While it's easy to sort whole numbers like 1, 3, and 8 by size, fractions can be difficult to measure at first glance. If each minor number, or denominator, is the same, you can order them as whole numbers, for example: 1/5, 3/5, and 8/5. Otherwise, you can alter your list of fractions to use the same denominator without changing the size of any of them. It will get easier with practice and you can also learn a few tricks when comparing two fractions or when solving “improper” fractions with a very large top number like 7/3.
Steps
Method 1 of 3: Order any number of fractions
Step 1. Find the common denominator for all the fractions
Use some of the methods that we will present to find the denominator, or the lowest number of a fraction, that you can use to rewrite all the fractions in the list so you can easily compare them. This is called the common denominator, or the lowest common denominator, if it is the smallest possible:
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Multiply all the different denominators. For example, if you are comparing 2/3, 5/6, and 1/3, multiply the two different denominators: 3 x 6 =
Step 18.. This method is simple, but generally provides a much larger number than the other methods, which may be more difficult to work with.
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If not, list the multiples of each denominator in a separate column until you see the number repeating in all the columns. Use that number. For example, to compare 2/3, 5/6, and 1/3, list a few multiples of 3: 3, 6, 9, 12, 15, 18. Then, list the multiples of 6: 6, 12, 18. Like
Step 18. appears in both lists, you will have to use that number.
Step 2. Convert each fraction so that it uses the common denominator
Remember: if you multiply the top and bottom of a fraction by the same number, the fraction will still be the same size. Use this technique on each fraction, one by one, so that the bottom number of each of the fractions is the common denominator. Test it for 2/3, 5/6, and 1/3 using the 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. Use the number above to order the fractions
Now that they all have the same denominator, it will be easy to compare them. Use the top number, or numerator, to order them from least to greatest. By sorting the fractions in the previous example, we got the following: 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 them back to their original form. You can do this by remembering how you transformed each fraction or by dividing the top and bottom of each fraction again:
- 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: Order Two Fractions Using Cross Multiplication
Step 1. Write two fractions next to each other
For example, compare the fraction 3/5 and the fraction 2/3. Write them side by side on a page: 3/5 on the left and 2/3 on the right.
Step 2. Multiply the top number of the first fraction with the bottom number of the second fraction
In our example, the top number, or numerator, of the first fraction (3/5) is
Step 3.. The bottom number, or denominator, of the second fraction (2/3) is also e
Step 3.. Multiply them: 3 x 3 =?
This method is called cross multiplication, because the numbers are multiplied diagonally from each other
Step 3. Write the answer next to the first fraction
Write the product, or the answer to the multiplication problem, next to the first fraction of the page. In our example: 3 x 3 = 9, so you will have to write
Step 9. next to the first fraction located on the left hand side of the page.
Step 4. Multiply the top number of the second fraction by the bottom number of the first
To find out which fraction is greater, we will have to compare our previous answer with the answer to another multiplication problem. Multiply these two numbers. For our example (comparing 3/5 and 2/3), multiply 2 x 5.
Step 5. Write this answer next to the second fraction
Write the answer to this second multiplication problem next to the second fraction. In this example, the answer is 10.
Step 6. 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 greater than the other, then the fraction next to that cross product will also be greater than the other fraction. In our example, since 9 is less than 10, this means that 3/5 must be less than 2/3.
Remember that you should always write the cross product next to the fraction whose number above you have used
Step 7. Understand why it works
To compare two fractions, you usually have to transform them so that they have the same denominator (the bottom of a fraction). Secretly, this is what cross multiplication does! It only skips the step of having to write the denominators, since when the two fractions have the same, you will only have to compare the two numbers above. This is our example (3/5 vs 2/3) written without the cross multiplication “shortcut”:
- 3/5 = (3x3) / (5x3) = 9/15
- 2/3 = (2x5) / (3x5) = 10/15
- 9/15 is less than 10/15
- Therefore 3/5 is less than 2/3
Method 3 of 3: Order Fractions Greater Than One
Step 1. Use this method for those fractions whose top numbers are equal to or greater than the bottom numbers
If a fraction has a top number, or numerator, greater than the bottom number, or denominator, it will be greater than one. 8/3 is an example of this type of fraction. You can also use this method for those fractions whose numerators and denominators are the same, such as 9/9. These two types of fractions are examples of improper fractions.
You can still use the other methods for these fractions, but this method will help you make sense of those fractions and you may be able to do them more quickly
Step 2. Convert each improper fraction to a mixed number
Convert them to a mixture of whole numbers and fractions. Sometimes you can do it mentally, for example 9/9 = 1. However, other times you have to do long divisions to find out how many times the numerator goes evenly with the denominator. The remainder of that long division problem, if there is one, will “stay” 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. Order the mixed numbers by whole numbers
Now that you no longer have improper fractions, you will have a better idea of the size of each number. Ignore the fractions for now and classify the fractions into groups of whole numbers.
- 1 is the smallest
- 2 + 2/3 and 2 + 1/6 (we still don't know which is greater)
- 4 + 3/4 is the greatest
Step 4. If necessary, compare the fractions in each group
If you have multiple 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 greater. You can use any of the methods that we have mentioned in the other sections to do it. Below we will present an example in which 2 + 2/3 and 2 + 1/6 are compared where fractions are converted 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
When you have classified the fractions in each group of mixed numbers, you can sort the entire list: 1, 2 + 1/6, 2 + 2/3, 4 + 3/4.
Step 6. Convert the mixed numbers back to their original fractions
Keep the same order, but undo the changes you have made and write the original improper fractions: 9/9, 8/3, 13/6, 19/4.
Advice
- When ordering a large number of fractions, it can be helpful if you compare and order smaller groups of 2, 3, or 4 fractions at a time.
- While figuring out the lowest common denominator is helpful so that you can work with smaller numbers, any common denominator will do. Try to order 2/3, 5/6, and 1/3 with a common denominator of 36 and see if you get the same result.
- If all the numerators are the same, you can order them in the reverse order of the denominator, for example: 1/8 <1/7 <1/6 <1/5. Imagine it is a pizza: if you go from 1/2 to 1/8, you will be cutting the pizza into 8 slices and not 2, so your slice will be much smaller.