Comparing Fractions 5/6 And 4/5 With Common Denominators

In the realm of mathematics, comparing fractions is a fundamental skill. This article will delve into a practical example: ordering the fractions 5/6 and 4/5. We will walk through the process of finding a common denominator and then using the symbols <, =, or > to accurately compare these fractions. Whether you're a student grappling with fraction comparisons or simply seeking a refresher, this guide will provide a clear and comprehensive understanding of the steps involved.

Understanding the Importance of Common Denominators

Before we dive into comparing 5/6 and 4/5, let's first understand why common denominators are crucial. A denominator, the bottom number in a fraction, represents the total number of equal parts into which a whole is divided. When fractions have different denominators, it's like comparing slices from different-sized pizzas – you can't easily tell which slice is bigger without adjusting them to be comparable. To accurately compare fractions, we need to express them with the same denominator, essentially dividing the wholes into the same number of equal parts.

Finding a common denominator allows us to directly compare the numerators, which represent the number of parts we have. The fraction with the larger numerator, when both fractions share a common denominator, represents the larger quantity. This principle forms the bedrock of fraction comparison, making it a vital concept to grasp for various mathematical operations and real-world applications.

To illustrate, imagine trying to compare 1/2 and 1/3. It's difficult to visually determine which is larger without a common reference point. However, if we convert them to equivalent fractions with a common denominator of 6 (3/6 and 2/6, respectively), the comparison becomes straightforward: 3/6 is clearly larger than 2/6. This simple example highlights the power and necessity of common denominators in accurately comparing fractions.

The process of finding a common denominator typically involves identifying the least common multiple (LCM) of the original denominators. The LCM is the smallest number that is a multiple of both denominators. Using the LCM as the common denominator ensures that we're working with the smallest possible equivalent fractions, simplifying the comparison process and reducing the risk of errors. In the subsequent sections, we will apply this principle to the fractions 5/6 and 4/5, demonstrating the practical steps involved in finding a common denominator and comparing the fractions.

Step 1: Finding the Least Common Multiple (LCM)

To rewrite 5/6 and 4/5 with a common denominator, the first crucial step involves identifying the least common multiple (LCM) of their denominators, 6 and 5. The LCM is the smallest positive integer that is divisible by both numbers. There are several methods to find the LCM, including listing multiples and prime factorization. Let's explore both approaches to solidify our understanding.

Listing Multiples: This method involves listing out the multiples of each denominator until a common multiple is found. Multiples of 6 are 6, 12, 18, 24, 30, 36, and so on. Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, and so on. By comparing the lists, we can see that the smallest multiple they share is 30. Therefore, the LCM of 6 and 5 is 30.

Prime Factorization: This method involves breaking down each denominator into its prime factors. The prime factorization of 6 is 2 x 3, and the prime factorization of 5 is simply 5 (as 5 is a prime number). To find the LCM using prime factorization, we take the highest power of each prime factor that appears in either factorization. In this case, we have 2, 3, and 5 as prime factors. The highest power of each is 2^1, 3^1, and 5^1. Multiplying these together, we get 2 x 3 x 5 = 30. Again, we arrive at the LCM of 30.

Understanding the LCM is paramount because it becomes our common denominator. Using the LCM ensures that we're working with the smallest possible equivalent fractions, making the subsequent comparison simpler. Once we have the LCM, we can proceed to rewrite the original fractions with this common denominator, setting the stage for a direct comparison of their numerators.

In our example, having determined that the LCM of 6 and 5 is 30, we now know that our common denominator will be 30. This means we need to convert both 5/6 and 4/5 into equivalent fractions with a denominator of 30. The next step involves figuring out what to multiply the numerator and denominator of each fraction by to achieve this transformation. This process will be detailed in the following section, further clarifying the method of comparing fractions.

Step 2: Rewriting Fractions with the Common Denominator

Now that we've established that the least common multiple (LCM) of 6 and 5 is 30, our next task is to rewrite the fractions 5/6 and 4/5 so that they both have a denominator of 30. This involves finding equivalent fractions, which represent the same value but have different numerators and denominators. To do this, we need to determine what number to multiply both the numerator and the denominator of each fraction by to get a denominator of 30.

For the fraction 5/6, we ask ourselves: What number multiplied by 6 equals 30? The answer is 5. Therefore, we multiply both the numerator and the denominator of 5/6 by 5: (5 x 5) / (6 x 5) = 25/30. This means that 5/6 is equivalent to 25/30. We have successfully rewritten 5/6 with the common denominator of 30.

Next, we turn our attention to the fraction 4/5. We ask a similar question: What number multiplied by 5 equals 30? The answer is 6. So, we multiply both the numerator and the denominator of 4/5 by 6: (4 x 6) / (5 x 6) = 24/30. This means that 4/5 is equivalent to 24/30. We have now rewritten 4/5 with the common denominator of 30.

By rewriting both fractions with a common denominator, we have transformed them into a format that allows for direct comparison. The fractions 25/30 and 24/30 represent the same quantities as 5/6 and 4/5, respectively, but their common denominator makes it easy to see which fraction is larger. This step is crucial because it eliminates the ambiguity of comparing fractions with different denominators, enabling us to confidently determine the relationship between their values.

Now that we have the equivalent fractions 25/30 and 24/30, we are well-positioned to compare them. The next step will involve examining the numerators of these fractions and using the appropriate symbol (<, =, or >) to express the relationship between 5/6 and 4/5. This final comparison will solidify our understanding of fraction ordering and provide a clear answer to the original problem.

Step 3: Comparing the Fractions

With the fractions 5/6 and 4/5 rewritten as 25/30 and 24/30, respectively, we can now compare the fractions directly. The key to comparing fractions with a common denominator lies in examining their numerators. The fraction with the larger numerator represents the larger quantity, as both fractions are now divided into the same number of equal parts (30ths in this case).

Looking at our equivalent fractions, we have 25/30 and 24/30. The numerators are 25 and 24. Since 25 is greater than 24, we can conclude that 25/30 is greater than 24/30. This comparison is straightforward because the denominators are the same, allowing us to focus solely on the numerators to determine the relative size of the fractions.

Therefore, since 25/30 is greater than 24/30, we can also say that the original fraction 5/6 is greater than the original fraction 4/5. This is because 25/30 and 24/30 are simply equivalent representations of 5/6 and 4/5, respectively. The relationship between the equivalent fractions holds true for the original fractions as well.

To express this relationship mathematically, we use the