MISSION · Number sense
Comparing and Ordering Fractions
Why bigger numbers can mean a smaller fraction
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LEVEL 2 · LEARN
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The idea in one sentence
You cannot compare fractions by their digits — you have to make the pieces the same size first, or compare both to something you already know.
The trap
Which is bigger, \(\frac{1}{3}\) or \(\frac{1}{8}\)?
Eight is bigger than three, so \(\frac{1}{8}\) looks like the larger fraction. It isn’t. Cutting a cake into 8 pieces gives smaller pieces than cutting it into 3, so \(\frac{1}{3} > \frac{1}{8}\).
A bigger denominator means smaller pieces. That single sentence undoes the most common error in fractions, and it’s worth saying aloud every time the digits tempt you.
Three ways to compare
Same bottom — compare the tops. \(\frac{3}{7}\) and \(\frac{5}{7}\) are made of identical pieces, so more pieces wins: \(\frac{5}{7}\) is bigger.
Same top — compare the bottoms, backwards. \(\frac{2}{5}\) and \(\frac{2}{9}\) are the same number of pieces, but fifths are bigger than ninths, so \(\frac{2}{5}\) wins.
Neither matches — make a common denominator. Recut both into the same size pieces, then count.
\[\frac{3}{5} \text{ and } \frac{5}{8} \quad \Rightarrow \quad \frac{24}{40} \text{ and } \frac{25}{40}\]Twenty-five beats twenty-four, so \(\frac{5}{8}\) is bigger — barely. Without recutting, nothing about 3, 5, 5, and 8 would have told you.
Benchmarks are faster
For a quick answer, compare each fraction to \(\frac{1}{2}\).
A fraction is more than a half when the top is more than half the bottom. \(\frac{5}{9}\) is over a half because 5 is more than half of 9. \(\frac{4}{9}\) is under.
So \(\frac{5}{9} > \frac{4}{9}\) — which the same-bottom rule also gives — but more usefully, \(\frac{5}{9} > \frac{3}{7}\) without any recutting, because one is over a half and the other is under.
Zero and one work as benchmarks too. \(\frac{7}{8}\) is nearly 1, while \(\frac{1}{8}\) is nearly 0, and no arithmetic is needed to place them.
Ordering a list
Sort by comparing in pairs, or convert everything to one common denominator and sort the tops. For a mixed list, benchmarks usually split it quickly: separate the ones under a half from the ones over, then sort within each group.
Common mix-ups
- Comparing denominators as though bigger means more. Bigger bottom, smaller pieces.
- Comparing tops when the bottoms differ. \(\frac{3}{4}\) beats \(\frac{5}{8}\) despite the smaller top.
- Changing only one fraction when finding a common denominator.
- Forgetting that equal fractions exist. \(\frac{2}{4}\) and \(\frac{3}{6}\) are neither bigger nor smaller than each other.
Check yourself
Which is larger, \(\frac{3}{4}\) or \(\frac{7}{10}\)? And put \(\frac{1}{3}\), \(\frac{3}{5}\), \(\frac{1}{6}\) in order, smallest first.
Twentieths give \(\frac{15}{20}\) against \(\frac{14}{20}\), so \(\frac{3}{4}\) wins. For the list: \(\frac{1}{6}\), \(\frac{1}{3}\), \(\frac{3}{5}\) — the last is the only one over a half.
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