A fraction represents a part (or portion)of the whole thing. It consists of two integers separated by a slash or a line. The number above the line is called the numerator while the number below theslash is referred to as the denominator. Ordering fraction is the process of arranging the fraction from least to greatest or greatest to smallest.

Ordering fractions is crucial in mathematicsbecause it helps us compare and rank fractions based on their sizes.In this article, we will explore the concept of ordering fractions and learn how to arrange abunchof fractions in different ways with the help of examples.

What is Ordering Fractions?

Ordering fractions means arranging the fractions in ascending order (from lowest to highest) or descending order (from largest to lowest). This helps in understanding which fraction is greater or lesser compared to others.

Let's learn with a real-life example! Imagine a birthday cake to be divided among siblings, with younger ones getting smaller pieces and older ones getting larger pieces. Parents would naturally get even bigger portions. Ordering fractions allows us to divide the cake fairly according to these relative sizes.

Different methods can be used to arrange fractions, but some common methods will be discussed in the next section.

Ordering fractions Methods

There are two different methods for ordering fractions:
1. Common Denominator Method
2. Conversion to Decimal

Common Denominator Method

Follow the following steps to arrange the fraction through this method.
• Calculate the least common multiple (LCM) of each denominator of the given fractions.
• Convert each fraction to the common denominator by multiplying the numerator and denominator by the appropriatefactor to ensure they all have the same least common denominator (LCD).
• Arrange the fraction by comparing the numerators of the obtained fractions.
• The fraction with larger numerators is larger than other fractions.
Consider an example to understand the above steps.

Example
Order the fractions 1/2, 2/9, 5/6, and 4/3 from least to greatest.

Solution:
Given Fraction: 1/2, 2/9, 5/6, 4/3

Step 1: Calculate the least common multiple (LCM) of the denominators (2, 9, 6, and 3).

L.C.M = 2 × 3 × 3 =18

Step 2: Convert each fraction to the common denominator (18).
1/2 = (1/2) × (9/9) = 9/18
2/9 = (2/9)× (2/2) = 4/18
5/6 = (5/6)× (3/3) = 15/18
4/3 = (7/3)× (6/6) = 42/18

Step 3:Order the fractions based on their numerators
Now, all fraction has a common denominator, it is easier to arrange the fractions from least to greatest by comparing their numerators.
Therefore, the ordered fractions from least to greatest are 2/9, 1/2, 5/6, and 4/3.

Conversion to Decimal

To order the fractions in ascending or deciding order by converting them into decimals, follow the given simple steps:
• First, convert the given fractionsinto decimal form by dividing the numerator by the denominator.
• Compare the digits at the greatest place value (thousands, hundreds, tens, and ones). If they are the same, move to the next digit to the right.
• Match each decimal to its corresponding fraction to find the ordered list of fractions.

Example:
Arrange the fractions 1/7, 3/5, 4/9, and 2/3 from greatest to least.

Solution:

Step 1: Convert the given fractions (1/7, 3/5, 4/9, and 2/3) into decimals.
1/7 = 0.143
3/5 = 0.6
4/9 = 0.444
2/3 = 0.667

Step 2:
Look at the following place value chart for the above decimal numbers.

Step 3: Compare the digits at the greatest place (ones). As digits at the place of ones are the same. Now move at the tenth place. Again, the digits are the same here.

Step 4: At the hundredth place 6 is greater than 0. So, 0.667 is greater than 0.6.

Step 5: 1 is the smallest digit at tenth so it is the smallest decimal number than all others.
So arranging the decimal numbers from greatest to least are 0.667, 0.6, 0.444, and 0.143.

Step 6: Match each decimal to its corresponding fraction to find the ordered list of fractions.
Thus the fractions in deciding order (greatest to least) are 2/3, 3/5, 5/9, and 1/7.

An ordering fractions calculator can also be used to arrange fractions from least to greatest and vice versa to get results with steps according to the above-mentioned techniques.

Mistakes to Avoid
There is a risk of being carried away or making simple mistakes when ordering fractions. One common mistake is forgetting to find a common denominator before comparing the fractions. Remember, without a common denominator, it is like comparing apples and oranges.

More Solved Examples of Ordering Fractions
Let us learn how to order fractions through the step-by-step solved examples.

Example 1: Order thefollowing Fractions from Least to Greatest.
3/5, 1/3, 2/7
Solution:

Step 1: Find the least common multiple.
The least common multiple (LCM) for 5, 3, and 7 is 105.

Step 2: Express every fraction with the common denominator.
3/5 = (3/5)× (105/105) = 63/105
1/3 = (1/3) × (105/105) = 35/105
2/7 = (2/7) × (105/105) = 30/105

Step 3: Arrange the given fractions from least to greatest.
Now that all fractions have the same denominator, compare their numerators:
1. 30/105 (2/7)
2. 35/105 (1/3)
3. 63/105 (3/5)
Therefore, the order from least to greatest is 2/7, 1/3, 3/5.

Example 2: Arrange the fractions 4/9, 5/6, 7/12 from Greatest to Least.

Solution:

Step 1:Convert each fraction to a decimal.

4/9 ≈ 0.4444
5/6 ≈ 0.8333
7/12 ≈ 0.5833
Step 2: Arrange the decimals from greatest to least.
0.8333 (5/6)
0.5833 (7/12)
0.4444 (4/9)
Thus, the given fraction in descending order (or from greatest to least) is 5/6, 7/12, 4/9.

Conclusion

In this article, we learned that ordering fractions involves arranging them from least to greatest or greatest to least. We explored the steps to understand how to arrange fractions and discussed how to avoid mistakes while ordering them. Numerous examples have been covered to help you become an expert in ordering fractions.

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