Ordering Rational Numbers 1/4, 5/6, 7/11, 10/24 In Ascending Order

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Rational numbers, as a fundamental concept in mathematics, are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Understanding how to compare and order these numbers is crucial for various mathematical operations and problem-solving scenarios. In this article, we will delve into the process of arranging the rational numbers 1/4, 5/6, 7/11, and 10/24 in ascending order. This involves finding a common ground for comparison, such as a common denominator, and then ordering the fractions based on their numerators. This seemingly simple task highlights the importance of foundational mathematical skills and their application in real-world scenarios. We will explore the steps involved in detail, providing a clear and concise guide to mastering the art of ordering rational numbers.

Understanding Rational Numbers

Before diving into the ordering process, it's essential to have a solid understanding of what rational numbers are. Rational numbers encompass all numbers that can be written as a fraction, where both the numerator and the denominator are integers. This includes fractions, integers, terminating decimals, and repeating decimals. The ability to represent numbers in fractional form allows for precise comparisons and calculations, making rational numbers a cornerstone of mathematical analysis. In our specific case, we have the rational numbers 1/4, 5/6, 7/11, and 10/24. Each of these can be expressed as a fraction, and our goal is to arrange them from the smallest to the largest. To do this effectively, we need to find a common denominator, which will serve as a standard unit for comparison. This involves determining the least common multiple (LCM) of the denominators, a crucial step in ordering rational numbers.

Finding a Common Denominator

The key to comparing and ordering fractions lies in finding a common denominator. A common denominator allows us to directly compare the numerators and determine the relative sizes of the fractions. The most efficient way to find a common denominator is to determine the least common multiple (LCM) of the denominators. In our case, the denominators are 4, 6, 11, and 24. To find the LCM, we can use several methods, including listing multiples or prime factorization. Let's use prime factorization:

  • 4 = 2 x 2
  • 6 = 2 x 3
  • 11 = 11
  • 24 = 2 x 2 x 2 x 3

The LCM is the product of the highest powers of all prime factors present in the numbers. So, LCM(4, 6, 11, 24) = 2^3 x 3 x 11 = 264. Therefore, 264 is the common denominator we will use to compare our fractions. This step is critical as it provides a uniform base for comparison, ensuring that we can accurately arrange the rational numbers in ascending order. With the common denominator in hand, we can now proceed to convert each fraction to its equivalent form with the denominator of 264.

Converting Fractions to a Common Denominator

With the common denominator of 264 established, the next step is to convert each fraction (1/4, 5/6, 7/11, and 10/24) into an equivalent fraction with this denominator. This conversion involves multiplying both the numerator and the denominator of each fraction by a factor that will result in the denominator becoming 264. Let's perform this conversion for each fraction:

  • 1/4: To get a denominator of 264, we multiply both the numerator and the denominator by 66 (264 / 4 = 66). So, 1/4 becomes (1 x 66) / (4 x 66) = 66/264.
  • 5/6: To get a denominator of 264, we multiply both the numerator and the denominator by 44 (264 / 6 = 44). So, 5/6 becomes (5 x 44) / (6 x 44) = 220/264.
  • 7/11: To get a denominator of 264, we multiply both the numerator and the denominator by 24 (264 / 11 = 24). So, 7/11 becomes (7 x 24) / (11 x 24) = 168/264.
  • 10/24: To get a denominator of 264, we multiply both the numerator and the denominator by 11 (264 / 24 = 11). So, 10/24 becomes (10 x 11) / (24 x 11) = 110/264.

Now that all fractions have the same denominator, we can easily compare their numerators to determine their relative sizes. This process of converting fractions to a common denominator is a fundamental skill in working with rational numbers and is essential for ordering them accurately.

Comparing Fractions with a Common Denominator

Once the fractions share a common denominator, comparing them becomes straightforward. The fraction with the smallest numerator is the smallest, and the fraction with the largest numerator is the largest. We now have the fractions 66/264, 220/264, 168/264, and 110/264. By observing the numerators, we can easily arrange them in ascending order:

  • 66/264 is the smallest.
  • 110/264 is the next smallest.
  • 168/264 follows.
  • 220/264 is the largest.

This direct comparison is only possible because the denominators are the same, highlighting the importance of finding a common denominator. This step is a crucial bridge between converting the fractions and finally ordering them in the correct sequence. With the numerators now clearly ranked, we can proceed to express the fractions in their original form, completing the process of arranging the rational numbers in ascending order.

Ordering the Fractions

Now that we have compared the fractions with a common denominator, we can order the original fractions based on the order of their numerators. We found that the numerators in ascending order are 66, 110, 168, and 220, which correspond to the fractions 66/264, 110/264, 168/264, and 220/264. Converting these back to their original forms, we get:

  • 66/264 = 1/4
  • 110/264 = 10/24
  • 168/264 = 7/11
  • 220/264 = 5/6

Therefore, the rational numbers in ascending order are 1/4, 10/24, 7/11, and 5/6. This final step brings the entire process to a conclusion, providing a clear and concise answer to the initial question. The ability to accurately order rational numbers is not only a fundamental mathematical skill but also a valuable tool in various real-world applications, from comparing prices to analyzing data.

Conclusion

In conclusion, arranging rational numbers in ascending order involves several key steps: understanding what rational numbers are, finding a common denominator, converting fractions to that common denominator, comparing the numerators, and finally, ordering the original fractions. This process, as demonstrated with the numbers 1/4, 5/6, 7/11, and 10/24, highlights the importance of foundational mathematical skills in problem-solving. By following these steps, anyone can confidently compare and order rational numbers, whether in academic settings or practical situations. The ability to work with fractions and understand their relative values is a crucial skill that extends far beyond the classroom, making it an essential component of mathematical literacy.