While it's easy to sort whole numbers (such as 1, 3, and 8), arranging fractions in ascending order can sometimes be confusing. If the number in the denominator is the same, you can arrange the fractions taking into account only the numerator, ordering them as you would with whole numbers (e.g. 1/5, 3/5 and 8/5). Otherwise, you must transform all fractions to the same denominator, without changing the value of the fraction. It becomes simple with practice and you can learn a couple of tricks to use when you only have to compare two fractions or you find yourself with improper fractions, i.e. with a numerator greater than the denominator, such as 7/3.
Steps
Method 1 of 3: Order any number of fractions
Step 1. Find the common denominator for all fractions
Use one of these methods to find the denominator to use to rewrite each fraction of the list, so you can compare them. It is called "common denominator" or "lowest common denominator" if it is the lowest possible.
- Multiply the different denominators together. For example, if you are comparing 2/3, 5/6 and 1/3, multiply the two different denominators: 3 x 6 = 18. This method is very simple, but still much more effective than other methods where it can be more difficult. work.
- Or list the multiples of each denominator in a separate column, until you meet the same number common to each column, then use this number. For example, if you are comparing 2/3, 5/6 and 1/3, list some multiples of 3: 3, 6, 9, 12, 15, 18. You can list those of 6: 6, 12, 18. Since appears 18 in both lists, use that number (you could also use 12, but in the example below we'll assume you're using 18).
Step 2. Convert each fraction to use the common denominator
Remember that if you multiply the numerator and denominator by the same number, the resulting fraction is equivalent to the given one, that is, it represents the same quantity. Use this technique for each fraction, one by one, so that each is expressed with the common denominator. Try it with 2/3, 5/6 and 1/3, using 18 as the common denominator:
- 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 numerator to reorder the fractions
Now that they all have the same denominator, it's easy to compare them. Take their numerators into account to arrange them from smallest to largest. Sorting the previous fractions, we get: 6/18, 12/18, 15/18.
Step 4. Return each fraction to its original form
Keep the fractions in the same order, but restore them to how they were initially. You can do this by remembering how each fraction has been transformed or by simplifying the numerator and denominator of each fraction:
- 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 the fraction 3/5 with the fraction 2/3. Write them side by side 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
In our example, the numerator of the first fraction (3/5) is 3. The denominator of the second fraction (2/3) is again 3. Multiply them together: 3 x 3 = 9.
This method is called "cross multiplication", because the numbers are multiplied along diagonal lines that cross
Step 3. Write your answer on the paper, next to the first fraction
In our example, 3 x 3 = 9, so you have to 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 need to compare the previous answer with the result of another product. Multiply these two numbers together. In our example (comparison between 3/5 and 2/3), multiply 2 and 5 together.
Step 5. Write the result of this second multiplication next to the second fraction
In this example, the answer is 10.
Step 6. Compare the values of the two “cross products”
The results of the multiplication calculations in this method are called “cross products”. If one cross product is larger than another, then the fraction next to that cross product is also greater than the other fraction. In our example, since 9 is less than 10, it means that 3/5 must be less than 2/3.
Remember: always write the cross product next to the fraction whose numerator you used
Step 7. Try to understand why it works
To compare two fractions, they typically transform to give them the same denominator. Actually, this is just what cross multiplication does! Just avoid writing the denominators, as once the two fractions have the same denominator, you will only have to compare the two numerators. Here is our own example (3/5 vs 2/3) written without the "shortcut" of cross multiplication:
- 3/5 = (3x3) / (5x3) = 9/15
- 2/3 = (2x5) / (3x5) = 10/15
- 9/15 is less than 10/15
- Consequently, 3/5 is less than 2/3.
Method 3 of 3: Sorting Fractions Greater than One
Step 1. Use this method for fractions with a numerator equal to or greater than the denominator
If a fraction has a numerator (the number above the fraction line) greater than the denominator (the number below), 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 fractions are examples of "improper fractions".
You can still use the other methods for these fractions. This method helps make sense of these fractions, however, and may be faster
Step 2. Convert any improper fraction to a mixed number
Change them all into whole numbers and fractions. Sometimes you may be able to do this in your head. For example, 9/9 = 1. Otherwise you will have to use long divisions to find how many times the denominator is in the numerator. The remainder, if any, is left in the form of a fraction. For instance:
- 8/3 = 2 + 2/3
- 9/9 = 1
- 19/4 = 4 + 3/4
- 13/6 = 2 + 1/6
Step 3. Sort the mixed numbers by whole number
Now that you have no more improper fractions, you can better understand the magnitude of each number. For now, ignore fractions and order them into integer groups:
- 1 is the smallest
- 2 + 2/3 and 2 + 1/6 (we still don't know which is the greater of the two)
- 4 + 3/4 is the largest
Step 4. If necessary, compare the fractions in each group
If you have multiple mixed numbers with the same integer, 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 presented in the other sections. Here is an example comparing 2 + 2/3 and 2 + 1/6, converting the 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 your entire list of mixed numbers
Once you have sorted out the fractions in each group of mixed numbers, you can sort the whole list: 1, 2 + 1/6, 2 + 2/3, 4 + 3/4
Step 6. Convert the mixed numbers to their original fractions
Keep the same order, but cancel the changes made and write the numbers as improper fractions of origin: 9/9, 13/6, 8/3, 19/4.
Advice
- When you have to sort a large number of fractions, it can be helpful to compare and sort smaller groups of 2, 3, or 4 fractions at a time.
- While agreeing that the lowest common denominator is useful for working with smaller numbers, any common denominator will do. Try sorting 2/3, 5/6 and 1/3 using 36 as the common denominator and see if you get the same result.
- If the numerators are all the same, you can put the denominators in reverse order. For example, 1/8 <1/7 <1/6 <1/5. Think of a pizza: if you go from 1/2 to 1/8, you cut the pizza into 8 slices instead of 2 and the single slice you spot is much smaller.
