In plain words
How do you compare fractions with the same numerator?
Compare the size of the pieces. When the numerators match, both fractions have the same number of pieces, so the one with bigger pieces (the smaller denominator) is bigger. For example, 3/5 > 3/8, because fifths are bigger than eighths.
- The big idea
- When the top numbers match, both fractions have the same number of pieces, so compare the pieces. Fewer parts in the whole means bigger pieces: 3/5 > 3/8.
- The common confusion
- With the same top number, the fraction with the bigger bottom number is bigger (3/8 > 3/5, because 8 > 5), or the two are equal because both have 3 pieces. It often sounds like this: “(three fifths) and (three eighths) both have 3 pieces, so they’re equal.”
Why it works
Take two bars of exactly the same size. Cut one into 5 equal pieces and the other into 8 equal pieces.
Shade 3 pieces on each bar. Both fractions have the same number of pieces, so the only question is how big the pieces are.
Fifths are bigger than eighths, because the same whole is shared into fewer pieces. Three bigger pieces cover more than three smaller ones, so 3/5 > 3/8.
When two fractions have the same top number, the one with the smaller bottom number is bigger. Like every comparison, this only works when both fractions are of the same whole.
It is the same idea as 1/5 > 1/8, repeated three times: three fifths is three copies of one fifth, and three eighths is three copies of one eighth.
A worked example
Two identical pizzas: one is cut into 5 equal slices and the other into 8. Take 3 slices from each. The slices from the pizza cut into 5 are bigger, so those 3 slices are more pizza: 3/5 > 3/8.
Why this confusion is so common
It is natural to think a bigger number means a bigger amount, so 3/8 can look bigger than 3/5. Children who notice that both fractions have a 3 may decide they are equal. Both ideas focus on the digits rather than the pieces. The key question is what the bottom number tells us about the size of each piece.
Try this away from the screen
Fold one paper strip into 4 equal parts and an identical strip into 8 equal parts. Color 3 parts of each strip and lay the strips side by side. Ask: both strips have 3 colored parts, so why is one colored length longer?
Helpful things to say
| Instead of | Try |
|---|---|
| “The smaller the bottom number, the bigger the fraction.” | “Both have 3 pieces. Which pieces are bigger, and how do you know?” |
| “They both have 3, so they’re the same.” | “Three of what? Are the pieces the same size?” |
Practice questions
For grown-ups helping out: the answers are tucked away so learners can try first.
Two same-size pies, one cut into 6 equal slices and one into 9. Which is more pie: 4/6 or 4/9?
Show the answer
Answer: 4/6. Both amounts are 4 slices. Sixths are bigger than ninths, because the pie is shared into fewer slices, so 4/6 > 4/9.
Which is bigger: 2/5 or 2/3?
Show the answer
Answer: 2/3. Both fractions are 2 steps from 0. Thirds are longer steps than fifths, so 2/3 lands further right: 2/3 > 2/5.
Which is longer: three 1/4 pieces or three 1/6 pieces?
Show the answer
Answer: Three 1/4 pieces. Three pieces each time, so compare the pieces: a 1/4 piece is longer than a 1/6 piece. That makes 3/4 longer than 3/6.
Which of these is the smallest: 5/6, 5/12 or 5/8?
Show the answer
Answer: 5/12. All three have 5 pieces. Twelfths are the smallest pieces, because the whole is cut into the most parts, so 5/12 is the smallest.
Questions people ask
Why is 3/5 bigger than 3/8?
Both fractions have 3 pieces of the same whole. Fifths are bigger than eighths, because the whole is shared into fewer pieces, so three fifths is more than three eighths.
Is a fraction with a bigger denominator always smaller?
Only when the numerators are the same and the wholes are the same size. 3/8 is less than 3/5, but 7/8 is more than 3/5: it is just one eighth short of a whole.
How is this different from comparing 2/7 and 5/7?
With the same bottom number, the pieces are the same size, so you compare how many there are. With the same top number, the count is the same, so you compare the size of the pieces.
Words to know
- Numerator
- The top number of a fraction. It counts how many of the equal parts we have.
- Denominator
- The bottom number of a fraction. It tells how many equal parts make one whole.
- Unit fraction
- A fraction with 1 on top, such as 1/2, 1/3 or 1/8. It is one equal part of the whole.
- Whole
- The one complete thing, length or group that a fraction is a part of.
Curriculum connections
We don’t assume any particular curriculum. For teachers in the United States, here’s how this slice lines up with the Common Core State Standards:
- 3.NF.A.3.d
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Related slices
The Bigger Denominator Trap
Is (one sixth) bigger than (one eighth)?
Why 1/8 is smaller than 1/6, even though 8 is bigger than 6. · About 3 minutes
Same Bottom, Compare the Top
How do you compare fractions with the same denominator?
Why 5/7 is bigger than 2/7: the pieces are the same size, so the one with more pieces is bigger. · About 3 minutes
Compare Without Guessing
Which is bigger, (three eighths) or (four sevenths)?
How to compare fractions like 3/8 and 4/7 by asking which side of one half each one is on. · About 4 minutes
The Bigger Numerator Trap
Which is bigger, (three eighths) or (two thirds)?
Why 2/3 is bigger than 3/8, even though 3 is bigger than 2. · About 3 minutes
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