Introduction to Ordering fractions: Explained with properties, Applications, and calculations

Ordering fractions is a well-known technique in mathematics that is used to order integers, percentages, and decimals. The arrangement of different terms could be increasing (smaller to larger) or decreasing (larger to smaller). 

The ordering fractions is a well-known way to sort data in a specific order based on some criteria. Now we’ll discuss the definition, properties, and applications of ordering fractions with the help of examples.

What is ordering fraction?

In number theory, the ordering fractions is used to represent the given data either in increasing order (smaller to larger) or decreasing order (larger to smaller). 

In simple words, Ascending (increasing) and descending (decreasing) order are two main terms for ordering fractions that are used in mathematics and other fields to describe the arrangement of numbers in a particular order.

Increasing order (Ascending order)

The order of given comma-separated terms from least to greatest (A.K.A smallest to largest) refers to ascending order. In other words, the increasing order from top to bottom or left to right is known as ascending order. 

For example

Data set	Ascending Order
½, 5/6, 12/3, 18/12, 10/15, 100/7	½, 10/15, 5/6, 18/12, 12/3, 100/7
2, 8, 3, 19, 21, 50, 10, 15	2, 3, 8, 10, 15, 19, 21, 50
10%, 9%, 2%, 12%, 19%, 6%, 1%	1%, 2%, 6%, 9%, 10%, 12%, 19%
1.23, 4.32, 1.12, 4.23, 5.12, 2.8	1.12, 1.23, 2.8, 4.23, 4.32, 5.12
10, 15, 5%, 18, 2.3, 15%, 100, 1.13	5%, 15%, 1.13, 2.3, 10, 15, 18, 100

Decreasing order (Descending Order)

The order of given comma-separated terms from greatest to least (A.K.A largest to smallest) refers to descending order. In other words, the decreasing order from bottom to top or right to left is known as descending order. 

For example

Data set	Ascending Order
½, 5/6, 12/3, 18/12, 10/15, 100/7	100/7, 12/3, 18/12, 5/6, 10/15, 1/2
1, 7, 4, 20, 24, 50, 10, 25	50, 25, 24, 20, 10, 7, 4, 1
3%, 19%, 12%, 22%, 9%, 16%, 1%	22%, 19%, 16%, 12%, 9%, 3%, 1%
2.23, 3.32, 1.12, 5.23, 5.12, 2.18	5.12, 5.23, 3.32, 2.18, 2.23, 1.12
3, 5, 15%, 108, 23.12, 25%, 100, 1.13	108, 100, 23.12, 5, 3, 1.13, 25%, 15%

These orders of ordering fractions are useful when we need to analyze data in a particular order. For example, if we have a list of students’ grades, we might want to sort them in ascending order to see who has the highest grade and who has the lowest. 

On the other hand, if we have a list of expenses, we might want to sort them in descending order to see where we spent the most money and where we spent the least.

Properties of ordering fractions

1. In ordering fractions, the arrangements of smaller to larger and larger to smaller are opposite to each other.
2. The order of arrangement is important in mathematical computations such as finding the maximum and minimum values, sorting data, and identifying patterns.
3. The order of arrangement can also be used to compare and rank numbers.

Techniques of ordering fractions

There are two well-known techniques to order the fractions:

1. Make the same denominator 
2. Find the decimal value 

Let’s discuss the above two techniques.

Make the same denominator

Make the same denominator is a well-known technique of ordering fractions in which you have to take the denominators of all the fractions. After that, you have to find the least common multiple of those denominators.

Then make all the fraction’s denominators equal to that least common multiple by taking the suitable integer to multiply the numerator and denominator. After that arrange the similar fractions on the basis of the numerators. 

You have to write the corresponding inputs in ascending and descending orders for the sake of the accurate arrangement of the given fractions. 

Find the decimal value 

Find the decimal value is another technique of ordering fractions in which you have to divide the numerator by denominators to get the result in decimals. After that arrange the decimals of fractions on the basis of the smaller and larger terms.

You have to write the corresponding inputs in ascending and descending orders for the sake of the accurate arrangement of the given fractions. 

Applications of ordering fractions

Applications	Explanation
Finance	In finance, ascending and descending order are used to rank financial assets, such as stocks (bonds) based on their performance.
Computer science	In computer science, sorting algorithms are used to arrange data in ascending or descending order.
Sports	In sports, teams are often ranked based on their performance, which can be represented in ascending or descending order.

How to calculate the ordering fractions problems?

To arrange a set of numbers, fractions, percentages, or decimals in ascending order and descending order, you can use the following steps:

• Identify the smallest (largest) value in the set.
• Move the smallest (largest) value to the beginning or end of the set, respectively.
• Repeat step 1 and 2 with the remaining numbers, fractions, percentages, or decimals until the set is completely arranged in ascending or descending order.

Here are some examples of ordering fractions.

Example 1: Make the same denominator

Arrange 1/3, 6/5, 12/7, 9/6, 12/8, 10/12, 15/10

Solution 

Step 1: Write the given comma-separated proper and improper fractions and take the denominators of each fraction. 

Set of fractions = 1/5, 6/5, 12/7, 9/6, 12/8, 10/12, 15/10

Denominators of given fractions = 3, 5, 7, 6, 8, 12, 10

Step 2: Now find the least common multiple (LCM) of the above set of denominators with the help of prime factorization (P.F).

P.F of 3	3
P.F of 5	5
P.F of 7	7
P.F of 6	2 x 3
P.F of 8	2 x 2 x 2
P.F of 12	2 x 2 x 3
P.F of 10	2 x 5

C.F = 2, 2, 3, 5

N.C.F = 2, 7

LCM = product of C.F x the product of N.C.F

LCM = (2 x 2 x 3 x 5) x (2 x 7)

LCM = 60 x 14 = 840

Step 3: Now use the least common multiple to bring like fractions. 

1/3	Multiply numerator and denominator by 260	260/840
6/5	Multiply numerator and denominator by 188	1128/840
12/7	Multiply numerator and denominator by 120	1440/840
9/6	Multiply numerator and denominator by 140	1260/840
12/8	Multiply numerator and denominator by 105	1260/840
10/12	Multiply numerator and denominator by 80	800/840
15/10	Multiply numerator and denominator by 84	1260/840

Step 4: Now arrange the above like fractions from smaller to larger and write the original terms.

Smaller to larger	260/840, 800/840, 1128/840, 1260/840, 1260//840, 1260/840, 1440/840
Original (given) fractions	1/3, 10/12, 6/5, 9/6, 12/8, 15/10, 12/7

Step 5: Arrange the similar fractions from larger to smaller and write the original terms.

Larger to smaller	1440/840, 1260/840, 1260//840, 1260/840, 1128/840, 800/840, 260/840
Original (given) fractions	12/7, 15/10, 9/6, 12/8, 6/5, 10/12, 1/3

You can take assistance from the ordering fractions calculator to find the arrangement of a given fraction to avoid these lengthy calculations. 

Example 2: Find the decimal value

Find the arrangement of the given fractions from least to greatest and vice versa.

26/5, 25/6, 11/5, 40/8, 15/2, 27/3, 2/4, 72/12

Solution 

Step 1: Write the given comma separated fractions.

Given fractions = 25/5, 24/6, 10/5, 40/8, 15/2, 27/3, 2/4, 72/12

Step 2: Now find the decimal value by dividing each numerator by the denominator.

Fractions	Decimals
26/5	5.2
25/6	4.16
11/5	2.2
40/8	5
15/2	7.5
27/3	9
2/4	0.5
72/12	6

Step 3: Now arrange the above list decimals from least to greatest and take their corresponding fractions.

Ascending order	0.5, 2.2, 4.16, 5, 5.2, 6, 7.5, 9
Original (given) fractions	2/4, 11/5, 25/6, 40/8, 26/5, 72/12, 15/2, 27/3

Step 4: Now arrange the above list of decimals from greatest to least and take their corresponding fractions.

Descending Order	9, 7.5, 6, 5.2, 5, 4.16, 2.2 0.5
Original (given) fractions	27/3, 15/2, 72/12, 26/5, 40/8, 25/6, 11/5, 2/4

Conclusion

Ordering fractions is very essential in mathematics and daily life dealing such as writing grades of the student to check the top-ranked student. Now you can arrange the fractions in increasing or decreasing orders by taking help from this post. 

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