Practice Maths

L42 — Comparing Fractions

Key Terms

common denominator
A shared denominator for two or more fractions. Used to compare or add fractions. E.g. to compare 12 and 13, convert to 36 and 26.
LCD (lowest common denominator)
The smallest number that is a multiple of all the denominators. E.g. LCD of 4 and 6 is 12.
benchmark fraction
A familiar fraction used as a reference point for estimating. Common benchmarks: 0, 14, 12, 34, 1.
cross-multiplication
A shortcut for comparing ab and cd: compare a×d with b×c. The larger cross-product corresponds to the larger fraction.
mixed number
A number written as a whole number plus a proper fraction. E.g. 234 = 2 + 34.
improper fraction
A fraction where the numerator is greater than or equal to the denominator. E.g. 114 = 234.

Comparing Fractions

To compare fractions, convert to a common denominator (LCD) and compare the numerators. Alternatively, use cross-multiplication: compare ab and cd by checking if a×d is less than, equal to, or greater than b×c.

Benchmark fractions help estimate quickly: a fraction close to 0 has a much smaller numerator than denominator; a fraction close to 1 has a numerator nearly equal to the denominator.

Hot Tip: When numerators are equal, a larger denominator means smaller pieces — so the fraction is smaller. For example, 18 < 13 because eighths are smaller pieces than thirds.

Worked Example 1 — Compare 34 and 56

Step 1: Find the LCD of 4 and 6. LCD = 12.

Step 2: Convert: 34 = 912    56 = 1012

Step 3: Compare: 9 < 10, so 34 < 56

Worked Example 2 — Order 23, 14, 58, 35 from smallest to largest

Step 1: LCD of 3, 4, 8, 5 = 120.

Step 2: Convert: 23 = 80120,   14 = 30120,   58 = 75120,   35 = 72120

Answer: 14 < 35 < 58 < 23

The Piece-Size Trap

A common misconception: students sometimes think a bigger denominator means a bigger fraction. But the opposite is true! If you cut a pizza into 8 slices, each slice is smaller than if you cut it into 4 slices. So 18 is smaller than 14, even though 8 > 4.

When comparing fractions, the denominator tells you the size of each piece. The numerator tells you how many pieces you have. Both matter when comparing.

Method 1: Common Denominator

The most reliable method: convert both fractions so they have the same denominator, then compare the numerators. Whoever has more pieces (bigger numerator) is the larger fraction.

Compare 34 and 56: LCD of 4 and 6 = 12. Convert: 34 = 912, 56 = 1012. Since 10 > 9, we get 34 < 56.

This method always works and explains exactly why one fraction is larger.

Method 2: Cross-Multiplication

A faster method for comparing two fractions: cross-multiply and compare the results. For ab vs cd: compare a×d with b×c.

Compare 34 and 56: 3×6 = 18 and 4×5 = 20. Since 18 < 20, we get 34 < 56. Matches our earlier result!

This works because cross-multiplication is secretly finding equivalent fractions with the same denominator, all at once.

Remember: When using cross-multiplication, the cross-product on the left corresponds to the fraction on the left. If the left product is bigger, the left fraction is bigger. Keep track of which is which!

Benchmark Fractions: Quick Estimation

For quick comparisons or ordering, benchmark fractions (0, 14, 12, 34, 1) are incredibly useful. You can often estimate at a glance:

  • A fraction where the numerator is much smaller than the denominator (like 19) is close to 0.
  • A fraction where numerator ≈ denominator÷2 (like 49) is close to 12.
  • A fraction where numerator ≈ denominator (like 89) is close to 1.

This helps you quickly check: “Does my answer make sense? Is this fraction bigger or smaller than 12?”

Common Mistake: When ordering multiple fractions, students sometimes compare pairs in the wrong order and get confused. The safest approach: always convert ALL fractions to a common denominator, then order them by numerator from smallest to largest.

  1. Compare each pair using <, > or =

    Write <, > or = in each box.

    1. 12 34
    2. 23 46
    3. 35 58
    4. 56 79
    5. 47 35
    6. 710 34
    7. 29 14
    8. 512 38

  2. Order each set from smallest to largest by finding a common denominator

    1. 12, 13, 14
    2. 34, 23, 56
    3. 25, 38, 14
    4. 710, 23, 34, 56
    5. 16, 29, 14, 310
    6. 58, 712, 1120
    7. 49, 37, 512
    8. 910, 89, 78, 1112

  3. Describe the position and explain your reasoning

    1. Place 35 on a number line from 0 to 1. Explain how you know where it goes.
    2. Which fraction is closer to 1: 78 or 56? Show working.
    3. Name two fractions that lie between 14 and 12.
    4. True or false: 45 is closer to 1 than 34. Justify your answer.

  4. Benchmark estimation — is each fraction closer to 0, ½ or 1?

    1. 19
    2. 511
    3. 78
    4. 37
    5. 910
    6. 213
    7. 49
    8. 1112

  5. Real-world comparisons

    1. A banana bread recipe uses 34 cup of sugar. A muffin recipe uses 58 cup. Which recipe uses more sugar?
    2. Sam ran 710 km and Jo ran 23 km. Who ran further? By how much?
    3. Maya completed 56 of her homework and Leo completed 79. Who has done more? Write your comparison using > or <.
    4. Three students each drank a different fraction of a water bottle: Ava drank 25, Ben drank 38 and Cam drank 512. Rank them from most to least.

  6. Compare mixed numbers

    Write <, > or = in each box. The whole number parts are equal, so compare only the fraction parts.

    1. 123 134
    2. 212 238
    3. 356 379
    4. 425 437
    5. 514 529
    6. 21112 256

  7. Convert mixed numbers and order

    Convert each set to improper fractions, then order from smallest to largest.

    1. 113, 54, 116
    2. 214, 94, 238
    3. 116, 158, 179
    4. 312, 227, 325

  8. Fractions on a number line

    1. Plot the following fractions on a number line from 0 to 2 and label each: 14, 34, 112, 74.
    2. A number line has 0 and 1 marked. At what position would you place 58? Describe how you find the position.
    3. What fraction is exactly halfway between 14 and 34?
    4. What fraction is exactly halfway between 25 and 45?

  9. Fraction puzzle and reasoning

    1. I am thinking of a fraction with denominator 12 that is greater than 12 but less than 34. Find all possible fractions (in simplest form or with denominator 12).
    2. A student says 37 > 49 because 3×9 = 27 and 7×4 = 28, so 27 < 28. Is the student correct? Explain the cross-multiplication method.
    3. Three people each ate a different fraction of an identical pizza: Ali ate 38, Bo ate 512, and Cal ate 26. Who ate the most? What fraction of the pizza was eaten altogether?
    4. Arrange in increasing order by finding a common denominator: 57, 710, 1115, 34.

  10. Fractions in context

    1. In a class vote, 512 of students voted for Option A and 38 voted for Option B. Which option got more votes? What fraction voted for neither?
    2. A swimming coach notes that Ella swam 34 of the race length before resting, and Finn swam 79 before resting. Who rested later in the race?
    3. Container A is 23 full and holds 1.5 litres when full. Container B is 34 full and holds 2 litres when full. Which container currently holds more liquid? Calculate the actual amounts.
    4. Write your own real-world scenario where you would need to compare two fractions with different denominators. Solve it.
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