Comparing fractions

Which is bigger, $\frac{3}{4}$ or $\frac{5}{6}$? It's not obvious — the bigger numerator might lose to the smaller denominator. Year 4 gives us three reliable tools for deciding.

Case 1 — same bottoms (denominators)

When the denominators match, the slice size is the same. So the bigger top wins.

Both fractions are pieces of a pizza cut into 5 slices. Having 4 slices beats having 2 slices.

Case 2 — same tops (numerators)

When the numerators match, you have the same number of pieces, but the pieces are different sizes. The bigger the bottom, the smaller each piece is.

Both fractions mean "3 pieces". But $\frac{1}{4}$ is a bigger slice than $\frac{1}{8}$, so 3 quarters beat 3 eighths.

⚠️ A bigger denominator means smaller slices. This is the most common Year-4 mistake — children see "8 is bigger than 4" and think $\frac{3}{8}>\frac{3}{4}$. It's the other way round.

Case 3 — different tops and different bottoms

The trick: rewrite both fractions with the same denominator using equivalent fractions, then use Case 1.

Which is bigger, $\frac{2}{3}$ or $\frac{3}{4}$?

Find a denominator both 3 and 4 divide into — the smallest is 12.

• $\frac{2}{3}=\frac{2\times 4}{3\times 4}=\frac{8}{12}$
• $\frac{3}{4}=\frac{3\times 3}{4\times 3}=\frac{9}{12}$

Now both have denominator 12. Compare the tops: 8 < 9. So $\frac{2}{3}<\frac{3}{4}$.

How to pick a common denominator

The smallest "common bottom" of two fractions is called the lowest common denominator. A simple rule of thumb:

• If one denominator divides the other (like 3 divides 6), use the bigger denominator.
• Otherwise, multiply the two denominators (3 and 4 → 12).

Year 4 sticks to small denominators (2, 3, 4, 5, 6, 8, 10, 12), so finding a common one is rarely tricky.

Compare with a benchmark — $\frac{1}{2}$

A quick trick that often works: compare each fraction to $\frac{1}{2}$.

Which is bigger, $\frac{2}{5}$ or $\frac{4}{7}$?

• $\frac{2}{5}$: half of 5 is 2.5, so 2 out of 5 is less than half.
• $\frac{4}{7}$: half of 7 is 3.5, so 4 out of 7 is more than half.

That's enough — $\frac{2}{5}<\frac{1}{2}<\frac{4}{7}$, so $\frac{2}{5}<\frac{4}{7}$.

A worked example

Put these fractions in order from smallest to largest: $\frac{3}{8}$, $\frac{1}{2}$, $\frac{5}{8}$, $\frac{3}{4}$.

Rewrite everything with denominator 8:

• $\frac{1}{2}=\frac{4}{8}$
• $\frac{3}{4}=\frac{6}{8}$
• $\frac{3}{8}$ and $\frac{5}{8}$ stay as they are.

Now order the tops: 3, 4, 5, 6. Answer:

What's next

• Worked examples — adding, subtracting, mixed numbers
• Equivalent fractions — the tool that makes comparing work
• Back to the introduction

Try it out

• ⚖️ Compare fractions with different denominators
• 🟰 Equivalent fractions