Content

• Steps
• Method 1 of 3: Arbitrary Number of Fractions
• Method 2 of 3: Two Fractions (Crosswise Multiplication)
• Method 3 of 3: Incorrect Fractions
• Tips

Ordering fractions in ascending order (from lowest to highest) can be misleading because, unlike whole numbers (1, 3, 8), fractions include a numerator and denominator. It is easy to arrange fractions if they have the same denominators, for example, 1/5, 3/5, 8/5; otherwise, it is necessary to bring all fractions to a common denominator. This article will show you how to order two fractions, any number of fractions, and improper fractions (7/3).

Steps

Method 1 of 3: Arbitrary Number of Fractions

1. 1 Find common denominator, which will allow you to arrange any number of fractions. You can find just the common denominator, or the lowest common denominator (LCN). To do this, use one of the following methods:
   • Multiply the different denominators. For example, if you are ordering the fractions 2/3, 5/6, 1/3, multiply two different denominators: 3 x 6 = 18. This is an easy way, but in most cases you will not find an NOZ.
   • Or write down the multiples of each denominator, and then choose a number that appears in all lists of multiples. In our example, multiples of 3 are numbers: 3, 6, 9, 12, 15, 18; multiples of 6 are numbers: 6, 12, 18. Since the number 18 occurs in both lists, this is the common denominator of these fractions (here NOZ = 6, but we will work with the number 18).
2. 2 Bring each fraction to a common denominator. To do this, multiply the numerator and denominator of the fraction by a number equal to the result of dividing the common denominator by the denominator of a particular fraction (remember that when the numerator and denominator are multiplied by the same number, the value of the fraction does not change).For our example, bring the fractions 2/3, 5/6, 1/3 to a common denominator of 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
3. 3 Order the fractions according to their numerators (lowest to highest). In our example, the correct order would be 6/18, 12/18, 15/18.
4. 4 Without changing the order of the fractions, rewrite them in their original form. To do this, simplify them by dividing the numerator and denominator by the appropriate number.
   • 6/18 = (6 ÷ 6)/(18 ÷ 6) = 1/3
   • 12/18 = (12 ÷ 6)/(18 ÷ 6) = 2/3
   • 15/18 = (15 ÷ 3)/(18 ÷ 3) = 5/6
   • Answer: 1/3, 2/3, 5/6

Method 2 of 3: Two Fractions (Crosswise Multiplication)

1. 1 Write down two fractions next to each other. For example, order the fractions 3/5 and 2/3. Write 3/5 on the left and 2/3 on the right.
2. 2 Multiply the numerator of the first fraction by the denominator of the second fraction. In our example, multiply the numerator of the first fraction (3) by the denominator of the second fraction (3): 3 x 3 = 9.
   • This method is called "cross-multiplication" because you are multiplying the numbers on the diagonal.
3. 3 Write your result near the first fraction. In our example, write 9 around 3/5 (left).
4. 4 Multiply the numerator of the second fraction by the denominator of the first fraction. In our example: 2 x 5 = 10.
5. 5 Write the result around the second fraction. In our example, write 10 around 2/3 (right).
6. 6 Compare the two results. In our example, 9 is less than 10, so the fraction near 9 (3/5) is less than the fraction near 10 (2/3).
   • Always write the result of multiplication next to the fraction, namely above its numerator.
7. 7 Explanation of the stated method. To arrange two fractions, it is necessary to bring them to a common denominator. So cross-multiplication brings two fractions to a common denominator! Here we simply do not write the denominators, since they are the same, but immediately compare the numerators of the fractions. Here's our example without cross-multiplication:
   • 3/5 = (3x3) / (5x3) = 9/15
   • 2/3 = (2x5) / (3x5) = 10/15
   • So 3/5 is less than 2/3.

Method 3 of 3: Incorrect Fractions

1. 1 An irregular fraction is a fraction in which the numerator is greater than or equal to the denominator, for example, 8/3 or 9/9 (that is, the value of the fraction is equal to or greater than one).
   • You can use other methods for improper fractions. However, the described method is simple and fast.
2. 2 Convert each improper fraction to a mixed number. Mixed number is a type of improper fraction notation that includes whole and fractional parts. You can do this mentally (for example, 9/9 = 1) or long division. The whole result of division is written to the whole part of the mixed number, and the remainder is written to the numerator of the fractional part (the denominator does not change). For example:
   • 8/3 = 2 + 2/3
   • 9/9 = 1
   • 19/4 = 4 + 3/4
   • 13/6 = 2 + 1/6
3. 3 First, sort the mixed numbers by their whole parts (forget about fractional parts for a while).
   • 1 is the smallest number.
   • 2 + 2/3 and 2 + 1/6 - here we do not know which of these mixed numbers is greater.
   • 4 + 3/4 is the largest mixed number.
4. 4 If two mixed numbers have the same whole parts, compare their fractional parts, bringing the latter to a common denominator. In our example, for the mixed numbers 2 + 2/3 and 1/6 + 2, compare the fractional parts:
   • 2/3 = (2x2) / (3x2) = 4/6
   • 1/6 = 1/6
   • 4/6 is more than 1/6
   • 2 + 4/6 is more than 2 + 1/6
   • 2 + 2/3 is greater than 2 + 1/6
5. 5 Sort the mixed numbers in ascending order. In our example: 1, 2 + 1/6, 2 + 2/3, 4 + 3/4.
6. 6 Without changing the order of the mixed numbers, convert them back to improper fractions. In our example: 9/9, 8/3, 13/6, 19/4.

Tips

• If you are given a lot of fractions, compare and order them by breaking them into small groups (2, 3, 4 fractions).
• If the fractions have the same numerators, then write them in order, starting with the largest denominator, for example, 1/8 1/7 1/6 1/5.
• It is perfectly acceptable to compare fractions by simply reducing them to a common denominator (that is, looking for the lowest common denominator is not necessary). Try to arrange the fractions 2/3, 5/6, 1/3 using a common denominator of 36, and you will get the same result.