In mathematics, the ordering fractions are widely used to perform calculations of complex mathematical and statistical problems. There are two ways to order decimals, numbers, and fractions. One is the least to greatest and the other is greatest to least.

Ordering fraction is a well-known technique that is helpful in determining the addition, subtraction, and comparison of two or more fractions. In this lesson, we will learn the term ordering fractions along with its types and solved examples.

**What is ordering fractions?**

In mathematics, the ordering fraction is a well-known technique that is used to arrange decimals, percentages, and fractions either in ascending order or descending order. The term ascending order means the list of numbers goes from the smaller one to the larger one.

The ascending order is also known as least to greatest (arrangement of numbers from least one to greatest one). While the term descending order means the list of numbers goes from the larger one to the smaller one.

The descending order is also known as greatest to least (arrangement of numbers from greatest one to least one). The numbers, decimals, percentages, and fractions can be arranged in ascending and descending orders.

**Decimals**

The numbers in which the natural number, whole number, real number, etc. are involved along with or without decimal points are said to be decimals. The arrangement of decimals in ascending and descending orders is quite easy as compared to percentages and fractions.

**Percentages**

The numbers that are taken as something from the whole or used as the hundredth part of the objects are referred to as percentages. It is denoted by the “%” sign. It can be arranged in ascending order or descending order by changing it into fractions and after that in decimals.

**Fractions**

A term that is written in the form of a/b where “a” is the numerator and “b” is the denominator is said to be a fraction. The fractions can be proper, improper, and mixed. For example, 2/3, ¾, 6/2, 9/4, 11/12, 15/3, 2 ¼, 12 3/6, etc.

**Ordering fractions techniques**

To arrange the fractions and percentages either in ascending order or descending order, there are two well-known techniques. These techniques are:

- Making like fractions
- Fractions to decimal conversion

A least to greatest calculator is a helpful tool to arrange the fractions according to the above techniques within no time. Let us briefly describe these techniques along with examples

**Making like fractions**

Making like fraction is a well-known and most commonly used technique in which you have to make all the denominators of the fractions the same by taking the least common multiple of the given denominators either by prime factorization or a list of multiplication methods.

Here are a few steps to make like fractions and arrange them in ascending or descending orders.

- First of all, take denominators from the given list of fractions.
- Calculate the least common multiple of the denominators.
- Multiply and divide all the fractions with a suitable number to make all the denominators equal to the lcm.
- After that arrange the fractions in ascending or descending orders according to the numerator of the fractions.
- Write the corresponding fractions.

Let us take an example of this technique of ordering fractions to understand it more accurately.

**Example **

Arrange the given fractions in ascending and descending orders by using the making-like fraction method.

1/2, 5/3, 7/4, 11/6, 12/8, 10/9, 14/10

**Solution **

**Step I:** First of all, take denominators from the given list of fractions.

1/2, 5/3, 7/4, 11/6, 12/8, 10/9, 14/10

Denominators = 2, 3, 4, 6, 8, 9, 10

**Step II:** Now take the multiples of these denominators and take out common multiples and select the least one.

Fraction’s denominators = 2, 3, 4, 6, 8, 9, 10

Now write the multiplication tables of the above numbers.

2 times table = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, …

3 times table = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, …

4 times table = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, …

6 times table = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, …

8 times table = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, …

9 times table = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, …

10 times table = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, …

LCM of 2, 3, 4, 6, 8, 9, 10 = 360

**Step III:** Multiply and divide each fraction with a suitable digit to make all the denominators equal to 360.

1/2 = 1 * 180 / 2 * 180 = 180/360

5/3 = 5 * 120 / 3 * 120 = 600/360

7/4 = 7 * 90 / 4 * 90 = 630/360

11/6 = 11 * 60 / 6 * 60 = 660/360

12/8 = 12 * 45 / 8 * 45 = 540/360

10/9 = 10 * 40 / 9 * 40 = 400/360

14/10 = 14 * 36 / 10 * 36 = 504/360

**Step IV:** Arrange the like fractions in ascending order and write their corresponding terms.

Ascending order = 180/360, 400/360, 504/360, 540/360, 600/360, 630/360, 660/360

Their corresponding terms are:

Ascending order = 1/2, 10/9,** **14/10, 12/8, 5/3, 7/4, 11/6

**Step V:** Similarly, Arrange the like fractions in descending order and write their corresponding terms.

Descending order = 660/360, 630/360, 600/360, 540/360, 504/360, 400/360, 180/360

Their corresponding terms are:

Descending order = 11/6, 7/4, 5/3, 12/8, 14/10, 10/9,** **1/2

**Faction to decimal conversion**

There is another method for arranging the fractions in ascending and descending is to divide the fractions and get the result in decimals and arrange them. Let us take an example of this technique of ordering fractions to understand it more accurately.

**Example **

Arrange the given terms in ascending and descending orders by using converting fractions in the decimals method.

100/4, 66/3, 108/36, 19%, 14%, 12 * 11/7, 11 * 1/2

**Solution **

**Step I:** Take the given data values.

100/4, 66/3, 108/36, 19%, 14%, 12 * 11/7, 11 * ½

**Step II:** First of all, divide the proper and improper fractions and get the result in decimals.

100/4 = 50/2 = 25

66/3 = 33/1 = 33

108/36 = 54/18 = 27/9 = 3

**Step III:** Now convert the percentages into a fraction and find the decimals.

19% = 19/100 = 0.19

14% = 14/100 = 7/50 = 0.14

**Step IV:** Deal with the mixed fractions and make them in form of fractions and find the decimal.

12 * 11/7 = 95/7 = 12.142

11 * 1/2 = 22/2= 11

**Step V:** Arrange the decimals from least to greatest and write the corresponding terms of each decimal value.

Ascending order = 0.14, 0.19, 3, 11, 12.142, 25, 33

The corresponding terms are:

Ascending order = 14%, 19%, 108/36, 11 * 1/2, 12 * 11/7, 100/4, 66/3

**Step VI:** Arrange the decimals from greatest to least and write the corresponding terms of each decimal value.

Descending order = 33, 25, 12.142, 11, 3, 0.19, 0.14

The corresponding terms are:

Descending order = 66/3, 100/4, 12 * 11/7, 11 * ½, 108/36, 19%, 14%,

**Conclusion**

In this post, we have discussed all the basics to arrange decimals, numbers, fractions, and percentages along with examples and solutions. Now you can easily arrange the fractions either in ascending or descending orders just by learning the above post.