Fractions play a vital role in the field of mathematics and daily life dealings. These can include finding proportions, interpreting data, and solving equations. One of the essential skills about fractions is to order them either in ascending order or descending order.
The arrangement of fractions plays an important role in comparing magnitudes, identifying patterns, and simplifying the calculations. In this post, we will explain ascending and descending orders of fractions with their techniques and examples with solutions.
What is ascending order of fractions?
Ascending order is a sequence of arranging the fractions from lower numbers to higher. This arrangement of the fractions refers to increasing order that comes from the least to the greatest. It is very essential to compare the magnitudes of the fractions from smaller to larger.
The ascending order of fractions is very useful for the comparison and identification of the smallest value of the set of fractions. The below picture will let you know how the terms increase from lower to higher.
What is descending order of fractions?
Descending order is a sequence of arranging the fractions from higher numbers to lower. This arrangement of the fractions refers to decreasing order that comes from the greatest to the least. It is very essential to compare the magnitudes of the fractions from larger to smaller.
The descending order of fractions is very useful for the comparison and identification of the largest value of the set of fractions. The below picture will let you know how the terms decrease from the higher to lower.
Techniques of Ordering Fractions
There are several techniques of ordering fractions either in ascending order or descending order. Below are a few well-known techniques for ordering fractions.
- Common Denominator Method
This method is helpful when the fractions are given with the different denominators also known as applicable for the unlike fractions. According to this technique, you have to find the common denominator of the fractions.
For this, you have to take the LCM of all the denominators with the list of multiples or prime factorization method. Then make all the denominators equivalent to the common denominator by multiplying the numerator and denominator with the suitable integer.
In other words, you have to make the like fractions. Then compare the numerators of the like fractions to order them in ascending order and descending order.
- Cross Multiplication Method
This method of ordering fractions is helpful when the fractions are given with the same denominators also known as applicable for the like fractions. According to this technique, you have to multiply the numerator of one fraction with the denominator of another fraction to compare the magnitudes of these fractions.
Here are a few steps to order the fractions with the help of the cross-multiplication method.
- Take two fractions like fraction A and fraction B with the same denominator.
- Multiply the numerator of fraction A with the denominator of fraction B. This will give you the first product.
- Multiply the numerator of fraction B with the denominator of fraction A. This will give you the second product.
- Compare both the product and the higher result will be the higher fraction than the other.
- Converting to Decimal Method
This method of ordering fractions is helpful when the fractions are given with the same and different denominators also known as applicable for both like and unlike fractions. According to this technique, you have to find the quotient of each fraction by dividing the numerator by the denominator.
Here are a few steps to order the fractions with the help of converting to decimals method.
- First of all, take the fraction and divide the numerator by the denominator to find the decimal value.
- After converting all the fractions to decimals, you have to compare them.
- Arrange the ascending order or descending order on the basis of the decimal representation.
- The decimal with the smallest value is the smaller fraction and the decimal with the larger value is the larger fraction.
How to arrange fractions?
Below are a few examples to order fractions from least to greatest or greatest to least manner.
Example 1: For the common denominator method
Use the common denominator method to order the fractions in ascending and descending orders.
3/2, 7/4, 2/5, 7/6, 2/7, 3/9, 5/10
Solution
Step 1: First of all, take the denominators of the given fractions to find the common denominator.
Given fractions = 3/2, 7/4, 2/5, 7/6, 2/7, 3/9, 5/10
Denominators of Given fractions = 2, 4, 5, 6, 7, 9, 10
Step 2: Now find the least common multiple of the above set of denominators with the help of prime factorization.
| List of multiples of 2 | 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40 … |
| List of multiples of 4 | 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52 … |
| List of multiples of 5 | 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65 … |
| List of multiples of 6 | 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, … |
| List of multiples of 7 | 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, … |
| List of multiples of 9 | 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, … |
| List of multiples of 10 | 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, … |
LCM (2, 4, 5, 6, 7, 9, 10) = 1260
Step 3: Now make like fractions by making the common denominator of all the fractions 1260.
| 3/2 | 3 x 630 / 2 x 630 | 1890/1260 |
| 7/4 | 7 x 315 / 4 x 315 | 2205/1260 |
| 2/5 | 2 x 252 / 5 x 252 | 504/1260 |
| 7/6 | 7 x 210 / 6 x 210 | 1470/1260 |
| 2/7 | 2 x 180 / 7 x 180 | 360/1260 |
| 3/9 | 3 x 140 / 9 x 140 | 420/1260 |
| 5/10 | 5 x 126 / 10 x 126 | 630/1260 |
Step 4: Now order the common denominator fractions on the basis of the numerator from least to greatest.
| Ascending order | 360/1260, 420/1260, 504/1260, 630/1260, 1470//1260, 1890/1260, 2205/1260 |
| Corresponding fractions | 2/7, 3/9, 2/5, 5/10, 7/6, 3/2, 7/4 |
Step 5: Now order the common denominator fractions on the basis of the numerator from greatest to least.
| Descending Order | 2205/1260, 1890/1260, 1470/1260, 630/1260, 504/1260, 420/1260, 260/1260 |
| Corresponding fractions | 7/4, 3/2, 7/6, 5/10, 2/5, 3/9, 2/7 |
You can take assistance from the least to the greatest calculator to find the arrangement of a given fraction in no time.
Example 2: For Converting to decimals
Use the “Converting to decimals” technique to arrange the given fractions in ascending and descending orders.
12/5, 16/2, 18/3, 24/12, 25/3, 2/6, 20/6, 10/4
Solution
Step 1: First of all, take the given set of fractions.
Set of fractions = 12/5, 16/2, 18/3, 24/12, 25/3, 2/6, 20/6, 10/4
Step 2: Now convert the given fractions to decimals.
| Fractions | Decimals |
| 12/5 | 2.4 |
| 16/2 | 8 |
| 18/3 | 6 |
| 24/12 | 2 |
| 25/3 | 8.33 |
| 2/6 | 0.33 |
| 20/6 | 3.33 |
| 10/4 | 2.5 |
Step 3: Now arrange the above list decimals from least to greatest and take their corresponding fractions.
| Ascending order | 0.33, 2, 2.4, 2.5, 3.33, 6, 8, 8.33 |
| Corresponding fractions | 2/6, 24/12, 12/5, 10/4, 20/6, 18/3, 16/2, 25/3 |
Step 4: Now arrange the above list of decimals from greatest to least and take their corresponding fractions.
| Descending Order | 8.33, 8, 6, 3.33, 2.5, 2.4, 2, 0.33 |
| Corresponding fractions | 25/3, 16/2, 18/3, 20/6, 10/4, 12/5, 24/12, 2/6 |
Example 3: For Cross Multiplication Method
Use the “cross multiplication” technique to arrange the given fractions in ascending and descending orders.
12/4 and 16/6
Solution
Step 1: First of all, take the given set of fractions and Name them.
12/4 and 16/6
A = 12/6
B = 16/6
Step 2: Multiply the numerator of fraction A with the denominator of fraction B.
Product 1 = 12 x 6
Product 1 = 72
Step 3: Multiply the numerator of fraction B with the denominator of fraction A.
Product 2 = 16 x 6
Product 2 = 96
Step 4: Now take the greater product as a greater fraction.
Fraction B is greater than fraction A
12/4 < 16/6
Wrap Up
Now you can take the basics of ordering fractions with its techniques. The order of the fractions could be taken either with any one of the above methods. You have to explore the solved examples above to understand them.