Which is more: 2/3, or 3/5?
Two pizzas, two friends, two different portions. Who has more? With whole numbers you see it at once. With fractions you need a moment to think — but you do not need to count long columns. Today you will learn three fast ways.
Way 1: same denominators. When the parts are the same size (the same name at the bottom), the number of parts decides: 3/8 < 5/8.
Way 2: same numerators. When you take the same number of parts, the size of the part decides: 2/3 > 2/5, because thirds are bigger pieces than fifths.
Way 3: compare with a half. Lots of pairs are split by the question “is it more, or less than a half?” 3/5 is more than a half, 2/5 is less — 完了, you do not even have to convert anything.
These three views will solve most comparisons you meet. And when they are not enough, a picture helps — drawing two identical rectangles takes ten seconds.
Jonáš and Tereza each ordered a pizza the same size. Jonáš ate 2/3 of his, Tereza 3/5. “I ate more, three pieces is more than two,” Tereza claims. Jonáš draws two identical rectangles on a napkin. He splits the first into 3 parts and colours 2, the second into 5 parts and colours 3. He puts them together: Jonáš’s coloured piece is longer. “Thirds are bigger pieces than fifths. Two big ones beat three small ones.” Tereza squints at the picture: “Fine. But next time I draw, your rectangles are wonky.”
A fraction is more than a half when the numerator is more than half the denominator: 3/5 → 3 is more than 2.5 → it is over a half.
Three views you use to compare fractions
1. Same denominators → compare the numerators. 5/8 and 3/8: the parts (eighths) are the same size, so the one who has more of them wins. 5/8 > 3/8. The simplest case.
2. Same numerators → compare the denominators the other way round. 2/3 and 2/5: in both cases I take 2 parts, but thirds are bigger pieces than fifths (a whole split into fewer parts = bigger parts). So 2/3 > 2/5. Watch — a bigger denominator means a smaller fraction, when the numerators are the same.
3. A half as a mark. Compare each fraction with 1/2: the numerator against half the denominator. For 3/5 half the denominator is 2.5 and the numerator is 3 → more than a half. For 3/8 half the denominator is 4 and the numerator is 3 → less than a half. So 3/5 > 3/8, without converting anything.
When none of that works: draw two identical rectangles, split and colour. Or — you will learn this in the next lesson — convert both fractions to the same denominators. But a picture is enough surprisingly often and it never lies, if the wholes are the same size.
Compare 4/9 and 7/9
The denominators are the same — both fractions talk about ninths, so about parts the same size.
The number of parts decides: 4 < 7.
Result: 4/9 < 7/9.
Picture it: a cake into 9 pieces. Who has 7 pieces has more than the one with 4 pieces. When the parts are the same, it is ordinary comparing of whole numbers.
Compare 3/4 and 3/10
The numerators are the same — in both cases I take 3 parts. The size of the part decides.
Quarters: a whole into 4 parts → big pieces. Tenths: a whole into 10 parts → small pieces.
Three big pieces > three small pieces, so 3/4 > 3/10.
Check through a half: 3/4 is over a half (3 > 2), 3/10 is well under it (3 < 5). Both views say the same.
Compare 5/8 and 2/5 through a half
Different denominators, different numerators — I use the 1/2 mark.
5/8: half of 8 is 4. Numerator 5 > 4, so 5/8 is more than a half.
2/5: half of 5 is 2.5. Numerator 2 < 2.5, so 2/5 is less than a half.
One fraction is over a half, the other under it: 5/8 > 2/5. No converting, no drawing — just two fast questions.
Do not compare fractions digit by digit. 3/5 is not more than 2/3 just because 3 > 2 and 5 > 3. A fraction is a share of two numbers, not two separate numbers. Always say which of the three views you will use — or draw.
Practice: now you do it 🎲
Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.
Compare fractions. As the answer write only the sign: <, > or =.
On paper
By each comparison write which view you used: same denominators, same numerators, or a half.
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Compare: 5/12 and 7/12, then 4/5 and 4/7, then 5/9 and 3/7. By each pair write the sign and the view you used.
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Draw two identical rectangles and decide with a picture: which is more, 2/3, or 3/5? (That is Jonáš and Tereza’s row — settle it.)
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Order from smallest: 1/2, 3/8, 5/8, 1/4. Hint: they can all be converted to eighths, or use a half as a mark.
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Invent a fraction that is bigger than 1/3 and smaller than 1/2. Check it by comparing with both.
Now you 💪
- I can compare two fractions with the same denominator without thinking.
- I know why 2/3 is more than 2/5 — and I can say it in a sentence about the size of the parts.
- I can tell at a glance whether a fraction is over or under a half.
Done when: You compare two fractions with one of the three views (denominators, numerators, a half) and in a pinch you help yourself with a picture of two identical wholes.
What to take from this lesson
- Same denominators: the numerator decides. 5/8 > 3/8.
- Same numerators: the size of the part decides — a bigger denominator = a smaller fraction. 2/3 > 2/5.
- The 1/2 mark: the numerator against half the denominator. One over, the other under → 完了.
- A picture of two identical rectangles does not lie. The wholes must be the same size.
- A fraction is not compared digit by digit — it is one fact, not two.
© 2026 Ing. Martin Polak / AlgoRhino · コンテンツ利用規約