How to Order Fractions (with Examples)

Ordering fractions comes down to finding the Least Common Multiple (LCM) within the denominators in a series of fractions. We’ll walk you through a simple way of finding the LCM, so you can order your fractions with ease and solve any math problem that comes your way. Plus, we'll share other methods for ordering fractions and provide tips, tricks, and example problems.

How to Order Fractions from Least to Greatest

Find the Least Common Multiple (LCM) among the fractions, then convert the fractions so they all have the same denominator. Next, order the converted fractions from least to greatest, using the numerators as a guide. Divide by the same number you multiplied by to convert the fractions to their original forms.

Section 1 of 5:

Ordering Fractions from Least to Greatest

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$Step 1 Identify the least common multiple for the fractions.$

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Identify the least common multiple for the fractions. According to math tutor David Jia, the first thing you have to do is find a common denominator.^{[1]
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David Jia
Academic Tutor

Expert Interview. 23 February 2021.} List out the multiples of each denominator in the problem until you find a number that all the fractions have in common.^{[2]
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- Let’s say you’re comparing three fractions: $5/8$ , $2/3$ , and $3/4$ . You’d list of several multiples of 8 (8, 16, 24), 3 (3, 6, 9, 12, 15, 18, 21, 24), and 4 (4, 8, 12, 16, 20, 24). In this example, 24 would be the least common denominator.
- Alternative: Multiply all the denominators together to get the LCM. For example, if you’re comparing $2/3$ , $5/6$ , and $1/3$ , you’d multiply 6 and 3 to get 18 as the LCM (you’d only need to multiply by 3 once).
$Step 2 Convert any mixed numbers to improper fractions.$

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Convert any mixed numbers to improper fractions. Mixed numbers are a combination of whole numbers and fractions, like $32/3$ $32/3$ . To order them properly with other fractions, you need to convert them to an improper fraction (a fraction with a higher numerator than denominator):
- Separate the whole number from the fraction. In our example, that would be 3 and 2/3.
- Multiply the whole number by the denominator of the fraction. In our example, you’d multiply $3/1$ by $3/3$ , giving you $9/3$ .
- Add the numerators of the improper fraction and leftover fraction together. In our example, $9/3+2/3$ would become $11/3$ .
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$Step 3 Adjust the fractions so they all have the same denominator.$

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Adjust the fractions so they all have the same denominator. Now that you know the LCM, multiply the numerator and denominator by the same number—whichever number turns the denominator into the LCM. In our example, for instance, $5/8$ $5/8$ would be multiplied by $3/3$ $3/3$ to make $15/24$ $15/24$ , with 24 being the LCM.^{[3]
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- To continue with the previous example, we’d multiply $2/3$ by $8/8$ to get $16/24$ and $3/4$ by $6/6$ to get $18/24$ .
$Step 4 Use the numerators to order the fractions from least to greatest.$

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Use the numerators to order the fractions from least to greatest. Now that the denominators are the same, list the fractions from the smallest numerator to the largest numerator. You’ve now successfully ordered fractions!
- The correct order of fractions from our previous example would be: $5/8(15/24),2/3(16/24),3/4(18/24)$ .
$Step 5 Divide by the original number to convert the fractions to their original numbers.$

According to Jia, you can “simplify by dividing the top and bottom by whatever their common factors are.”^{[4]
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David Jia
Academic Tutor

Expert Interview. 23 February 2021.} Let’s go back to the example of $5/8$ , which we converted to $15/24$ by multiplying the original fraction by $3/3$ . Now that we’ve successfully ordered the fractions, we can convert $15/24$ by dividing both the numerator and denominator by 3, which gives us $5/8$ again.

Section 2 of 5:

Ordering Fractions with the Same Numerator

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$Order the fractions from the largest denominator to the smallest denominator.$
Order the fractions from the largest denominator to the smallest denominator. The denominator (bottom number) describes how many pieces a whole is broken into. The larger the number, the more pieces there are—and, more importantly, the smaller those pieces are. To order fractions with identical numerators (top numbers) but different denominators from least to greatest, work your way from the fractions with the largest denominators to the fractions with the smallest denominators.^{[5]
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- Picture two circles, with one divided into 10 pieces and the other divided into 5 pieces. A $1/10$ slice would be smaller than a $1/5$ slice.

Section 3 of 5:

Ordering Fractions with the Same Denominator

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$Order the fractions from the smallest numerator to the largest numerator.$
Order the fractions from the smallest numerator to the largest numerator. When the denominators (bottom numbers) are identical, all you have to do is organize the numerators (top numbers) in numerical order. So, the fraction with the smallest numerator would go on one end, while the fraction with the largest numerator would go on the opposite side.^{[6]
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- Imagine you have two fractions: $2/8$ and $5/8$ . The $2/8$ fraction would be considered smaller, since it incorporates fewer total pieces of the pie.

Section 4 of 5:

Ordering Fractions with Cross Multiplication

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$Step 1 Write the two fractions next to each other.$

For example, let’s compare the fraction $3/5$ and the fraction $2/3$ . Write these next to each other on the page: $3/5$ on the left, and $2/3$ on the right.
$Step 2 Multiply the top of the first fraction with the bottom of the second fraction.$

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Multiply the top of the first fraction with the bottom of the second fraction. In our example, the top number, or numerator, of the first fraction ( $3/5$ $3/5$ ) is 3. The bottom number, or denominator, of the second fraction ( $2/3$ $2/3$ ) is also 3. Multiply these together: $3x3=?$ $3x3=?$ ^{[7]
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- This method is called cross-multiplication because you multiply numbers in a diagonal line across from each other.
$Step 3 Write your answer next to the first fraction.$

Write the product, or answer to your multiplication problem, next to the first fraction on the page. In our example, $3x3=9$ , so you would write 9 next to the first fraction, on the left side of the page.
$Step 4 Multiply the top of the second fraction with the bottom of the first.$

To find out which fraction is larger, we'll need to compare our answer above with the answer to another multiplication problem. Multiply these two numbers together. For our example (comparing 3/5 and 2/3), multiply $2x5$ together.^{[8]
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$Step 5 Write this answer next to the second fraction.$

In this example, the answer is 10.
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Compare the values of the two cross-products. The answers to the multiplication problems in this method are called cross-products. If one cross-product is larger than the other, then the fraction next to that cross-product is also larger than the other fraction. In our example, because 9 is less than 10, this means $3/5$ $3/5$ must be less than $2/3$ $2/3$ .^{[9]
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- Remember, always write the cross-product next to the fraction whose top number you used.
$Step 7 Order the fractions based on your results.$

Cross-multiply fractions all the fractions within the series to figure out which ones are the biggest and smallest. Place the fractions in correct order to finish the puzzle.

Section 5 of 5:

Example Problems

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9/10, 2/5, 1/2, 4/5 The LCM for this problem is 10, which would be your target denominator for all the fractions in this puzzle. Upon conversion, the fractions would read as: 9/10, 2/5, 1/2, 8/10. From there, it’s a simple matter of reordering the fractions using the numerators as a reference.
- Answer: 2/5, 1/2, 4/5, 9/10
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2/5, 1/3, 2/4, 3/5 The LCM of these fractions is 60, so your first step is converting all the fractions so they have 60 as a denominator. From there, order the numerators from least to greatest, which gives you 1/3 (20/60), 2/5 (24/60), 2/4 (30/60), 3/5 (36/60)
- Answer: 1/3, 2/5, 2/4, 3/5
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3/4, 7/8, 3/16 Start by calculating the LCM, which is 16. Adjust 3/4 and 7/8 so they share this denominator, transforming these fractions into 12/16 and 14/16. From there, order the numbers from least to greatest using the numerators, and then convert them back to their original form.
- Answer: 3/16, 3/4, 7/8

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Question

How can I tell which fraction is greater?

David Jia
Math Tutor
David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.

David Jia

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Expert Answer

Look for a common denominator between the fractions, then multiply the numerators and denominators by the same value. Then you can see which fraction is greater just by the numerator.

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Question

Which is the lowest: 3/5, 3/4, 4/7, or 2/3?

Jasmine Tipping

Community Answer

4/7 is the lowest, then 3/5, 2/3 and 3/4.

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Can I convert them into decimals while ordering them?

Community Answer

Yes you can, the order will be the same. Just make sure to convert back to fractions for your final answer if the original numbers are given as fractions.

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