Fractions · Grade 5

How do you compare two fractions and put them in order?

If the denominators match, the larger fraction is the one with the larger numerator. If the numerators match, the larger one is the fraction with the smaller denominator, because the whole was cut into fewer pieces. In any other case, expand both to a common denominator and then compare numerators.

Learning objectives

When one of the two matches

Matching denominators is the easy case. The parts are the same size, so you simply count who took more of them: five sevenths is larger than three sevenths.

Matching numerators is the confusing case. We took the same number of parts, but in the fraction with the smaller denominator each part is larger — so three quarters is larger than three fifths.

A common denominator

When neither matches, bring both fractions to the same denominator. Then the parts are the same size and comparing goes back to counting numerators.

The common denominator can simply be the product of the two denominators, and that always works. The smallest one that fits is tidier, and you find it among the multiples of the larger denominator.

Quick checks

Before any calculation, it is worth asking whether each fraction is more or less than a half. A fraction is more than a half when its numerator exceeds half its denominator, and that settles any pair falling on opposite sides of it.

Another check is the distance from one whole. Six sevenths and nine tenths are both close to a whole, but the first is a seventh short and the second only a tenth — and since a tenth is smaller, the second is larger.

Worked examples

  1. Which is larger: 3 over 4 or 5 over 8?

    1. The denominators differ, so expand to a common one
    2. 8 is a multiple of 4, so the common denominator is 8
    3. Expand the first: 3 over 4 is 6 over 8
    4. Compare numerators: 6 is greater than 5

    Answer: Three quarters is larger

  2. Which is larger: 2 over 5 or 2 over 7?

    1. The numerators match, so compare by denominator
    2. 5 is less than 7, so each fifth is larger than each seventh
    3. We took the same number of parts, and they are bigger

    Answer: Two fifths is larger

  3. Order from smallest to largest: 1 over 2, 2 over 3, 3 over 8

    1. A usable common denominator is 24
    2. Expand: 12 over 24, 16 over 24, 9 over 24
    3. Compare numerators: 9, then 12, then 16

    Answer: 3 over 8, then a half, then 2 over 3

Common mistakes

Comparing numerators when the denominators differ
Numerators only count when the parts are the same size. Two thirds is larger than three eighths, even though two is less than three.
Assuming a larger denominator means a larger fraction
A larger denominator means smaller parts. When the numerators match, the fraction with the smaller denominator wins.
Expanding only one of the two
After expanding, both fractions must sit on the same denominator. Expanding one alone compares two different units.

What to remember

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