Comparing Fractions with the Same Denominator - Third Grade Mathematics

When two fractions share the exact same denominator, comparing them is as natural as counting whole numbers. Because the denominators are identical, the size of each fractional piece is exactly the same; you only need to determine who has more pieces! In this chapter, you will master comparing fractions with like denominators using visual models, number lines, and mathematical symbols.

The Rule for Comparing Fractions with the Same Denominator

When the denominators are the same, compare the numerators: The fraction with the greater numerator represents the greater quantity!

Rule: If denominators are equal, then:
Larger Numerator = Larger Fraction!

Why is this rule always true? Think about slices of pizza: Suppose two pizzas are both cut into 8 equal slices (eighths).
- Person A takes 3 slices (3/8).

- Person B takes 5 slices (5/8).
Because every slice is the exact same size (1/8), 5 slices is obviously more than 3 slices! Therefore: 5/8 > 3/8.

Visual Model (Same size pieces):
3/8: [ * ] [ * ] [ * ] [   ] [   ] [   ] [   ] [   ]  (3 pieces)
5/8: [ * ] [ * ] [ * ] [ * ] [ * ] [   ] [   ] [   ]  (5 pieces)

5/8 has 2 more pieces than 3/8.

Comparing on a Number Line

On a number line, both fractions belong on the exact same scale with identical interval spacing. The fraction that is further to the right (farther from 0) is always the greater number.

Comparing 2/6 and 5/6:

0       1/6     2/6     3/6     4/6     5/6     6/6 (1)
+-------+-------+-------+-------+-------+-------+
                ^                       ^
               2/6                     5/6
           (Closer to 0)          (Closer to 1)

Because 5/6 is farther to the right on the number line than 2/6, we write: 5/6 > 2/6 or 2/6 < 5/6.

Ordering Sets of Fractions with Common Denominators

Ordering three or more fractions with common denominators is simple: arrange their numerators in order!

Example: Order from least to greatest:
5/8,   1/8,   7/8,   3/8

Step 1: Check denominators. All are 8 (eighths).
Step 2: Look at numerators: 1, 3, 5, 7.
Step 3: Arrange from least to greatest:
1/8  <  3/8  <  5/8  <  7/8

Chapter Practice Exercises

  1. Compare using >, <, or =: a. 2/4 ___ 3/4 b. 5/6 ___ 1/6 c. 3/8 ___ 7/8 d. 4/5 ___ 2/5 e. 6/6 ___ 4/6
  2. Order these fractions from least to greatest: 4/10, 1/10, 9/10, 6/10
  3. Order these fractions from greatest to least: 5/6, 2/6, 6/6, 1/6
  4. Aiden and Zoe had identical chocolate bars. Aiden ate 3/6 of his bar, and Zoe ate 4/6 of her bar. a. Who ate more chocolate? b. Write a comparison sentence using < or >.
  5. Draw a number line from 0 to 1 divided into fourths. Plot 1/4 and 3/4. Use the number line to explain why 3/4 > 1/4.

Solutions and Step-by-Step Answers

  1. Comparisons: a. 2/4 < 3/4 (2 is less than 3). b. 5/6 > 1/6 (5 is greater than 1). c. 3/8 < 7/8 (3 is less than 7). d. 4/5 > 2/5 (4 is greater than 2). e. 6/6 > 4/6 (6 is greater than 4).
  2. Least to greatest: 1/10 < 4/10 < 6/10 < 9/10.
  3. Greatest to least: 6/6 > 5/6 > 2/6 > 1/6.
  4. Aiden vs. Zoe: a. Both bars were split into sixths. Zoe ate 4 pieces while Aiden ate 3 pieces, so Zoe ate more chocolate. b. Comparison sentence: 4/6 > 3/6 (or 3/6 < 4/6).
  5. Number line partitioned into fourths: Tick marks at 0, 1/4, 2/4, 3/4, 1. 1/4 is 1 hop from 0, while 3/4 is 3 hops from 0. Since 3/4 is located further to the right toward 1, 3/4 > 1/4.