Comparing & Ordering Unlike Fractions - Fourth Grade Mathematics
When chefs adjust recipes from different culinary traditions, when carpenters select timber cuts from mixed blueprints, or when sports statisticians analyze batting averages, they regularly need to compare fractions that have neither the same numerator nor the same denominator. These are unlike fractions. While benchmark fractions provide fast mental estimates, determining exact mathematical relationships between fractions that sit close together—such as 3/5 and 5/8—requires rigorous, systematic methods. In this chapter, you will master the two universal algebraic methods for comparing and ordering unlike fractions: converting to common denominators and using cross-multiplication, giving you the power to arrange any set of fractions in ascending or descending order with complete certainty.
Method 1: The Common Denominator Strategy
The gold standard for comparing unlike fractions is converting both fractions into equivalent forms sharing a common denominator.
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| COMPARING 3/5 AND 5/8 USING A COMMON DENOMINATOR |
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| Step 1: Find a common denominator: |
| Multiply the denominators: 5 x 8 = 40. |
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| Step 2: Convert both fractions to equivalent fractions with denominator 40: |
| Convert 3/5: (3 x 8) / (5 x 8) = 24/40 |
| Convert 5/8: (5 x 5) / (8 x 5) = 25/40 |
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| Step 3: Compare the numerators: |
| Since 24 < 25, we have 24/40 < 25/40. |
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| Conclusion: 3/5 < 5/8 (3/5 is less than 5/8). |
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Why Common Denominators Never Fail
Notice how close 3/5 and 5/8 are: 3/5 is 0.60, and 5/8 is 0.625. Benchmark estimation cannot reliably distinguish between fractions that are separated by only 1/40 of a unit! Converting to a common denominator levels the playing field, making the comparison as simple as comparing the whole numbers 24 and 25.
Method 2: The Cross-Multiplication Method
Cross-multiplication is a lightning-fast shortcut for comparing two fractions that is mathematically derived directly from the common denominator method.
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| THE CROSS-MULTIPLICATION SHORTCUT |
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| Compare: 4/7 vs 5/9 |
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| Step 1: Multiply bottom-right denominator (9) by top-left numerator (4): |
| 4 x 9 = 36 <-- Write 36 above the left fraction (4/7) |
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| Step 2: Multiply bottom-left denominator (7) by top-right numerator (5): |
| 5 x 7 = 35 <-- Write 35 above the right fraction (5/9) |
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| Step 3: Compare the cross-products: |
| 36 > 35 |
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| Conclusion: 4/7 > 5/9 |
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Why Cross-Multiplication Works
Why is this shortcut mathematically valid? If you converted 4/7 and 5/9 to common denominators, the common denominator would be 7 x 9 = 63. The new numerator for 4/7 would be 4 x 9 = 36 (36/63). The new numerator for 5/9 would be 5 x 7 = 35 (35/63). Cross-multiplication simply calculates the new numerators (36 and 35) while skipping the step of writing the common denominator 63! Because both fractions share the denominator 63, comparing 36 and 35 gives the exact comparison.
Ordering Sets of Three or More Unlike Fractions
When ordering a set of three or more fractions from least to greatest, converting all fractions to a single common denominator is the most reliable strategy.
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| ORDERING THREE FRACTIONS: 2/3, 3/4, AND 5/6 |
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| Step 1: Find the Least Common Multiple (LCM) of 3, 4, and 6: |
| Multiples of 3: 3, 6, 9, 12, 15... |
| Multiples of 4: 4, 8, 12, 16... |
| Multiples of 6: 6, 12, 18... |
| LCM = 12 |
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| Step 2: Convert all fractions to twelfths: |
| 2/3 = (2 x 4) / (3 x 4) = 8/12 |
| 3/4 = (3 x 3) / (4 x 3) = 9/12 |
| 5/6 = (5 x 2) / (6 x 2) = 10/12 |
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| Step 3: Order by numerators (Least to Greatest): |
| 8/12 < 9/12 < 10/12 |
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| Final Ordered Sequence: 2/3 < 3/4 < 5/6 |
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Chapter Practice Exercises
Exercise 1: Compare 5/7 and 7/10 using cross-multiplication. Show both cross-products and state the final inequality symbol.
Exercise 2: Compare 4/9 and 5/12 by finding the least common denominator. Show the converted fractions and compare.
Exercise 3: Arrange the following four fractions in order from least to greatest: 1/2, 2/5, 5/8, 3/10.
Exercise 4: Three students competed in a swimming race. Alex completed 7/12 of the laps, Bailey completed 5/8 of the laps, and Charlie completed 2/3 of the laps. Who completed the greatest fraction of laps?
Exercise 5: A student cross-multiplies to compare 2/5 and 3/8. The student multiplies 2 x 3 = 6 and 5 x 8 = 40, concluding that 3/8 is much larger. Explain the student's mistake and provide the correct cross-multiplication steps.
Solutions and Step-by-Step Explanations
Solution 1: To compare 5/7 and 7/10 by cross-multiplication: multiply 5 by 10 to get 50 (above the left fraction), and multiply 7 by 7 to get 49 (above the right fraction). Comparing the cross-products: 50 > 49. Therefore, 5/7 > 7/10.
Solution 2: Multiples of 9: 9, 18, 27, 36. Multiples of 12: 12, 24, 36. The least common denominator is 36. Converting 4/9: (4 x 4) / (9 x 4) = 16/36. Converting 5/12: (5 x 3) / (12 x 3) = 15/36. Comparing numerators: 16 > 15, so 16/36 > 15/36. Therefore, 4/9 > 5/12.
Solution 3: To order 1/2, 2/5, 5/8, 3/10, find a common denominator for 2, 5, 8, 10: the least common multiple is 40. Convert 1/2: (1 x 20) / (2 x 20) = 20/40. Convert 2/5: (2 x 8) / (5 x 8) = 16/40. Convert 5/8: (5 x 5) / (8 x 5) = 25/40. Convert 3/10: (3 x 4) / (10 x 4) = 12/40. Ordering numerators from least to greatest: 12/40 (3/10) < 16/40 (2/5) < 20/40 (1/2) < 25/40 (5/8). The ordered sequence is: 3/10 < 2/5 < 1/2 < 5/8.
Solution 4: We find a common denominator for 12, 8, and 3: the least common multiple is 24. Alex (7/12): (7 x 2) / (12 x 2) = 14/24. Bailey (5/8): (5 x 3) / (8 x 3) = 15/24. Charlie (2/3): (2 x 8) / (3 x 8) = 16/24. Comparing the numerators: 16/24 > 15/24 > 14/24. Charlie completed the greatest fraction of laps (2/3).
Solution 5: The student multiplied horizontally across numerators (2 x 3 = 6) and denominators (5 x 8 = 40) instead of cross-multiplying diagonally. Cross-multiplication requires multiplying each numerator by the denominator of the opposite fraction. For 2/5 vs 3/8: left cross-product is 2 x 8 = 16; right cross-product is 5 x 3 = 15. Since 16 > 15, 2/5 is actually greater than 3/8 (2/5 > 3/8).