Whether you are learning how to order fractions for the first time, preparing for a standardized test, or brushing up on your math skills, you’ve come to the right place!
In this kid-friendly guide, we’ll walk you through the simple steps of arranging fractions from least to greatest using easy-to-follow explanations, practical examples, and fun exercises to practice what you’ve learned.
Enjoy!
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Simply put, a fraction is a part of a whole. It’s written as two numbers, one over the other, separated by a line, like so: \( \Large \frac{1}{2}\).
The top number is called the numerator, and it tells us how many parts we have.
The bottom number is the denominator, which tells us how many equal parts the whole is divided into.
For instance, in the fraction \( \Large \frac{3}{4}\), the 3 (numerator) means we have 3 parts, and the 4 (denominator) shows that the whole is divided into 4 equal parts.
There are four types of fractions:
To show you how to order fractions from least to greatest, we’ll start with the easier case—when the denominators are the same—and then move on to the slightly more complex case of fractions with different denominators.
When fractions have the same denominator, comparing them is easy peasy.
This is because the same number of parts is being divided into SMALLER (not larger) pieces.
The fraction with the smallest numerator is the least, and the fraction with the largest numerator is the greatest.
Let’s see an example!
We want to order \( \Large \frac{2}{7}\), \( \Large \frac{5}{7}\), \( \Large \frac{3}{7}\), and \( \Large \frac{1}{7}\).
So, the fractions from least to greatest are: \( \Large \frac{1}{7}\), \( \Large \frac{2}{7}\), \( \Large \frac{3}{7}\), and \( \Large \frac{5}{7}\).
Why does this work?
Since all the fractions are divided into the same size parts, the numerators tell us how many of those parts we have. Smaller numerators mean fewer parts, so the fraction is smaller.
When fractions have different denominators, we can’t compare them straight away.
Why?
Because the different denominators (such as 3 in \( \Large \frac{1}{3}\) and 6 in \( \Large \frac{1}{6}\)) mean the fractions are divided into different number of pieces. To compare them fairly, we need to make the denominators match.
To show you the steps of ordering fractions with different denominators, let’s use this example:
\( \Large \frac{2}{3}\), \( \Large \frac{4}{6}\), \( \Large \frac{1}{4}\), and \( \Large \frac{3}{6}\).
The lowest common denominator (LCD) is the smallest number that all the denominators can divide into evenly.
Now think, what is the lowest common denominator in our case?
We’re working with these fractions: \( \Large \frac{2}{3}\), \( \Large \frac{5}{6}\), \( \Large \frac{1}{4}\), and \( \Large \frac{3}{6}\).
So, the LCD is 12.
Since our LCD is 12, we need to rewrite each fraction so it has 12 as its denominator.
To do this, we ask: What number do we multiply each denominator by to get 12?
To keep the fraction equivalent, we multiply both the numerator and the denominator by the same number.
Let’s go through each fraction:
\( \Large \frac{2}{3}\): To make the denominator 12, we multiply 3 by 4. We multiply the numerator by the same number.
\( \Large \frac{2}{3}\) x \( \Large \frac{4}{4}\) = \( \Large \frac{2×4}{3×4}\) = \( \Large \frac{8}{12}\)
\( \Large \frac{5}{6}\): To make the denominators 12, we multiply 6 by 2. We do the same with the numerator.
\( \Large \frac{5}{6}\) x \( \Large \frac{2}{2}\) = \( \Large \frac{5×2}{6×2}\) = \( \Large \frac{10}{12}\)
\( \Large \frac{1}{4}\): To make the denominator 12, multiply 4 by 3. Multiply the numerator 1 by 3 as well.
\( \Large \frac{1}{4}\) x \( \Large \frac{3}{3}\) = \( \Large \frac{1×3}{4×3}\) = \( \Large \frac{3}{12}\)
\( \Large \frac{1}{4}\): To make the denominator 12, multiply 6 by 2. Multiply the numerator 3 by 2 as well.
\( \Large \frac{3}{6}\) x \( \Large \frac{2}{2}\) = \( \Large \frac{3×2}{6×2}\)
Now our fractions are:
\( \Large \frac{8}{12}\), \( \Large \frac{10}{12}\), \( \Large \frac{3}{12}\), and \( \Large \frac{6}{12}\).
Now that all the fractions have the same denominator, we can compare the numerators and put the fractions in order.
\( \Large \frac{3}{12}\), \( \Large \frac{6}{12}\), \( \Large \frac{8}{12}\), and \( \Large \frac{10}{12}\).
Finally, we write the fractions back in their original form in the same order.
\( \Large \frac{1}{4}\), \( \Large \frac{3}{6}\), \( \Large \frac{2}{3}\), and \( \Large \frac{5}{6}\).
When we order mixed numbers, the first part to compare is the whole number. A larger whole number always means a larger mixed number. For example, 3\( \Large \frac{1}{4}\) is always larger than 2\( \Large \frac{3}{4}\).
If the whole numbers are the same, we then compare the fractional parts. Let’s walk through an example step by step:
Order 2\( \Large \frac{1}{3}\) , 1\( \Large \frac{3}{4}\), and 2\( \Large \frac{5}{6}\) from smallest to largest.
First, we find the LCD for \( \Large \frac{1}{3}\) and \( \Large \frac{5}{6}\). The smallest number both 3 and 6 can divide evenly into is 6. Therefore, LCD is 6.
Next, we rewrite the fractions with 6 as LCD.
To get the denominator of 6, we multiply \( \Large \frac{1}{3}\) by \( \Large \frac{2}{2}\):
\( \Large \frac{1}{3}\) x \( \Large \frac{2}{2}\) = \( \Large \frac{1x2}{3x2}\) = \( \Large \frac{2}{6}\)
Since \( \Large \frac{5}{6}\) already has a denominator of 6, it stays the same.
Now, we can compare fractions easily.
\( \Large \frac{2}{6}\) is smaller than \( \Large \frac{5}{6}\).
Now we can put the mixed numbers in order:
In order from smallest to largest:
1\( \Large \frac{3}{4}\), 2\( \Large \frac{1}{3}\), 2\( \Large \frac{5}{6}\).
You Might Also Like: How Do We Convert Mixed Numbers to Improper Fractions?
To compare unit fractions (fractions with a numerator of 1), there’s no need for multiple steps—just remember this simple rule:
The larger the denominator, the smaller the fraction.
For example, if we compare \( \Large \frac{1}{2}\), \( \Large \frac{1}{4}\), and \( \Large \frac{1}{8}\):
The largest denominator is 8, so \( \Large \frac{1}{8}\) is the smallest fraction.
In order from smallest to largest, the fractions are:
\( \Large \frac{1}{8}\), \( \Large \frac{1}{4}\), and \( \Large \frac{1}{2}\).
Great job following along so far!
Learning how to order fractions takes practice, and the more you practice, the easier it gets.
Let’s go through a few more examples together to make sure you’re ready to handle any fraction ordering question that comes your way.
Order the following fractions from least to greatest:
\( \Large \frac{3}{4}\), \( \Large \frac{2}{5}\), \( \Large \frac{7}{10}\), \( \Large \frac{1}{2}\)
Step 1: Find the Lowest Common Denominator (LCD)
The denominators are 4,5,10, and 2.
The smallest number all these denominators divide into evenly is 20. So, the LCD is 20.
Step 2: Rewrite the Fractions with the LCD
Next, rewrite each fraction so it has 20 as the denominator.
Step 3: Arrange the Fractions
Now that all the fractions have the same denominator, compare the numerators:
\( \Large \frac{8}{20}\), \( \Large \frac{10}{20}\), \( \Large \frac{14}{20}\), \( \Large \frac{15}{20}\)
In order from smallest to largest, the fractions are:
\( \Large \frac{2}{5}\), \( \Large \frac{1}{2}\), \( \Large \frac{7}{10}\), and \( \Large \frac{3}{4}\).
Order the following mixed numbers from least to greatest:
2\( \Large \frac{1}{3}\), 3\( \Large \frac{1}{2}\), 2\( \Large \frac{3}{4}\), 4\( \Large \frac{1}{5}\)
Step 1: Compare the Whole Numbers
The whole numbers are: 2,3,2, and 4.
Since mixed numbers with whole numbers 3 (3\( \Large \frac{1}{2}\)) and 4 (4\( \Large \frac{1}{5}\)) are larger, they will come last in the order.
Now, we just need to find the order of the numbers with a whole number of 2 (2\( \Large \frac{1}{3}\) and 2\( \Large \frac{3}{4}\)).
Step 2: Compare the Fractional Parts
Now let’s compare the fractional parts of the mixed numbers with a whole number of 2.
The fractions are \( \Large \frac{1}{3}\) and \( \Large \frac{3}{4}\).
So, 2\( \Large \frac{1}{3}\) is smaller than 2\( \Large \frac{3}{4}\).
Step 3: Arrange the Mixed Numbers
In order from least to greatest, the mixed numbers are:
2\( \Large \frac{1}{3}\), 2\( \Large \frac{3}{4}\), 3\( \Large \frac{1}{2}\), 4\( \Large \frac{1}{5}\)
Ready to test your skills? Try ordering these fractions and mixed numbers from least to greatest.
Once you’re done, check your answers below.
Mastering how to order fractions often comes with dilemmas along the way. Here are answers to some of the most common questions our students in San Diego ask when learning this topic.
If two fractions have the same numerator, the fraction with the larger denominator is smaller.
This is because the same number of parts is being divided into smaller pieces.
For example, \( \Large \frac{2}{5}\) is smaller than \( \Large \frac{2}{3}\), because fifths are smaller than thirds.
Improper fractions (e.g. \( \Large \frac{7}{4}\)) are always larger than proper fractions (e.g. \( \Large \frac{3}{4}\)), as improper fractions represent values greater than one whole.
Simply check if the proper fraction is less than 1—if it is, the improper fraction will always be larger.
Not always! You can also turn fractions into decimals by dividing the top number by the bottom number.
For example, 23≈0.67 and 34=0.75.
But to do this, you might need a calculator, while the common denominator method lets you solve it using just your brain!
Mathnasium of La Jolla is a math-only learning center for K-12 students of all skill levels, including elementary schoolers, in and near La Jolla neighborhood in San Diego.
Using a proprietary teaching approach called the Mathnasium Method™, our specially trained math tutors offer face-to-face instruction in an engaging and supportive group environment to help students master any math class and topic, including ordering fractions.
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Whether your student is looking to catch up, keep up, or get ahead in their math class, schedule an assessment, and enroll at Mathnasium of La Jolla today!
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