algebra and beyond. Remember that simplifying fractions is often a crucial step in determining whether fractions are indeed 'like' or not. Consistent practice and a thorough understanding of the underlying principles will pave your way to mastering this important mathematical skill. This will empower you to confidently tackle more advanced mathematical concepts in the future.