Concept 4 of 9
The Matchmaker: Comparing with Common Denominators
Today's big idea
Rebuild both fractions onto a shared bottom, and then the bigger top wins.
2/3 and 3/4 become 8/12 and 9/12, so 3/4 is the bigger one.
How it works
👀 What's going on?
Which is bigger, 2/3 or 3/4? You cannot decide by looking at the tops. One pizza is cut into 3 and the other into 4, so the slices are not the same size to begin with.
💡 Here's the trick
Look at the two bars above. Both get rebuilt into twelfths, which makes every slice identical. 2/3 turns into 8/12 and 3/4 turns into 9/12. Now the tops can be compared directly, and 9 beats 8, so 3/4 is the bigger fraction.
🪄 How to do it
- 1.Find a number both bottoms divide into, as 12 does for the picture above.
- 2.Rename each fraction to that bottom by multiplying its top and bottom.
- 3.Compare the new tops, and the bigger top wins.
✍️ Try it
Which is bigger, 2/5 or 1/3? This time both bottoms have to change.
The smallest number that 5 and 3 both divide into is 15. That turns 2/5 into 6/15 and 1/3 into 5/15. Six beats five, so 2/5 is bigger. Neither fraction kept its original bottom, and that is perfectly normal.
⚠️ Watch out
You might be thinking a bigger bottom means a bigger fraction, since 5 is more than 3. It works the other way round. A bigger bottom cuts the whole into more pieces, so every piece comes out smaller.
🌟 Why this matters
Two shops sell rice in different bag sizes and you want the better deal. Two classes each finish part of a reading challenge. Any comparison like that needs the pieces measured at the same size first.
Exam tip: Multiplying the two bottoms together always gives a bottom that works, even when it is not the smallest one.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Which is bigger: 3/4 or 5/6?”
- 1
Find a common bottom
LCM of 4 and 6 is 12.
- 2
Convert 3/4
Multiply top & bottom by 3: 3/4 = 9/12.
- 3
Convert 5/6
Multiply top & bottom by 2: 5/6 = 10/12.
- 4
Compare the tops ✓
10 > 9 → 5/6 is bigger
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Tap a scaler (×1 to ×4) for each fraction until BOTH bottoms match. Then the bars are cut into the same-size pieces and you can compare them fairly.
2/3 × 1/1 = 2/3
3/4 × 1/1 = 3/4
Bottoms don't match yet…
Quick check
✅ VERIFICATION
Which is greater, 3/8 or 2/5?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Two Ways to Say It
1 3/4 and 7/4 are the same pizza counted differently.