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Lesson 5 of 9

Fractions · Lesson 5 of 9

Comparing and Ordering Fractions

“Compare shares by reasoning about their units, then order fractions using a common denominator.”

Learning Objectives

• Compare fractions with the same denominator or the same numerator. • Use equivalent fractions to compare different fractional units. • Choose a common denominator using common multiples. • Arrange fractions in ascending and descending order. • Justify a comparison with a sharing model or a number line.

First ask what stays the same

Which group gives each child more chikki: one chikki shared by two children, or five chikkis shared by eight? The shares are 1/2 and 5/8. Simply looking at the numerator or denominator cannot settle this pair. We need to compare the amounts. Some comparisons become easy when either the number of pieces or the size of each piece stays the same.

If the denominator is the same, the fractional unit is the same. Four sevenths and five sevenths count copies of 1/7, so five sevenths is larger. In a sharing story, keeping the number of children unchanged while increasing the food supply increases each child’s share. Thus 1/5 < 2/5 and 3/7 < 4/7.

4/75/7
Same-sized units, different counts— Both strips use sevenths. The second contains one more shaded seventh.
Example — equal numbers of children

Problem
Compare 4 glasses of juice shared among 7 children with 5 glasses shared among 7 children.

  1. 1.The shares are 4/7 and 5/7 of a glass.
  2. 2.Both are measured in seventh-glasses. Five such units exceed four.
  3. 3.Therefore 5/7 > 4/7, so the group with five glasses gets the larger share.

If the numerator is the same, imagine sharing the same food supply among different numbers of children. More children give smaller shares. Four chikkis among seven children gives more per child than four chikkis among eight. Therefore 4/7 > 4/8. This extends the unit-fraction comparison: the counted amount of food remains fixed while the number of recipients changes.

Example — equal food supplies

Problem
Explain why 1/2 is smaller than 4/7.

  1. 1.Rewrite 1/2 as 4/8 by multiplying both numbers by 4.
  2. 2.Now compare 4/8 with 4/7: four chikkis shared by seven children give larger shares than four chikkis shared by eight.
  3. 3.So 4/7 > 4/8 = 1/2. Equivalence makes the comparison easier without changing either amount.

Use the same fractional unit

With different denominators and numerators, compare equivalent fractions that use the same fractional unit. For instance, one half is four eighths. Five eighths is then clearly larger than one half because five eighth-pieces exceed four eighth-pieces. This method works because equivalence preserves size and a shared denominator gives equal-sized pieces to count.

Example — two chikki groups

Problem
Which is more: 1 chikki among 2 children or 5 chikkis among 8 children?

  1. 1.The first share is 1/2; the second is 5/8.
  2. 2.Convert the first to eighths: 1/2 = 4/8.
  3. 3.Since 5 > 4, 5/8 > 4/8 = 1/2. Each child in the second group receives more.

A common denominator is a number that both denominators divide exactly. It is a common multiple of them. The product of the denominators always works, though a smaller common multiple often reduces the calculation. We multiply each numerator by the same factor used to change its denominator, so each original fraction keeps its value.

Definition
Common denominator

A denominator used to express two or more fractions with the same fractional unit. It can be chosen as a common multiple of their original denominators.

Example — three quarters and seven tenths

Problem
Compare 3/4 and 7/10 using two choices of common denominator.

  1. 1.The product 4 × 10 = 40 is a common multiple. Multiply 3/4 by 10/10 to get 30/40, and 7/10 by 4/4 to get 28/40.
  2. 2.Thirty fortieths exceed twenty-eight fortieths, so 3/4 > 7/10.
  3. 3.A smaller common multiple is 20: 3/4 = 15/20 and 7/10 = 14/20.
  4. 4.The same comparison follows from 15 > 14. Both methods work because they give equal-sized units within each comparison.
Equivalent is not automatically comparable by numerators

3/4 = 6/8 and 7/10 = 21/30, but comparing 6 with 21 is not enough: eighths and thirtieths differ in size. The equivalent fractions must have the same denominator before their numerators alone decide the comparison.

Compare close fractions carefully

A picture can suggest which fraction is greater, but nearby amounts may look almost identical. A common denominator allows an exact comparison. This works for fractions below one, fractions above one, or a mixture. If a fraction represents a mixed number, you can first write it as a single fraction and then use the same method.

Example — four fifths and seven ninths

Problem
Which is greater, 4/5 or 7/9?

  1. 1.Use 45 as a common denominator, since 45 = 5 × 9.
  2. 2.Change 4/5 to 36/45 by multiplying both numbers by 9.
  3. 3.Change 7/9 to 35/45 by multiplying both numbers by 5.
  4. 4.Since 36 > 35, 4/5 > 7/9. The difference in their counts is only one forty-fifth.
Example — ninths and twenty-firsts

Problem
Compare 7/9 with 17/21.

  1. 1.The multiples 9 × 7 and 21 × 3 are both 63.
  2. 2.Write 7/9 = 49/63 and 17/21 = 51/63.
  3. 3.Since 49 < 51, 7/9 < 17/21. The denominator 63 is smaller than the product 189 and is sufficient.

There are two stages to the general comparison method. First express every amount with the same denominator. Then compare how many of those units each amount contains. You can check the conclusion on a number line: larger fractions lie farther to the right. A common denominator is a tool for making a comparison, not a requirement that the final fractions must keep that form.

Arrange several fractions in order

Ascending order means smallest to largest, while descending order means largest to smallest. Find one common unit for all the fractions, arrange their numerator counts in the required direction, and finally write the original fractions in that order. Keeping each equivalent fraction paired with its original name prevents an ordering mistake.

Example — ascending order

Problem
Arrange 7/10, 11/15, and 2/5 from smallest to largest.

  1. 1.Choose 30, a common multiple of 10, 15, and 5.
  2. 2.Write 7/10 = 21/30, 11/15 = 22/30, and 2/5 = 12/30.
  3. 3.The counts increase as 12 < 21 < 22.
  4. 4.Return to the original names: 2/5 < 7/10 < 11/15.
Example — descending order beyond one

Problem
Arrange 25/16, 7/8, 13/4, and 17/32 from largest to smallest.

  1. 1.Use denominator 32: the equivalent fractions are 50/32, 28/32, 104/32, and 17/32.
  2. 2.Descending numerator order is 104, 50, 28, 17.
  3. 3.Therefore 13/4 > 25/16 > 7/8 > 17/32. The two fractions above one correctly come before those below one.

Quiz

Quick check

Which is greater?

Quick check

Which is greater when four cakes are shared equally?

Quick check

To compare 3/4 and 7/10, which pair uses the same unit?

Quick check

Which comparison is correct?

Quick check

Which is the smallest of 7/10, 11/15, and 2/5?

Quick check

Which order is descending?

Practice Problems

Practice Problems
  1. Explain with equal shares why 1/5 < 2/5, 3/7 < 4/7, and 1/2 < 5/8.
  2. Compare 3/4 and 7/10 of a glass per child, then 4/7 and 5/7. Which comparison is immediate and why?
  3. Express each pair with a shared denominator: 7/2 and 3/5; 8/3 and 5/6; 3/4 and 3/5; 6/7 and 8/5; 9/4 and 5/2; 1/10 and 2/9; 8/3 and 11/4; 13/6 and 1/9.
  4. Compare and justify: 8/3 with 5/2; 4/9 with 3/7; 7/10 with 9/14; 12/5 with 8/5; 9/4 with 5/2.
  5. Arrange 19/24, 5/6, and 7/12 in ascending order.
  6. Arrange 25/16, 7/8, 13/4, and 17/32 in descending order. Then arrange 3/4, 12/5, 7/12, and 5/4 in descending order.
  7. A student claims 6/8 is less than 21/30 because 6 < 21. Explain the mistake and compare the fractions correctly.

Key Takeaways

Key Takeaways

• With the same denominator, compare the numerators. • With the same numerator, fewer equal recipients give a larger share. • For general comparisons, first use equivalent fractions with a common denominator. • Any common multiple works; a smaller suitable one can save effort. • Ascending means smallest first, and descending means largest first.

Previous · Lesson 4

Equivalent Fractions and Simplest Form

Next · Lesson 6

Adding Fractions