Comparing and Ordering Fractions

Why we compare fractions

In everyday life we often need to know which share is bigger. Did you eat more pizza than your friend? Is 3/4 of an hour longer than 2/3 of an hour? Comparing fractions answers questions like these, and ordering fractions lines several of them up from smallest to largest.

Remember what a fraction means: the denominator (bottom) tells you how many equal pieces the whole is cut into, and the numerator (top) tells you how many pieces you have. If you need a reminder, our Introduction to Fractions lesson explains the basics. There are three situations, and we will take them one at a time.

Same denominator: compare the numerators

When two fractions have the same denominator, the pieces are the same size. So you just look at how many pieces each one has — the numerators.

Rule: Same denominator? The bigger numerator wins.

Example 1 — Compare 3/8 and 5/8.

1. Both are eighths, so the pieces are equal in size.
2. Compare the tops: 5 is bigger than 3.
3. So 5/8 > 3/8 (5/8 is greater than 3/8).

Example 2 — Order 2/7, 6/7, 4/7 from smallest to largest.

1. All sevenths, so just order the numerators: 2, 4, 6.
2. So the order is 2/7, 4/7, 6/7.

Same numerator: compare the denominators

This case surprises people. When the numerators are the same, you might think the bottom numbers behave the same way — but they are the opposite. A bigger denominator means smaller pieces.

Rule: Same numerator? The smaller denominator wins, because the pieces are bigger.

Example 3 — Compare 1/3 and 1/5.

Imagine sharing one chocolate bar. If you share it between 3 people, each gets a big piece (1/3). If you share it between 5 people, each gets a smaller piece (1/5). So 1/3 > 1/5, even though 5 is a bigger number than 3.

Example 4 — Order 2/4, 2/9, 2/5 from smallest to largest.

1. Same numerator (2), so the bigger the denominator, the smaller the fraction.
2. Denominators in order from largest to smallest: 9, 5, 4.
3. So smallest to largest is 2/9, 2/5, 2/4.

Different numerators and denominators: find a common denominator

The trickiest case is when both numbers differ, like 2/3 and 3/4. You cannot compare them directly because the pieces are different sizes. The fix is to rewrite both with a common denominator — a number both denominators divide into — so the pieces match. This uses Equivalent Fractions.

Example 5 — Compare 2/3 and 3/4.

1. Find a common denominator for 3 and 4. The smallest is 12.
2. Rewrite each with denominator 12:
3. 2/3 = (2×4)/(3×4) = 8/12
4. 3/4 = (3×3)/(4×3) = 9/12
5. Now the pieces match. Compare the tops: 9 > 8.
6. So 3/4 > 2/3.

Example 6 — Order 1/2, 2/3, 3/4 from smallest to largest.

1. A common denominator for 2, 3 and 4 is 12.
2. Rewrite each: 1/2 = 6/12, 2/3 = 8/12, 3/4 = 9/12.
3. Order the numerators: 6, 8, 9.
4. So the order is 1/2, 2/3, 3/4.

A reference table

Fractions	Same...?	Rule used	Result
3/8 and 5/8	denominator	bigger numerator wins	5/8 > 3/8
1/3 and 1/5	numerator	smaller denominator wins	1/3 > 1/5
2/3 and 3/4	neither	common denominator (12)	8/12 < 9/12, so 3/4 > 2/3
3/5 and 7/10	neither	common denominator (10)	6/10 < 7/10, so 7/10 > 3/5

Why the denominator rule flips

It is worth pausing on the why, because the same-numerator case is where most mistakes happen. The denominator does not count pieces — it tells you how big each piece is. Cutting a whole into more pieces makes each piece smaller. So a 9 on the bottom means tiny pieces, while a 4 on the bottom means chunky pieces. That is why, with equal numerators, the fraction with the smaller denominator is the bigger fraction. Always think about the size of the pieces, not just the size of the numbers.

A practice activity

Cut three identical paper strips:

1. Fold the first into 2 equal parts and shade 1 part: that is 1/2.
2. Fold the second into 3 equal parts and shade 2 parts: that is 2/3.
3. Fold the third into 4 equal parts and shade 3 parts: that is 3/4.
4. Lay the strips on top of one another and compare the shaded lengths. You will see that 1/2 < 2/3 < 3/4.

Then try these on paper using <, > or = (answers below): (a) 4/9 ? 7/9, (b) 1/2 ? 1/8, (c) 2/5 ? 1/2, (d) 3/6 ? 1/2.

Answers: (a) 4/9 < 7/9, (b) 1/2 > 1/8, (c) 2/5 < 1/2 (since 4/10 < 5/10), (d) 3/6 = 1/2.

Where this leads

Comparing and ordering fractions builds directly on equivalent fractions and prepares you for adding, subtracting and converting fractions into decimals and percentages. The key habit is always to ask: are the pieces the same size? If not, make them match first.