Comparing Fractions

Share the same pizza between two people — one takes three of the eight slices, the other takes two. Who ended up with more? Comparing fractions settles everyday questions like this.

Which is bigger, \frac{3}{8} or \frac{2}{8}? Comparing two fractions just means asking which one covers more of the whole. Sometimes the answer jumps straight out at you; sometimes you need a little setup first. This page sorts the cases out so you always know what to do.

Every fraction has a top number (the numerator — how many pieces you have) and a bottom number (the denominator — how many equal pieces the whole was cut into). To compare two fractions, keep your eye on both numbers, because each one tells you something different.

Same denominator — compare the tops

This is the easy case. When two fractions share the same denominator, the pieces are exactly the same size. So whoever has more pieces wins — just compare the numerators:

\frac{3}{8} > \frac{2}{8}

Three eighths is more than two eighths, because three same-size pieces beat two of them. The bigger top number is the bigger fraction. It is just like counting: if every slice is one-eighth, then 3 slices is more cake than 2 slices.

\frac{5}{6} > \frac{3}{6} \qquad \frac{4}{10} < \frac{7}{10}

Same numerator — bigger bottom is smaller

Here is the surprising one, and the one that trips people up. If two fractions have the same numerator, the one with the bigger denominator is the smaller fraction:

\frac{1}{3} < \frac{1}{2} \qquad \frac{2}{5} > \frac{2}{9}

It makes sense once you picture it: cutting a whole into more pieces makes each piece smaller. One slice of a pie cut into three is less than one slice of the same pie cut into two. More pieces means thinner pieces — so a bigger bottom number means each piece is tinier.

\frac{1}{3} is bigger than \frac{1}{4}, even though 4 is bigger than 3. A bigger bottom means a smaller piece — so don't just compare the bottom numbers and pick the larger one.
You can only "compare the tops" when the bottoms already match. If the bottoms are different, the pieces are different sizes, so \frac{3}{4} vs \frac{5}{8} is not just "3 vs 5". Make the bottoms match first.

Different denominators — make them match

When the bottoms are different, the pieces are different sizes, so you can't compare the tops directly. The fix is to rewrite both as equivalent fractions with a common denominator, and then compare. To weigh up \frac{2}{3} against \frac{3}{4}, give them both a denominator of 12:

\frac{2}{3} = \frac{8}{12} \qquad \frac{3}{4} = \frac{9}{12}

Now the pieces are the same size, so we just compare the tops: \frac{8}{12} < \frac{9}{12}, which means \frac{2}{3} < \frac{3}{4}.

A quick way to find a common denominator: multiply the two bottoms together (here 3 \times 4 = 12). It is not always the smallest choice, but it always works.

Picture it: bars and number lines

Two pictures make every comparison obvious. The first is a fraction bar: draw two bars of the same length, shade each fraction, and the one with the longer shaded part is bigger.

The second is a number line from 0 to 1. Mark each fraction as a point; the one further to the right is bigger. On a 0-to-1 line, \frac{1}{2} sits right in the middle, while \frac{1}{3} sits to its left — closer to 0 — so it is smaller.

The comparison sign helps you remember the answer: the open mouth of < or > always faces the bigger number, like a hungry mouth pointing at the larger snack.

Three worked examples

Walk through one of each case:

Same bottom. Compare \frac{5}{6} and \frac{3}{6}. Bottoms match, so compare tops: 5 > 3, therefore \frac{5}{6} > \frac{3}{6}.
Same top. Compare \frac{2}{5} and \frac{2}{9}. Same numerator, so the bigger bottom is the smaller fraction: \frac{2}{9} is smaller, so \frac{2}{5} > \frac{2}{9}.
Different bottoms. Compare \frac{3}{4} and \frac{5}{8}. Make the bottoms 8: \frac{3}{4} = \frac{6}{8}. Now \frac{6}{8} > \frac{5}{8}, so \frac{3}{4} > \frac{5}{8}.

pizza pizza

Mia ate \frac{3}{8} of a pizza and Leo ate \frac{2}{8} of an identical pizza. The slices are the same size (both pizzas were cut into 8), so we just count slices: 3 beats 2. Mia ate more — \frac{3}{8} > \frac{2}{8}. Easy, because the bottoms matched.

cake

You get \frac{1}{4} of a birthday cake; your friend gets \frac{1}{3} of an identical cake. Same top number (one slice each!), but whose slice is bigger? The cake cut into 3 makes fatter slices than the cake cut into 4, so \frac{1}{3} > \frac{1}{4} — your friend's slice is bigger. Bigger bottom, smaller piece.

cookie cookie cookie

Three cookies shared between a few children: would you rather have \frac{2}{3} of the pile or \frac{3}{4}? The bottoms don't match, so rewrite both over 12: \frac{2}{3} = \frac{8}{12} and \frac{3}{4} = \frac{9}{12}. Nine twelfths beats eight twelfths, so take the \frac{3}{4} — more cookie for you.

See it: two random bars

Two bars of the same length, each cut into its own number of equal pieces and shaded. Step through it: the top fraction, then the bottom one, then the answer — the bigger fraction's bar lights up and the < / > sign points its open mouth at it. Press Refresh for a brand-new pair.

See it explained

Sal Khan compares fractions with different denominators by rewriting them so the pieces match.