Choose The Sequence That Orders The Fractions From Least To Greatest.A.$\[ \begin{tabular}{cccc} \frac{5}{8} & \frac{5}{6} & \frac{11}{12} & \frac{9}{16} \end{tabular} \\]B.$\[ \begin{array}{cccc} \frac{9}{16} & \frac{5}{8} & \frac{5}{6} &

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Introduction

Comparing fractions is an essential skill in mathematics, particularly when dealing with fractions in various forms. In this article, we will explore the process of comparing fractions and ordering them from least to greatest. We will use a step-by-step approach to ensure that you understand the concept clearly.

Understanding Fractions

Before we dive into comparing fractions, let's quickly review what fractions are. A fraction is a way of expressing a part of a whole as a ratio of two numbers. It consists of a numerator (the top number) and a denominator (the bottom number). For example, the fraction 3/4 can be read as "three-quarters" or "three out of four."

Comparing Fractions

Comparing fractions involves finding the least common multiple (LCM) of the denominators and then converting each fraction to have the same denominator. This allows us to compare the numerators directly.

Step 1: Find the Least Common Multiple (LCM) of the Denominators

To compare fractions, we need to find the LCM of the denominators. The LCM is the smallest multiple that all the denominators have in common.

  • For the fractions 5/8, 5/6, 11/12, and 9/16, we need to find the LCM of 8, 6, 12, and 16.
  • The prime factorization of each denominator is:
    • 8 = 2^3
    • 6 = 2 * 3
    • 12 = 2^2 * 3
    • 16 = 2^4
  • The LCM is the product of the highest power of each prime factor that appears in any of the denominators.
  • The LCM of 8, 6, 12, and 16 is 2^4 * 3 = 48.

Step 2: Convert Each Fraction to Have the Same Denominator

Now that we have the LCM, we can convert each fraction to have a denominator of 48.

  • For the fraction 5/8, we multiply the numerator and denominator by 6 to get 30/48.
  • For the fraction 5/6, we multiply the numerator and denominator by 8 to get 40/48.
  • For the fraction 11/12, we multiply the numerator and denominator by 4 to get 44/48.
  • For the fraction 9/16, we multiply the numerator and denominator by 3 to get 27/48.

Step 3: Compare the Numerators

Now that all the fractions have the same denominator, we can compare the numerators directly.

  • The numerators are 30, 40, 44, and 27.
  • The smallest numerator is 27, so the fraction 9/16 is the least.
  • The next smallest numerator is 30, so the fraction 5/8 is the next least.
  • The next smallest numerator is 40, so the fraction 5/6 is the next least.
  • The largest numerator is 44, so the fraction 11/12 is the greatest.

Conclusion

Comparing fractions involves finding the LCM of the denominators and then converting each fraction to have the same denominator. This allows us to compare the numerators directly. By following these steps, we can order fractions from least to greatest.

Discussion

  • What is the LCM of 8, 6, 12, and 16?
  • How do you convert each fraction to have a denominator of 48?
  • What is the order of the fractions from least to greatest?

Answer Key

  • The LCM of 8, 6, 12, and 16 is 48.
  • To convert each fraction to have a denominator of 48, multiply the numerator and denominator by the necessary multiple.
  • The order of the fractions from least to greatest is 9/16, 5/8, 5/6, 11/12.
    Frequently Asked Questions: Comparing Fractions =====================================================

Q: What is the least common multiple (LCM) of a set of numbers?

A: The least common multiple (LCM) of a set of numbers is the smallest multiple that all the numbers have in common. For example, the LCM of 8, 6, 12, and 16 is 48.

Q: How do I find the LCM of a set of numbers?

A: To find the LCM of a set of numbers, you need to find the prime factorization of each number and then multiply the highest power of each prime factor that appears in any of the numbers. For example, the prime factorization of 8 is 2^3, the prime factorization of 6 is 2 * 3, the prime factorization of 12 is 2^2 * 3, and the prime factorization of 16 is 2^4. The LCM is the product of the highest power of each prime factor that appears in any of the numbers, which is 2^4 * 3 = 48.

Q: How do I convert a fraction to have a denominator of a certain number?

A: To convert a fraction to have a denominator of a certain number, you need to multiply the numerator and denominator by the necessary multiple. For example, to convert the fraction 5/8 to have a denominator of 48, you need to multiply the numerator and denominator by 6, which gives you 30/48.

Q: How do I compare fractions with different denominators?

A: To compare fractions with different denominators, you need to find the least common multiple (LCM) of the denominators and then convert each fraction to have the same denominator. This allows you to compare the numerators directly.

Q: What is the order of the fractions from least to greatest?

A: The order of the fractions from least to greatest is determined by comparing the numerators of the fractions. The fraction with the smallest numerator is the least, and the fraction with the largest numerator is the greatest.

Q: Can you give an example of how to compare fractions?

A: Let's compare the fractions 5/8, 5/6, 11/12, and 9/16. To compare these fractions, we need to find the LCM of the denominators, which is 48. We then convert each fraction to have a denominator of 48:

  • 5/8 = 30/48
  • 5/6 = 40/48
  • 11/12 = 44/48
  • 9/16 = 27/48

Now that all the fractions have the same denominator, we can compare the numerators directly. The numerators are 30, 40, 44, and 27. The smallest numerator is 27, so the fraction 9/16 is the least. The next smallest numerator is 30, so the fraction 5/8 is the next least. The next smallest numerator is 40, so the fraction 5/6 is the next least. The largest numerator is 44, so the fraction 11/12 is the greatest.

Q: What are some common mistakes to avoid when comparing fractions?

A: Some common mistakes to avoid when comparing fractions include:

  • Not finding the least common multiple (LCM) of the denominators
  • Not converting each fraction to have the same denominator
  • Not comparing the numerators directly
  • Not considering the sign of the fractions (e.g., positive or negative)

Q: How do I practice comparing fractions?

A: You can practice comparing fractions by using online resources, such as fraction comparison games or worksheets. You can also practice by creating your own fraction comparison problems and solving them. Additionally, you can practice by comparing fractions in real-world situations, such as comparing the prices of different items or the amounts of different substances.