Comparing Fractions

rectangle three fourths blue nonroutine

Example 1: Drake rode his bike for three-fourths of a mile and Josh rode his bike for one-fourth of a mile. Which boy rode his bike farther?

Drake: line three fourths

Josh: line one fourth

Analysis:

3over4.gif  ?  1over4.gif

These fractions have like denominators, so we can compare the numerators.

Solution:

3over4.gif  >  1over4.gif

Since three is greater than one, three-fourths is greater than one-fourth. Therefore, Drake rode his bike farther.

When comparing two fractions with like denominators, the larger fraction is the one with the greater numerator. Let's look at some more examples of comparing fractions with like denominators.

Example 2:  Compare the fractions given below using the symbols <, > or =.
a)
1over3.gif  ?  2over3.gif   1over3.gif   2over3.gif
b)
3over2.gif  ?  1over2.gif   3over2.gif   1over2.gif
c)
3over4.gif  ?  7over4.gif   3over4.gif   7over4.gif
d)
6over6.gif  ?  5over6.gif   6over6.gif   5over6.gif
e)
4over3.gif  ?  5over3.gif   4over3.gif   5over3.gif
f)
5over5.gif  ?  5over5.gif   5over5.gif   5over5.gif

Example 3: Josephine ate three-fourths of a pie and Penelope ate two-thirds of a pie. If both pies are the same size, then which girl ate more pie?

Analysis:

3over4.gif  ?  2over3.gif

These fractions have unlike denominators (and unlike numerators). It would be easier to compare them if they had like denominators. We need to convert these fractions to equivalent fractions with a common denominator in order to compare them more easily.

Josephine: compare_example3_3over4.gif compare_example3_3over4_partb.gif

Penelope: compare_example3_2over3.gif compare_example3_2over3_partb.gif

Solution: compare_example3_partc.gif

Since nine-twelfths is greater than eight-twelfths, three-fourths is greater than two-thirds. Therefore, Josephine ate more pie.

The example above works out nicely! But how did we know to use 12 as our common denominator? It turns out that the least common denominator is the best choice for comparing fractions.

Definition: The least common denominator (LCD) of two or more non-zero denominators is the smallest whole number that is divisible by each of the denominators.

To find the least common denominator (LCD) of two fractions, find the least common multiple (LCM) of their denominators.

Remember that "..." at the end of each list of multiples indicates that the list goes on forever. Revisiting example 3, we found that the least common multiple of 3 and 4 is 12. Therefore, the least common denominator of two-thirds and three-fourths is 12. We then converted each fraction into an equivalent fraction with a denominator of 12, so that we could compare them.

Josephine: compare_example3_3over4.gif compare_example3_3over4_partb.gif

Penelope: compare_example3_2over3.gif compare_example3_2over3_partb.gif

Procedure: To compare fractions with unlike denominators, follow these steps:

1. Use the LCD to write equivalent fractions with a common denominator.

2. Compare the numerators: The larger fraction is the one with the greater numerator.

Let's look at some more examples of comparing fractions with unlike denominators.

Example 4:   compare_example4.gif
Analysis: Convert these fractions to equivalent fractions with a common denominator in order to compare them.
Step1:
Find the least common multiple (LCM) of 8 and 10.
multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72, 80,...
multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80,...
The LCM of 8 and 10 is 40.
Analysis: compare_example4_analysis.gif
Step 2:
Convert each fraction to an equivalent fraction with a denominator of 40.
compare_example4_5over8.gif   compare_example4_5over8_step2.gif
compare_example4_7over10.gif   compare_example4_7over10_step2.gif
Answer: compare_example4_answer.gif
Example 5:   compare_example5.gif
Analysis: Convert these fractions to equivalent fractions with a common denominator in order to compare them.
Step 1:
Find the least common multiple (LCM) of 6 and 4.
multiples of 6 are 6, 12, 18, 24, 30, 36,...
multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36,...
The LCM of 6 and 4 is 12.
Analysis: compare_example5_analysis.gif
Step 2:
Convert each fraction to an equivalent fraction with a denominator of 12.
compare_example5_5over6.gif   compare_example5_5over6_step2.gif
compare_example5_3over4.gif   compare_example5_3over4_step2.gif
Answer: compare_example5_answer.gif
Example 6:   compare_example6.gif
Analysis: Convert these fractions to equivalent fractions with a common denominator in order to compare them.
Step 1:
Find the least common multiple (LCM) of 9 and 3.
multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72,...
multiples of 3 are 3, 6, 9, 12, 18, 21, 24, 27, 30, 33, 36,...
The LCM of 9 and 3 is 9.
Analysis: compare_example6_analysis.gif
Step 2:
Convert each fraction to an equivalent fraction with a denominator of 9.
compare_example6_14over9.gif   compare_example6_14over9_step2.gif
compare_example6_5over3.gif   compare_example6_5over3_step2.gif
Answer: compare_example6_answer.gif

In this lesson, we have compared fractions with like denominators and with unlike denominators. Let's see what happens when we compare fractions with like numerators. Look at the shaded rectangles below.

rectangle one half red 1over2.gif

rectangle one third pink 1over3.gif

rectangle one fourth blue 1over4.gif

rectangle one fifth green 1over5.gif

The fractions above all have the same numerator. (Each of these fractions is called a unit fraction.) As the denominator gets larger, the fraction gets smaller. To compare fractions with like numerators, look at the denominators. The fraction with the smaller denominator is the larger fraction. Let's look at some examples.

compare_example7_again.gif
1over2.gif  >  1over3.gif

Since one-half has the smaller denominator, it is the larger fraction.

compare_example8a.gif
rectangle three fifths green 3over5.gif
rectangle three fourths blue 3over4.gif
3over5.gif  <  3over4.gif

Since three-fourths has the smaller denominator, it is the larger fraction.

compare_example9.gif
5/3 (five-thirds)

circle three thirds pink_2.gif

circle_two_thirds_pink_rotated_1.gif

5/4 (Five-fourths)

circle_four_fourths_blue_2.gif

circle_one_fourth_blue_2.gif

5over3_1.gif  >  5over4.gif

Since five-thirds has the smaller denominator, it is the larger fraction. Remember, when comparing fractions with like numerators, the fraction with the smaller denominator is the larger fraction. Let's look at some more examples of comparing fractions with like numerators.

Example 10:  Compare the fractions given below using the symbols <, > or =.
a)
1over5.gif  ?  1over3.gif   1over5.gif   1over3.gif
b)
3over4.gif  ?  3over5.gif   3over4.gif   3over5.gif
c)
5over3_1.gif  ?  5over2.gif   5over3_1.gif   5over2.gif
d)
2over7_1.gif  ?  2over9.gif   2over7_1.gif   2over9.gif
e)
6over6.gif  ?  6over5.gif   6over6.gif   6over5.gif
compare_example11.gif
3/3 (Three-thirds)

circle three thirds pink

4/4 (Four-fourths)

circle four fourths blue

3over3_1.gif  =  4over4_1.gif

Note that the improper fractions in example 11 are equivalent. This is because for each fraction, the numerator is equal to its denominator. So, each fraction is equivalent to 1. We have looked at many examples in this lesson. Let's try to summarize what we have learned.

R U L E S  F O R  C O M P A R I N G  F R A C T I O N S

Relationship: Like Denominators

How To Compare: Look at the numerators. The larger fraction is the one with the greater numerator.

Example: 

3over4.gif  >  1over4.gif

Relationship: Unlike Denominators

How To Compare: Convert each fraction to an equivalent fraction with a common denominator. The larger fraction is the one with the greater numerator.

​Example: 

compare_example4_answer_small.gif

Relationship: Like Numerators

How To Compare: Look at the denominators. The fraction with the smaller denominator is the larger fraction.

​Example:

2over7_1.gif  >  2over9.gif

Summary: In this lesson, we learned how to compare fractions with like denominators, with unlike denominators, and with like numerators. To compare fractions with unlike denominators, use the LCD to write equivalent fractions with a common denominator; then compare the numerators.


Exercises

In Exercises 1 through 5, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Note: To write the fraction two-thirds, enter 2/3 into the form.

1. Jill jogged for three-tenths of a mile and Jane jogged for seven-tenths of a mile. Which girl jogged farther?
 
 
ANSWER BOX:   

RESULTS BOX: 

2. A magazine sells one advertisement that is seven-eighths of a page and another advertisement that is five-sixths of a page. What is the LCD of these two fractions?
 
 
ANSWER BOX:   

RESULTS BOX: 

3. Which fraction from exercise 2 represents the larger advertisement? (Write your answer in lowest terms.)
 
 
ANSWER BOX:   

RESULTS BOX: 

4. Compare two-ninths and one-sixth by using the LCD to write equivalent fractions. Then write the smaller fraction in lowest terms.
 
 
ANSWER BOX:   

RESULTS BOX: 

5. Which is greater: nine-tenths or nine-ninths? (Write the fraction below.)
 
 
ANSWER BOX:   

RESULTS BOX: