Ordering and comparing fractions

Learning outcome

Compare two fractions, arrange several fractions in order, and handle negative or mixed forms using a method whose direction is justified.

Understanding comparison methods

Fractions with a common positive denominator count parts of the same size. Compare their numerators directly. For positive fractions with equal numerators, a larger denominator means smaller parts, so the fraction is smaller. Do not apply this second rule unchanged to negative fractions.

With unlike denominators, create equivalent fractions with a common denominator. The least common multiple is often convenient because it keeps the arithmetic small; any positive common multiple works.

For a/b and c/d with positive denominators, you may instead compare a × d and c × b. Multiplying both fractions by the same positive number b × d clears their denominators and preserves their order. This explains cross-multiplication; it is not a rule about choosing whichever original numerator looks larger.

Sign provides an immediate check. Every negative fraction is less than zero, and every positive fraction is greater than zero. Among negative fractions, the one with greater distance from zero is smaller. Signed numerators can also be compared directly after creating a common positive denominator.

For nonnegative mixed forms, compare whole parts first and fractional remainders if needed. Alternatively, convert the entire quantities to improper fractions. Reduction is helpful but is not required before a valid comparison.

Worked examples

Example 1: Compare 7/12 and 5/8.

Use denominator 24. Multiplying both parts of the first fraction by 2 gives 7/12 = 14/24; multiplying both parts of the second by 3 gives 5/8 = 15/24. Fourteen equal parts are fewer than fifteen, so 7/12 < 5/8.

Example 2: Order 3/4, 7/10 and 2/3.

A common denominator is 60. The equivalent fractions are 45/60, 42/60 and 40/60 respectively. Since 40 < 42 < 45, the increasing order is 2/3 < 7/10 < 3/4.

Example 3: Compare -5/6 and -7/9.

Use denominator 18: -5/6 = -15/18 and -7/9 = -14/18. Since -15 < -14, we obtain -5/6 < -7/9. The first fraction lies farther below zero; comparing only unsigned numerators would reverse the answer.

Common mistakes

Do not compare unlike-denominator numerators alone. Do not assume a larger denominator always means a larger fraction. Keep denominators positive before using the stated cross-product rule. Compare the whole mixed quantity, not just its fractional remainder.

Practice questions

1. Compare 9/14 and 5/7.
2. Arrange 4/9, 4/7 and 4/11 in increasing order.
3. Arrange -3/4, -2/3 and 1/12 in increasing order.
4. Compare 2 + 1/5 with 13/6.

Worked answers

1. Since 5/7 = 10/14, compare 9 with 10. Thus, 9/14 < 5/7.
2. The positive numerators match, so larger denominators give smaller values: 4/11 < 4/9 < 4/7.
3. The common-denominator forms are -9/12, -8/12 and 1/12. Therefore, -3/4 < -2/3 < 1/12.
4. Convert 2 + 1/5 = 11/5. Cross-products are 11 × 6 = 66 and 13 × 5 = 65. Since 66 > 65, (2 + 1/5) > 13/6.