Comparing & Ordering Fractions - Sixth Grade Mathematics
Comparing and ordering fractions requires evaluating fractional magnitudes, identifying equivalent denominators, and using benchmark numbers to reason about relative values.
Three Structural Cases of Fraction Comparison
When comparing two fractions, inspect their numerators and denominators to determine the most efficient comparison pathway.
+-------------------------------------------------------------------------+
| THREE PATHWAYS FOR COMPARING FRACTIONS |
+-------------------+-----------------------+-----------------------------+
| Case | Structural Condition | Comparison Rule |
+-------------------+-----------------------+-----------------------------+
| Same Denominators | Unit sizes are equal | Compare numerators directly.|
| | (e.g., 3/7 vs 5/7) | Greater numerator is larger.|
+-------------------+-----------------------+-----------------------------+
| Same Numerators | Same number of pieces | Compare denominators: |
| | (e.g., 4/5 vs 4/9) | SMALLER denominator means |
| | | LARGER pieces! 4/5 > 4/9 |
+-------------------+-----------------------+-----------------------------+
| Unlike Both | Different sizes and | Find common denominator or |
| | counts (3/8 vs 5/12) | compare to benchmark (1/2) |
+-------------------+-----------------------+-----------------------------+
The Same Numerator Inverse Rule
When two fractions have identical numerators, they represent the same number of pieces. However, a smaller denominator means the whole is divided into fewer, larger pieces. For example, comparing 3/4 and 3/8: both have 3 pieces, but fourths are twice as large as eighths. Therefore, 3/4 is strictly greater than 3/8. As denominators increase, the individual slice size decreases.
The Benchmark Comparison Strategy
Before computing complex common denominators, you can often compare fractions quickly using mental benchmarks such as 0, 1/2, and 1.
Comparing 3/8 and 5/9 using Benchmark 1/2:
- What is half of 8? Exactly 4. Since 3 < 4, the fraction 3/8 < 1/2.
- What is half of 9? Exactly 4.5. Since 5 > 4.5, the fraction 5/9 > 1/2.
Conclusion: Since 3/8 < 1/2 and 5/9 > 1/2, it follows that 3/8 < 5/9!
Closeness to One Whole
Fractions that are both greater than 1/2 can often be compared by examining how much they lack to complete one whole. Consider 5/6 and 7/8. The fraction 5/6 is missing 1/6 of a whole, while 7/8 is missing 1/8 of a whole. Because an eighth is smaller than a sixth (1/8 < 1/6), 7/8 is missing a smaller piece than 5/6. Therefore, 7/8 is closer to 1 whole, meaning 7/8 > 5/6.
The Least Common Denominator (LCD) Method
When benchmark reasoning is inconclusive, the universal method is to rewrite the fractions with a common denominator equal to the Least Common Multiple of the denominators.
Compare 5/6 and 7/9:
Step 1: Find LCM(6, 9): Multiples of 9: 9, 18. Since 18 is a multiple of 6, LCD = 18.
Step 2: Scale each fraction:
5/6 = (5 * 3) / (6 * 3) = 15/18
7/9 = (7 * 2) / (9 * 2) = 14/18
Step 3: Compare numerators: Since 15 > 14, it follows that 15/18 > 14/18.
Conclusion: 5/6 > 7/9.
The Cross-Multiplication Method
To compare any two fractions a/b and c/d rapidly, multiply diagonally: compute a * d and b * c. Place the product a * d above the first fraction, and the product b * c above the second fraction. The fraction with the greater diagonal product is the greater fraction. For 4/7 versus 5/9: 4 * 9 = 36 (left) and 7 * 5 = 35 (right). Since 36 > 35, 4/7 > 5/9.
Exercises
Exercise 1. Compare each pair of fractions using <, >, or =: (a) 7/12 and 7/15; (b) 4/9 and 5/8; (c) 9/10 and 11/12.
Exercise 2. Use the Least Common Denominator method to compare 5/12 and 7/16.
Exercise 3. Arrange the following four fractions in ascending order: 2/3, 5/12, 3/4, 1/2.
Solutions
Solution to Exercise 1. For (a) 7/12 and 7/15: both fractions have the same numerator (7). Because twelfths are larger than fifteenths (12 < 15), 7/12 > 7/15. For (b) 4/9 and 5/8: use benchmark 1/2; half of 9 is 4.5, so 4/9 < 1/2; half of 8 is 4, so 5/8 > 1/2; therefore, 4/9 < 5/8. For (c) 9/10 and 11/12: both are one unit fraction away from 1 whole; 9/10 lacks 1/10, while 11/12 lacks 1/12; since 1/12 is smaller than 1/10, 11/12 is closer to 1; therefore, 9/10 < 11/12.
Solution to Exercise 2. To compare 5/12 and 7/16, find the LCM of 12 and 16: multiples of 16 are 16, 32, 48 (which is divisible by 12); thus LCD = 48. Scale 5/12 by multiplying by 4/4: (5 * 4) / (12 * 4) = 20/48. Scale 7/16 by multiplying by 3/3: (7 * 3) / (16 * 3) = 21/48. Comparing the scaled numerators, 20 < 21, so 20/48 < 21/48. Therefore, 5/12 < 7/16.
Solution to Exercise 3. To arrange 2/3, 5/12, 3/4, and 1/2 in ascending order, find the LCD for denominators 3, 12, 4, and 2. The Least Common Multiple of 3, 12, 4, and 2 is 12. Convert each fraction to twelfths: 2/3 = 8/12; 5/12 = 5/12; 3/4 = 9/12; 1/2 = 6/12. Now order the numerators from least to greatest: 5/12 < 6/12 < 8/12 < 9/12. Substituting the original fractions gives the ascending order: 5/12 < 1/2 < 2/3 < 3/4.