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This article was co-written by David Jia. David Jia is a tutoring teacher and founder of LA Math Tutoring, a private tutoring facility based in Los Angeles, California. With over 10 years of teaching experience, David teaches a wide variety of subjects to students of all ages and grades, as well as college admissions counseling and prep for SAT, ACT, ISEE, etc. scoring 800 in math and 690 in English on the SAT, David was awarded a Dickinson Scholarship to the University of Miami, where he graduated with a bachelor’s degree in business administration. Additionally, David has worked as an instructor in online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.
This article has been viewed 113,131 times.
While sorting integers such as 1, 3, and 8 by large or small is simple, sorting fractions can seem difficult at first glance. If the denominators are the same, you can sort them as whole numbers, for example 1/5, 3/5 and 8/5. Otherwise, you can convert the fractions to the same denominator without changing their values. This gets easier with practice, and you may learn a few “tricks” when comparing two fractions, or when sorting “unordinary” fractions with a larger numerator than a denominator like 7/ 3.
Steps
Sort any number of fractions
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Find the common denominator for all fractions. Use one of the methods below to find a denominator that you can use to rewrite all the fractions in the list, then you can easily compare them. This method is called a common denominator , or least common denominator if it is the smallest possible denominator: [1] X Research Source
- Multiply different denominators together. For example, if you are comparing three fractions 2/3, 5/6 and 1/3, multiply the two different denominators: 3 x 6 = 18 . This is a simple method, but will usually result in a much larger number than the other methods.
- Or list the multiples of each denominator in a separate column until you find a common multiple between the columns. This is the number you are looking for. For example, comparing 2/3, 5/6, and 1/3, list several multiples of 3: 3, 6, 9, 12, 15, 18. Then list multiples of 6: 6, 12, 18. Since 18 appears in both lists, we will use this number. (You can also use 12, but 18 is assumed to be used in the examples below.)
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Transform each fraction so that it uses a common denominator. Remember, if you multiply both the numerator and the denominator by the same number, the fractional value will not change. Use this technique on each fraction so that the fractions all use a common denominator. Try 2/3, 5/6, and 1/3, using a common denominator of 18:
- 18 3 = 6, so 2/3 = (2×6)/(3×6)=12/18
- 18 6 = 3, so 5/6 = (5×3)/(6×3)=15/18
- 18 3 = 6, so 1/3 = (1×6)/(3×6)=6/18
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Use numerators to sort fractions. Now all fractions have the same denominator, so they are easy to compare. Use numerators to sort them from smallest to largest. Arrange the fractions above, we have: 6/18, 12/18, 15/18.
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Convert each fraction back to its original form. Keep their order, but convert each fraction back to its original form. You can do this by remembering how each fraction was previously transformed, or divide the numerator and denominator by the number you previously multiplied:
- 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”
Sort two fractions by cross multiplying
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Write two fractions side by side. For example, compare 3/5 and 2/3. Write these two fractions side by side: 3/5 on the left, and 2/3 on the right.
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Multiply the numerator of the first fraction by the denominator of the second. In our example, the numerator of the first fraction (3/5) is 3 . The denominator of the second fraction (2/3) is also 3 . Multiply them together: 3 x 3 = ?
- This method is called cross multiplication , because you multiply the numbers diagonally between two fractions.
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Write the result next to the first fraction. Write the product of the cross multiplication next to the first fraction. In this example, 3 x 3 = 9, so you would write 9 next to the first fraction, on the left side of the page.
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Multiply the numerator of the second fraction by the denominator of the first fraction. To find out which fraction is greater, we will have to compare the product above with the product of this multiplication. Multiply these two numbers together. In this example (compare 3/5 and 2/3), multiply 2 x 5 together.
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Write the result obtained next to the second fraction. Write the result of the second multiplication next to the second fraction. In this example, the answer is 10.
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Compare the value of two cross products. The result of these two multiplications is called the cross product . If one diagonal product is greater than the other, then the fraction next to that diagonal product is also larger than the other fraction. In the example above, since 9 is less than 10, 3/5 is less than 2/3.
- Remember, always write the cross product next to the numerator of the fraction you’re comparing.
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Understand the principle of this method. To compare two fractions, you usually have to convert them to a form with the same denominator. This is the principle of the cross-multiplication method! [2] X Research Source It only skips the step of writing the denominator, because when two fractions have the same denominator, all you have to do is compare the numerators. Here’s the same example (3/5 vs 2/3), written without the cross-multiplication “shortcut”:
- 3/5=(3×3)/(5×3)=9/15
- 2/3=(2×5)/(3×5)=10/15
- 9/15 is less than 10/15
- Therefore, 3/5 is less than 2/3
Sort fractions greater than 1
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Use this method for fractions with numerators equal to or greater than the denominator. If a fraction has a numerator greater than the denominator, then it is greater than one. 8/3 is an example of this type of fraction. You can also use this method for fractions with the same numerator and denominator, such as 9/9. Both of these fraction forms are examples of irregular fractions . [3] X Research Sources
- You can still use other methods for this type of fraction. However this method makes it easier to understand, and possibly faster.
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Convert each irregular fraction into a mixed number. Convert them to a combination of integers and fractions. Sometimes you can do mental math. For example, 9/9 = 1. In other cases, work out how many times the numerator is divisible by the denominator. The remainder in that division, if any, will be in the fractional part. For example:
- 8/3 = 2 + 2/3
- 9/9 = 1
- 19/4 = 4 + 3/4
- 13/6 = 2 + 1/6
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Sort mixed numbers by integers. Now that there are no irregular fractions, you will know clearly how big each number is. Skipping the fractions part for now, sort the fractions into groups by their integer part:
- 1 is the smallest
- 2 + 2/3 and 2 + 1/6 (we don’t know which is greater than the other)
- 4 + 3/4 is the largest
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If necessary, compare the fractions in each group. If you have multiple mixed numbers with the same integer part, such as 2 + 2/3 and 2 + 1/6, compare the fractional part of the number to see which is greater. You can use any of the methods above to do this. Here’s an example of comparing 2 + 2/3 and 2 + 1/6, converting fractions to common denominators:
- 2/3 = (2×2)/(3×2) = 4/6
- 1/6 = 1/6
- 4/6 is bigger than 1/6
- 2 + 4/6 is greater than 2 + 1/6
- 2 + 2/3 is greater than 2 + 1/6
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Use your results to sort the entire list of mixed numbers. Once you’ve sorted the fractions into each group of mixed numbers, you can sort the entire list: 1, 2 + 1/6, 2 + 2/3, 4 + 3/4.
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Convert mixed numbers back to their original fraction form. Keep the same order, but change the mixed numbers to the original irregular fractions: 9/9, 8/3, 13/6, 19/4.
Advice
- If the numerators are the same, you can sort in reverse order of the denominators. For example, 1/8 < 1/7 < 1/6 < 1/5. Think of a pizza: if you go from 1/2 to 1/8, that means you’ll cut the pie into 8 pieces instead of 2, and the piece you have now is much smaller.
- When sorting a large number of fractions, you should compare and sort small groups of 2, 3, or 4 fractions at a time.
- While the least common denominator helps you work with small numbers, any common denominator is useful. Try sorting 2/3, 5/6, and 1/3 using the common denominator of 36, and see if you get the same result.
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This article was co-written by David Jia. David Jia is a tutoring teacher and founder of LA Math Tutoring, a private tutoring facility based in Los Angeles, California. With over 10 years of teaching experience, David teaches a wide variety of subjects to students of all ages and grades, as well as college admissions counseling and prep for SAT, ACT, ISEE, etc. scoring 800 in math and 690 in English on the SAT, David was awarded a Dickinson Scholarship to the University of Miami, where he graduated with a bachelor’s degree in business administration. Additionally, David has worked as an instructor in online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math.
This article has been viewed 113,131 times.
While sorting integers such as 1, 3, and 8 by large or small is simple, sorting fractions can seem difficult at first glance. If the denominators are the same, you can sort them as whole numbers, for example 1/5, 3/5 and 8/5. Otherwise, you can convert the fractions to the same denominator without changing their values. This gets easier with practice, and you may learn a few “tricks” when comparing two fractions, or when sorting “unordinary” fractions with a larger numerator than a denominator like 7/ 3.
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