How do you compare fractions?

When the denominators are the same

5/8 means 5 parts of 1/8, and 3/8 means 3 parts of 1/8. Both use the same fractional unit, so the one with more units is bigger: 5/8 > 3/8. With the same denominator, just compare the numerators.

When the numerators are the same

Which is bigger, 1/5 or 1/9? Think of sharing one roti. Shared among 5 children, each piece is bigger than when it is shared among 9 children, so 1/9 < 1/5. The same idea works for any numerator. If 4 chikkis are shared among 7 children, each gets 4/7; shared among 8 children, each gets 4/8. More children, smaller share, so 4/7 > 4/8. And because 4/8 = 1/2, we also know that 4/7 > 1/2.

So with the same numerator, the fraction with the smaller denominator is bigger.

When both are different

Who gets more chikki: a child in a group where 1 chikki is shared between 2 children, or a child in a group where 5 chikkis are shared among 8? We must compare 1/2 and 5/8. Since 1/2 = 4/8, and 4/8 < 5/8, each child in the second group gets more.

Most of the time neither number matches, so we change both fractions to the same fractional unit first. Compare 3/4 and 2/3: both 4 and 3 divide 12, so 3/4 = 9/12 and 2/3 = 8/12.

9 twelfths are more than 8 twelfths, so 3/4 > 2/3.

Two steps to compare any fractions

1. Step 1: rewrite the fractions as equivalent fractions with the same denominator. A common multiple of the denominators works.
2. Step 2: compare the numerators, that is, count the fractional units.

Example 1: 4/5 or 7/9? 45 is a common multiple of 5 and 9. 4/5 = (4 × 9)/(5 × 9) = 36/45 and 7/9 = (7 × 5)/(9 × 5) = 35/45. So 4/5 > 7/9.

Example 2: 7/9 or 17/21? 63 is a common multiple of 9 and 21. 7/9 = (7 × 7)/(9 × 7) = 49/63 and 17/21 = (17 × 3)/(21 × 3) = 51/63. So 7/9 < 17/21.

Putting fractions in order

To arrange 1/2, 5/6 and 2/3 in ascending order, use the common denominator 6: 1/2 = 3/6, 5/6 = 5/6 and 2/3 = 4/6. So the order is 1/2 < 2/3 < 5/6. For descending order, simply reverse it: 5/6 > 2/3 > 1/2.

Fractions greater than 1

The same steps work for fractions greater than 1. Compare 8/3 and 5/2: with the denominator 6, 8/3 = 16/6 and 5/2 = 15/6, so 8/3 > 5/2. With mixed fractions you can compare the whole parts first: 8/3 = 2 2/3 and 5/2 = 2 1/2 have the same whole part, so compare 2/3 and 1/2, and 2/3 is bigger.

Comparing is also the first step before adding and subtracting fractions, which uses the same common-denominator idea.