An Introduction to Ordering Fractions Explained with Examples

In algebra, the ordering fraction is frequently used to create a sequence of natural numbers, whole numbers, percentages, fractions, and decimals either from least to greatest or greatest to least. The arrangement of numbers is also used in statistics to perform central tendency problems.

Various kinds of algebraic expressions need to be solved after the arrangement of terms. In this article, we will study the term ordering fractions and how to arrange the terms either in ascending or descending order with examples.

What are the ordering fractions?

In mathematics and statistics, a method or technique is used to arrange numbers, whole numbers, percentages, fractions, and decimals in increasing order, and decreasing order is said to be the ordering fractions.

The increasing order of the numbers, percentages, fractions, and decimals is known as ascending order. While the decreasing order of the numbers, whole numbers, percentages, fractions, and decimals is said to be the descending order.

All the kinds of fractions like a mixed fractions, proper fractions, and improper fractions can be written either in increasing order or decreasing order. Similarly, whole numbers, natural numbers, integers, decimals, and percentages can also be written in ascending or descending order.

The natural number is an infinite term of numbers starting from 1, while the whole number is starting from zero. Integers contain positive and negative numbers. Percentages are those numbers written along with a % sign.

The percentage is a part of the whole. For example, 19% means it is the nineteenth part of 100. It can also be written in the form of the fraction by dividing the term by 100 e.g., 12% = 12/100.

The terms written in the form of a/b where “a” and “b” are integers and b never be zero is said to be a fraction. While the upper term “a” is known as the numerator and the lower term is said to be the denominator.

Methods of arranging fractions, percentages, and decimals

To arrange the terms either in increasing order or decreasing order there are two well-known methods. The ordering fraction methods are:

  • Make the same denominator by using the LCM method or making like fraction method
  • Converting the fractions into decimals method

Let us study the above methods of ordering fractions briefly.

1.  Making like fractions method

Making like fractions methods is a frequently used method of ordering fractions. This method states that take the denominators of the fractions and determine the least common multiple of the denominator terms and make all the denominators equal to the LCM.

Follow the below steps to learn how to arrange the fractions in increasing or decreasing order.

  • First of all, take a set of proper, improper, or mixed fractions.
  • After that take the denominators of all the fractional terms.
  • Then find the least common multiple (LCM) of denominators of the fractions.
  • When you got the result of LCM, make all the denominators equal to the least common multiple and multiply the same digit by the numerator to balance the fraction.
  • Then arrange the terms by viewing the numerator of the fractions.
  • Then write the corresponding terms of the fractions

Below is a solved example of this method of ordering fractions to understand the concept accurately.

Example

Find the increasing and decreasing orders of the fractions by using like fraction methods.

1/4, 20/6, 12/8, 10/2, 30/10, 21/14, 32/12, 1/16

Solution

Step-i: First of all, take the set of proper and improper fractions.

1/4, 20/6, 12/8, 10/2, 30/10, 21/14, 32/12, 1/16

Step-ii: Take the denominators of the proper and improper fractions and determine the least common multiple by using the list of multiplies method.

Denominators of the proper and improper fraction = 4, 6, 8, 2, 10, 14, 12, 16

Least common multiple of 4, 6, 8, 2, 10, 14, 12, 16 = 1680

Step-iii: Make the like fractions.

1/4 = 1 * 420 / 4 * 420 = 420/1680

20/6 = 20 * 280 / 6 * 280 = 5600/1680

12/8 = 12 * 210 / 8 * 210 = 2520/1680

10/2 = 10 * 840 / 2 * 840 = 8400/1680

30/10 = 30 * 168 / 10 * 168 = 5040/1680

21/14 = 21 * 120 / 14 * 120 = 2520/1680

32/12 = 32 * 140 / 12 * 140 = 4480/1680

1/16 = 1 * 105 / 16 * 105 = 105/1680

Step-iv: Now write the above fraction from least to greatest and write corresponding given proper and improper fractions.

Least to greatest = 105/1680, 420/1680, 2520/1680, 2520/1680, 4480/1680, 5040/1680, 5600/1680, 8400/1680

Their corresponding terms are:

Least to greatest = 1/16, 1/4, 12/8, 21/14, 32/12, 30/10, 20/6, 10/2

Step-v: Now write the above fraction from greatest to least and write corresponding given proper and improper fractions

greatest to least = 8400/1680, 5600/1680, 5040/1680, 4480/1680, 2520/1680, 2520/1680, 420/1680, 105/1680

Their corresponding terms are:

Greatest to least = 10/2, 20/6, 30/10, 32/12, 21/14, 12/8, 1/4, 1/16

To avoid such larger and time-consuming calculations, you can use a least to greatest calculator.

2.  Converting the fractions into decimals method

In this method, you have to convert all the proper, improper, & mixed fractions and percentages into decimals and make an arrangement either in increasing or decreasing order.

  • First of all, take a set of proper, improper, or mixed fractions and percentages.
  • Divide the fractions to get the decimal result of the given terms.
  • Arrange the decimal in increasing or decreasing order.
  • Then write the corresponding terms of the fractions

Below is a solved example of this method of ordering fractions to understand the concept accurately.

Example

Find the increasing and decreasing orders of the ordering fractions by using Converting the fractions into decimals method.

160/8, 94/6, 12/72, 49%, 9%, 4 * 4/8, 3 * 2/4

Solution

Step-i: First of all, take the set of proper and improper fractions.

160/8, 94/6, 12/72, 49%, 9%, 4 * 4/8, 3 * 2/4

Step-ii: Convert the proper fractions and improper fractions into decimals.

160/8 = 80/4 = 40/2 = 20

94/6 = 47/3 = 15.67

12/72 = 6/36 = 3/18 = 1/6 = 0.167

Step-iii: Change the percentages into fractions and find the decimals

49% = 49/100 = 0.49

9% = 9/100 = 0.09

Step-iv: Now convert the mixed fractions into proper fractions and convert them into decimals.

4 * 4/8 = 36/8 = 18/4 = 9/2 = 4.5

3 * 2/4 = 14/4 = 7/2 = 3.5

Step-v: Now write the above fraction from least to greatest and write corresponding given proper and improper fractions.

least to greatest = 0.09, 0.167, 0.49, 3.5, 4.5, 15.67, 20

The corresponding terms are:

Ascending order = 9%, 12/72, 49%, 3 * 2/4, 4 * 4/8, 94/6, 160/8

Step-vi: Now write the above fraction from greatest to least and write corresponding given proper and improper fractions.

greatest to least = 20, 15.67, 4.5, 3.5, 0.49, 0.167, 0.09

The corresponding terms are:

Descending order = 160/8, 94/6, 4 * 4/8, 3 * 2/4, 49%, 12/72, 9%

Summary

Now you are witnessed that the ordering fraction is not a difficult topic. Now after reading the above post, you can easily arrange the numbers, fractions, and percentages either in an increasing order or decreasing order.

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