While it's easy to sort whole numbers like 1, 3, and 8 from smallest to largest, fractions can be difficult to measure at first glance. If the denominators are equal in all fractions compared, you can sort the fractions as if they were whole numbers. For example, 1/5, 3/5 and 8/5. Otherwise, you can change the list to get fractions with the same denominator, without changing their size. This becomes easier with practice, and you can learn some "tricks" such as comparing just two fractions, or when you're rating "inappropriate" fractions as 7/3.
Method 1 of 3: Sorting any number of fractions
Step 1. Find the lowest common denominator for all fractions.
Use one of these methods to find a common denominator, or lower number of a fraction, which you can use to rewrite each fraction in the list. This is called the 'common denominator', or the 'least common denominator' "if it is the lowest possible value:
- Multiply the different denominators together. For example, if you are comparing 2/3, 5/6 and 1/3, multiplying the two different denominators (3 x 6 = '18'), you get a common denominator. This is a simple method, but it can often result in a much larger number than the other methods.
- You can also list the multiples of each denominator in a separate column until you find a number that appears in every column. Use this number. For example, comparing 2/3, 5/6, and 1/3, let's list some multiples of 3: 3, 6, 9, 12, 15, and 18. Next, let's list multiples of 6: 6, 12, and 18. As the number '18' appears on both lists, use that number. (You can also use 12, but the following examples assume you are using 18).
Step 2. Convert each fraction so it can use the common denominator
Remember that if you multiply the numerator and denominator of a fraction by the same number, the resulting fraction is equivalent to the original. Try applying this method with 2/3, 5/6 and 1/3, with common denominator 18:
- 18 ÷ 3 = 6, so 2/3 = (2x6)/(3x6)=12/18
- 18 ÷ 6 = 3, so 5/6 = (5x3)/(6x3)=15/18
- 18 ÷ 3 = 6, so 1/3 = (1x6)/(3x6)=6/18
Step 3. Sort fractions by numerator
Now that they all have the same denominator, fractions can be easily compared. Use the 'numerator' of each fraction to sort them from smallest to largest. Ordering our examples above, we have: 6/18, 12/18, 15/18.
Step 4. Convert each fraction back to its original form
Keep the fractions in the same order, but convert each one to its original form. You can do this by remembering how each fraction was transformed or by dividing both the numerator and denominator of each fraction by the same number used in multiplication:
- 6/18 = (6 ÷ 6)/(18 ÷ 6) = 1/3
- 12/18 = (12 ÷ 6)/(18 ÷ 6) = 2/3
- 15/18 = (15 ÷ 3)/(18 ÷ 3) = 5/6
- The answer is "1/3, 2/3, 5/6".
Method 2 of 3: Sorting Two Fractions Using Cross-Multiplication
Step 1. Write the two fractions next to each other
For example, let's compare 3/5 and 2/3. Write down 3/5 on the left and 2/3 on the right of the sheet of paper.
Step 2. Multiply the numerator of the first fraction by the denominator of the second
In our example, the top number or numerator of the first fraction (3/5) is '3'. The lower number or denominator of the second fraction (2/3) is also '3'. Multiplying the two numbers, we have: 3 x 3 = ?
This method is called 'cross multiplication' because you multiply the numerator of one by the denominator of the other, forming an “X” between the two fractions
Step 3. Write the result next to the first fraction
In our example, 3 x 3 = 9, so you would write '9' next to the first fraction on the left side of the page.
Step 4. Multiply the numerator of the second fraction by the denominator of the first
To find out which fraction is larger, we will have to compare the answer obtained earlier with another result. For our example (3/5 and 2/3), let's multiply 2 x 5.
Step 5. Write this answer next to the second fraction
In this example, the answer is 10.
Step 6. Compare the values of the two products of cross multiplication
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 result is also larger than the other fraction. In our example, because 10 is greater than 9, 2/3 must be greater than 3/5.
Don't forget to write the cross product next to the fraction whose numerator you used
Step 7. Do you know why this works?
To compare two fractions, you usually need to transform them to give them the same denominator. And that's exactly what cross-multiplication does! That way you just need to compare the two numerators. Here's our same example (3/5 versus 2/3), written without the "trick" of cross multiplication:
- 9/15 is less than 10/15
- So 3/5 is less than 2/3.
Method 3 of 3: Ordering fractions greater than one
Step 1. This method is useful fractions with a numerator equal to or greater than the denominator
8/3 is an example of this type of fraction. You can also use this feature for fractions with the same numerator and denominator, such as 9/9. Both are examples of improper fractions.
You can still use other methods for these fractions. But this one in particular can help you get to the solution faster
Step 2. Convert each improper fraction into a mixed number
Turn them into a mix of whole numbers and fractions. Sometimes you might be able to do this in your head. For example, 9/9 = 1. At other times, it is better to use long division to find out how many times the denominator fits into the numerator. What is left of this division is "left over" as a fraction. For example:
- 8/3 = 2 + 2/3
- 9/9 = 1
- 19/4 = 4 + 3/4
- 13/6 = 2 + 1/6
Step 3. Work with whole numbers only
Now that there are no improper fractions, you will have a better idea of the value of each one. Ignore fractions for now and sort fractions into groups like whole numbers:
- 1 is the smallest
- 2 + 2/3 and 2 + 1/6 (we still don't know which is the biggest)
- 4 + 3/4 is the biggest of them all
Step 4. If necessary, compare the fractions from each group
If you have several mixed numbers with the same whole number, such as 2 + 2/3 and 2 + 1/6, compare the fraction part of the number to see which is larger. You can use any of the methods shown above to do this. Here is an example of comparing 2 + 2/3 and 2 + 1/6, converting fractions to the same denominator:
- 2/3 = (2x2)/(3x2) = 4/6
- 1/6 = 1/6
- 4/6 is greater than 1/6.
- 2 + 4/6 is greater than 2 + 1/6.
- 2 + 2/3 is greater than 2 + 1/6.
Step 5. Use the results to sort the entire list of mixed numbers
Once you have solved the fractions in each group of mixed numbers, you can sort your entire list: 1, 2 + 1/6, 2 + 2/3, 4 + 3/4.
Step 6. Convert the mixed numbers back to the original fractions
Keep the same order but undo the changes you made and write the numbers as the original improper fractions: 9/9, 8/3, 13/6, 19/4.
- When sorting a large number of fractions, it can be helpful to compare and sort into smaller groups of 2, 3, or 4 fractions at a time.
- Finding the lowest common denominator is helpful so you can work with smaller numbers, as any common denominator will work. Try sorting 2/3, 5/6 and 1/3 using a common denominator of 36, and see if you can get the same result.
- If the numerators are all the same, you can sort them in descending order of denominator. For example, 1/8 < 1/7 < 1/6 < 1/5. Think of it like a pizza: if you were to compare ½ to 1/8, you're comparing a pizza cut into 8 slices instead of 2.