Math Vale

🍰 Fraction Isles · Fractions & Proportion

Comparing Fractions

Decide which of two fractions is larger using common denominators, benchmarks, or cross-multiplication, and order a set of fractions.

In short

  • Fractions can only be compared piece by piece when the pieces are the same size, which is what a common denominator arranges.
  • With equal numerators the larger denominator gives the smaller fraction, because the whole was cut into more pieces.
  • Cross-multiplying a x d against c x b is exactly the common-denominator method with the shared denominator left out.
  • Benchmarks 0, 1/2 and 1 settle many comparisons instantly; the doubling test decides the 1/2 benchmark without dividing.
  • To order a set, group by benchmark first, then rewrite each group over one denominator.

Why you cannot just compare the top numbers

With whole numbers, bigger digits mean a bigger number. Fractions do not behave that way, because the two fractions may be counting different sized pieces.

3/10 has the larger numerator than 1/2, but 3/10 is much smaller. Ten pieces means each one is tiny; two pieces means each one is huge.

The same trap works in reverse: a bigger denominator makes a fraction smaller, not bigger, when the numerator stays the same. 1/3 is more than 1/8, because thirds are fatter slices than eighths.

  • Same denominator? Compare the numerators.
  • Same numerator? The smaller denominator wins.
  • Neither the same? You have work to do first.

Common denominators

The reliable method is to make the pieces the same size, then simply count them.

Compare 3/4 and 5/6. A denominator that both 4 and 6 divide is 12 — the least common denominator.

3/4 = 9/12 and 5/6 = 10/12.

Now the pieces match, so 10 twelfths beats 9 twelfths: 5/6 > 3/4.

Finding the least common denominator is the same job as finding the least common multiple of the two denominators. Multiplying the denominators together (4 x 6 = 24) always gives *a* common denominator, and it is never wrong — it just leaves bigger numbers to tidy up afterwards.

Writing both fractions over a common denominator is also how you would place them on the same number line.

Cross-multiplying is the same idea, faster

Comparing a/b with c/d over the common denominator b x d turns the numerators into a x d and c x b. The denominators are then identical, so only those two products matter.

That gives the shortcut: to compare a/b and c/d, compare a x d with c x b.

For 3/4 and 5/6: 3 x 6 = 18 and 5 x 4 = 20. Since 18 < 20, we get 3/4 < 5/6 — the same answer as before.

Keep the products on the correct sides. The product that used the numerator of the first fraction belongs to the first fraction.

This shortcut only works when both denominators are positive, which for fractions of a whole they always are.

Benchmarks: 0, 1/2 and 1

Often you do not need an exact comparison, only a decision. Benchmarks give it in a moment.

  • A fraction is less than 1/2 when the numerator is less than half the denominator. Doubling test: 2 x numerator < denominator.
  • A fraction is exactly 1/2 when doubling the numerator gives the denominator: 7/14, 9/18.
  • A fraction is less than 1 when the numerator is smaller than the denominator, and more than 1 when it is larger.

So 5/12 is below a half (10 < 12) while 7/12 is above it (14 > 12) — no common denominator needed. If one fraction is below 1/2 and the other above, you are already finished.

A second benchmark trick: a fraction close to 1 is easier to judge by its gap from 1. 11/12 is 1/12 short of a whole; 15/16 is only 1/16 short, so 15/16 is larger.

Ordering a whole set

To sort several fractions from smallest to largest, do not compare them all at once. Sort them into rough groups first, then settle the ties.

1. Split them by benchmark: below 1/2, at 1/2, above 1/2 but below 1, and 1 or more. 2. Inside each group, put every fraction over one common denominator and read off the numerators. 3. Write the answer using the original fractions, in the order you found — the rewritten ones were only scaffolding.

For 2/5, 3/4, 1/2 and 7/10: 2/5 is below a half, the rest are at or above. Over 20ths they are 8/20, 15/20, 10/20 and 14/20, so the order is 2/5, 1/2, 7/10, 3/4.

A number line drawn with all four marked on it is the best final check: the order you wrote should read left to right.

Worked examples

Example 1

Which is larger, 5/8 or 7/12?

  1. The denominators differ, so make the pieces match. The least common multiple of 8 and 12 is 24.
  2. 5/8 = 15/24, because 8 x 3 = 24 and 5 x 3 = 15.
  3. 7/12 = 14/24, because 12 x 2 = 24 and 7 x 2 = 14.
  4. Same sized pieces now: 15 twenty-fourths beats 14 twenty-fourths.
  5. So 5/8 > 7/12. Cross-multiplying agrees: 5 x 12 = 60 and 7 x 8 = 56.

Example 2

Put 3/8, 5/6, 1/2 and 2/3 in order from smallest to largest.

  1. Benchmark first: 3/8 is below 1/2 (2 x 3 = 6, which is less than 8). The other three are 1/2 or above.
  2. The least common denominator of 8, 6, 2 and 3 is 24.
  3. Rewrite them: 3/8 = 9/24, 1/2 = 12/24, 2/3 = 16/24, 5/6 = 20/24.
  4. Read the numerators in increasing order: 9, 12, 16, 20.
  5. So the order is 3/8, 1/2, 2/3, 5/6 — and 3/8 is indeed the only one below a half, as predicted.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Which symbol makes this true? 1/6 __ 3/4

  1. <
  2. >
  3. =

Answer: A. <

  1. Common denominator: 12.
  2. 1/6 = 2/12 and 3/4 = 9/12.
  3. Now the pieces are the same size, so just compare the counts.
  4. 1/6 < 3/4.

Problem 2

Difficulty 3 of 5

To compare 1/6 with 9/10 you rewrite both over the least common denominator 30. 1/6 = ?/30. What is the missing numerator?

Answer: 5

  1. 6 x 5 = 30, so the whole fraction is scaled by 5.
  2. 1 x 5 = 5.
  3. 1/6 = 5/30, the same number written with smaller pieces.

Problem 3

Difficulty 4 of 5

Is 12/24 less than, equal to, or greater than 1/2?

  1. less than 1/2
  2. equal to 1/2
  3. greater than 1/2

Answer: B. equal to 1/2

  1. Doubling test: 2 x 12 = 24, and the denominator is 24.
  2. 24 is equal to 24, so the fraction is exactly one half.
  3. So 12/24 is equal to 1/2.

Common mistakes

  • Comparing numerators alone and concluding that 3/10 is bigger than 1/2.
  • Assuming a bigger denominator means a bigger fraction, so claiming 1/8 > 1/3.
  • Cross-multiplying but attaching each product to the wrong fraction.
  • Changing the denominator to the common one without scaling the numerator to match.
  • Giving the answer as the rewritten fractions (9/24, 12/24, ...) instead of the originals the question asked about.

What you should be able to do

  • Compare two fractions with unlike denominators.
  • Use benchmarks such as 1/2 and 1 to reason about size quickly.
  • Order a set of fractions, decimals and mixed numbers.
  • Explain why a larger denominator means smaller pieces.

Where this fits in the curriculum

Common Core

  • 3.NF.A.3.D

    Grade 3 — Compare two fractions with the same numerator or the same denominator by reasoning about size.

  • 4.NF.A.2

    Grade 4 — Compare fractions with different numerators and denominators, using a common denominator or a benchmark.

  • 4.NF.C.7

    Grade 4 — Compare two decimals to hundredths by reasoning about their size.

    Ordering fractions and decimals in one list is not a Common Core standard on its own; it falls between 4.NF.A.2 (fractions) and 4.NF.C.7 (decimals).

Ontario

  • G5.B1.4

    Grade 5 — Compare and order fractions from halves to twelfths, including improper fractions and mixed numbers.

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