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Comparing Fractions



Comparing Fractions with Same Denominator
Comparing Fractions with Same Numerator
Common Denominator Method
Cross-Multiplication Method
Benchmark Fractions
Decimal Conversion Method
Comparing Mixed Numbers
Ordering Multiple Fractions
Number Line Visualization



Which Fraction Is Larger

Comparing fractions determines which of two values is larger, smaller, or whether they are equal. Several methods exist, each suited to different situations. Simple cases with matching numerators or denominators resolve by inspection, while more complex comparisons require finding common denominators, cross-multiplying, or converting to decimals.



Comparing Fractions with Same Denominator

When two fractions share a denominator, compare their numerators directly. The fraction with the larger numerator is the larger fraction.

58>38\frac{5}{8} > \frac{3}{8}


Both fractions represent eighths. Five eighths is more than three eighths because five pieces exceed three pieces of the same size.

This principle extends to any number of fractions with a common denominator. Arranging 29\frac{2}{9}, 79\frac{7}{9}, and 49\frac{4}{9} in order requires only sorting the numerators: 29<49<79\frac{2}{9} < \frac{4}{9} < \frac{7}{9}.

Comparing Fractions with Same Numerator

When two fractions share a numerator, compare their denominators inversely. The fraction with the smaller denominator is the larger fraction.

34>37\frac{3}{4} > \frac{3}{7}


Both fractions take three parts, but fourths are larger pieces than sevenths. Three large pieces exceed three small pieces.

This inverse relationship often surprises at first. The key insight is that larger denominators create smaller individual pieces. Dividing something into 100 parts produces much smaller pieces than dividing it into 2 parts.

Common Denominator Method

When neither numerator nor denominator matches, convert both fractions to equivalent fractions with a common denominator, then compare numerators.

To compare 34\frac{3}{4} and 56\frac{5}{6}, find a common denominator. The LCM of 4 and 6 is 12.

34=912and56=1012\frac{3}{4} = \frac{9}{12} \quad \text{and} \quad \frac{5}{6} = \frac{10}{12}


Since 10>910 > 9, we have 56>34\frac{5}{6} > \frac{3}{4}.

This method always works and is the most reliable approach when other shortcuts do not apply.

Cross-Multiplication Method

Cross-multiplication provides a shortcut that avoids explicitly finding a common denominator. To compare ab\frac{a}{b} and cd\frac{c}{d}, compute the cross products a×da \times d and b×cb \times c. The larger cross product indicates the larger fraction.

To compare 37\frac{3}{7} and 49\frac{4}{9}:

3×9=27and7×4=283 \times 9 = 27 \quad \text{and} \quad 7 \times 4 = 28


Since 28>2728 > 27, we have 49>37\frac{4}{9} > \frac{3}{7}.

The rule: multiply each numerator by the opposite denominator. The cross product stays on the same side as its numerator. If a×d>b×ca \times d > b \times c, then ab>cd\frac{a}{b} > \frac{c}{d}.

Benchmark Fractions

Benchmark fractions like 0, 12\frac{1}{2}, and 1 provide quick reference points for estimation and comparison without calculation.

To compare 38\frac{3}{8} and 59\frac{5}{9}: note that 38<12\frac{3}{8} < \frac{1}{2} (since 3<43 < 4) while 59>12\frac{5}{9} > \frac{1}{2} (since 5>4.55 > 4.5). Therefore 59>38\frac{5}{9} > \frac{3}{8} without any cross-multiplication.

Other useful benchmarks include 14\frac{1}{4} and 34\frac{3}{4}. With practice, recognizing whether a fraction is close to 0, near 12\frac{1}{2}, or approaching 1 speeds up many comparisons.

Decimal Conversion Method

Converting fractions to decimals allows direct numerical comparison. Divide the numerator by the denominator for each fraction.

To compare 58\frac{5}{8} and 711\frac{7}{11}:

58=0.625and711≈0.636\frac{5}{8} = 0.625 \quad \text{and} \quad \frac{7}{11} \approx 0.636


Since 0.636>0.6250.636 > 0.625, we have 711>58\frac{7}{11} > \frac{5}{8}.

This method is practical with a calculator or for fractions that produce simple decimals. It becomes cumbersome for fractions with long repeating decimals.

Comparing Mixed Numbers

Compare mixed numbers by first examining the whole parts. The mixed number with the larger whole part is larger overall.

518>4785\frac{1}{8} > 4\frac{7}{8}


Five wholes exceed four wholes regardless of the fractional parts.

When whole parts are equal, compare the fractional parts using any method from this page. For 3253\frac{2}{5} versus 3373\frac{3}{7}, cross-multiply the fractions: 2×7=142 \times 7 = 14 and 5×3=155 \times 3 = 15. Since 15>1415 > 14, we have 37>25\frac{3}{7} > \frac{2}{5}, so 337>3253\frac{3}{7} > 3\frac{2}{5}.

Ordering Multiple Fractions

To arrange several fractions in order, find a common denominator for all of them and compare the resulting numerators.

Order 23\frac{2}{3}, 34\frac{3}{4}, 56\frac{5}{6}, and 712\frac{7}{12} from least to greatest. The LCD of 3, 4, 6, and 12 is 12.

23=812,34=912,56=1012,712=712\frac{2}{3} = \frac{8}{12}, \quad \frac{3}{4} = \frac{9}{12}, \quad \frac{5}{6} = \frac{10}{12}, \quad \frac{7}{12} = \frac{7}{12}


Sorting by numerators: 7<8<9<107 < 8 < 9 < 10. The order is 712<23<34<56\frac{7}{12} < \frac{2}{3} < \frac{3}{4} < \frac{5}{6}.

Number Line Visualization

Placing fractions on a number line shows their relative positions visually. Fractions further to the right are larger.

The interval from 0 to 1 can be subdivided according to any denominator. Marking fourths at 14\frac{1}{4}, 24\frac{2}{4}, 34\frac{3}{4} and thirds at 13\frac{1}{3}, 23\frac{2}{3} shows that 13\frac{1}{3} lies between 14\frac{1}{4} and 24\frac{2}{4}, confirming 14<13<12\frac{1}{4} < \frac{1}{3} < \frac{1}{2}.

Number lines are particularly useful for developing intuition about fraction size and for understanding why certain comparison methods work. Seeing that 23\frac{2}{3} and 46\frac{4}{6} occupy the same point reinforces the concept of equivalent fractions.