Comparing Fractions with the Same Numerator - Third Grade Mathematics

Comparing fractions when the numerators are identical requires a special type of thinking that often catches beginners by surprise. When the numerators are the same, you are comparing the same number of pieces, but the pieces themselves are completely different sizes! In this chapter, you will master the denominator size rule and learn how to compare fractions like 3/4 and 3/8 with complete confidence.

The Counter-Intuitive Truth About Denominators

Here is the central question: If you have 1 slice of a cake cut into 3 pieces (1/3), or 1 slice of the same cake cut into 8 pieces (1/8), which slice is bigger?

The slice of 1/3 is MUCH bigger!
- Slicing a cake into only 3 pieces makes large, thick slices.

- Slicing a cake into 8 pieces makes thin, small slices.


Unit Fractions Visualized:
+---------------------------------------------------------------+
|                       1 Whole Cake                            |
+-----------------------+-----------------------+---------------+
|          1/3          |          1/3          |      1/3      |
+-------+-------+-------+-------+-------+-------+-------+-------+
|  1/8  |  1/8  |  1/8  |  1/8  |  1/8  |  1/8  |  1/8  |  1/8  |
+-------+-------+-------+-------+-------+-------+-------+-------+

Look at one piece of each:
One 1/3 piece is much larger than one 1/8 piece!

The Rule for Comparing Fractions with the Same Numerator

When two fractions have the same numerator, you have the same number of parts. Therefore, the fraction with the SMALLER denominator is the GREATER fraction!

Rule: If numerators are equal:
Smaller Denominator = Larger Piece = GREATER FRACTION!
Larger Denominator  = Smaller Piece = LESSER FRACTION!

Let us compare 3/4 and 3/8:
- Both fractions have the same numerator: 3 pieces.

- In 3/4, each piece is a fourth (1/4).

- In 3/8, each piece is an eighth (1/8).
Since fourths are much bigger than eighths, having 3 fourths gives you more than having 3 eighths! Therefore: 3/4 > 3/8.

Visual Comparison of 3/4 vs. 3/8:
3/4: [=== 1/4 ===][=== 1/4 ===][=== 1/4 ===][           ]  (Takes up most of bar!)
3/8: [=1/8=][=1/8=][=1/8=][     ][     ][     ][     ][     ]  (Takes up less than half!)

Comparing Different Numerators and Denominators

Notice the difference between the two comparison rules:
- Same Denominators (e.g., 3/8 vs. 5/8): Denominators are the same, so larger numerator wins (5/8 > 3/8).

- Same Numerators (e.g., 2/3 vs. 2/6): Numerators are the same, so smaller denominator wins (2/3 > 2/6).


Always check first whether the numerators match or the denominators match!

Chapter Practice Exercises

  1. Compare using >, <, or =: a. 1/3 ___ 1/5 b. 2/6 ___ 2/4 c. 3/8 ___ 3/4 d. 4/5 ___ 4/10 e. 5/6 ___ 5/8
  2. Order these fractions from least to greatest: 2/8, 2/3, 2/6, 2/4
  3. Order these fractions from greatest to least: 3/4, 3/10, 3/6, 3/3
  4. Jordan ran 2/3 of a mile. Caleb ran 2/5 of a mile. a. Who ran the greater distance? b. Explain your answer using the size of the fractional pieces.
  5. A student says: "3/8 must be bigger than 3/4 because 8 is bigger than 4." Explain why this student's reasoning is incorrect.

Solutions and Step-by-Step Answers

  1. Comparisons: a. 1/3 > 1/5 (thirds are larger pieces than fifths). b. 2/6 < 2/4 (fourths are larger pieces than sixths, so 2 fourths is greater). c. 3/8 < 3/4 (fourths are larger than eighths). d. 4/5 > 4/10 (fifths are larger than tenths). e. 5/6 > 5/8 (sixths are larger than eighths).
  2. Least to greatest: Largest denominator means smallest pieces: 2/8 < 2/6 < 2/4 < 2/3.
  3. Greatest to least: Smallest denominator means largest pieces: 3/3 > 3/4 > 3/6 > 3/10.
  4. Jordan vs. Caleb: a. Jordan ran the greater distance (2/3 > 2/5). b. Both boys ran 2 parts of a mile. But thirds (1/3) are larger pieces of a mile than fifths (1/5) because the mile is split into fewer total sections.
  5. The student is confusing whole numbers with denominators. In whole numbers, 8 is greater than 4. But in fractions, the denominator represents how many equal parts the whole is cut into. Cutting a whole into 8 pieces produces smaller pieces than cutting the same whole into 4 pieces. Therefore, 3 fourths is greater than 3 eighths.