In plain words
Which is bigger, 3/8 or 2/3?
2/3 is bigger. 3/8 is three small parts (eighths), which is less than half of the whole. 2/3 is two much bigger parts (thirds), which is more than half. So 2/3 > 3/8, even though 3 is bigger than 2.
- The big idea
- A fraction’s size depends on both numbers together: the top counts the parts, and the bottom sets how big each part is.
- The common confusion
- A larger numerator means a larger fraction, so 3/8 must be bigger than 2/3. It often sounds like this: “(three eighths) has a 3 and (two thirds) only has a 2, so (three eighths) is bigger.”
Why it works
A fraction uses two numbers that do different jobs. The bottom number (the denominator) says how many equal parts the whole is cut into, which sets the size of each part. The top number (the numerator) counts how many of those parts we have.
In 3/8, the whole is cut into 8 equal parts and we take 3. In 2/3, the same whole is cut into only 3 equal parts and we take 2. Each third is more than twice the size of each eighth.
Shade both on bars of the same length and line them up. 3/8 covers less than half of its bar, and 2/3 covers more than half. So 2/3 > 3/8, even though 3 is bigger than 2.
A bigger numerator only means a bigger fraction when the parts are the same size, as with 3/8 and 5/8. When the denominators differ, think about both numbers together: how many parts, and how big each part is.
One half is a handy checkpoint. Half of 8 is 4, so 3/8 is less than half. Half of 3 is one and a half, so 2/3 is more than half.
A worked example
Two friends have chocolate bars of the same size. Ali’s bar is snapped into 8 equal pieces and he eats 3. Bea’s bar is snapped into 3 equal pieces and she eats 2. Ali eats more pieces, but Bea eats more chocolate: 2/3 of a bar is more than 3/8 of a bar.
Why this confusion is so common
With whole numbers, a bigger number always means more, so it is natural to judge a fraction by its top number. Many children first compare fractions with the same denominator, such as 3/8 and 5/8, where the top number does decide. With different denominators, the size of the parts matters as much as how many there are. This is another face of whole-number bias, the habit of reading a fraction as if it were made of whole numbers.
Try this away from the screen
Cut two paper strips of the same length. Fold one into 8 equal parts (fold it in half three times) and color 3. Fold the other into 3 equal parts and color 2. Lay the strips side by side. Then ask your child to find a fraction with a top number bigger than 2 that is still smaller than 2/3, and check it with another strip.
Helpful things to say
| Instead of | Try |
|---|---|
| “Look at the top numbers: 3 is more than 2.” | “How many pieces are there, and how big is each piece?” |
| “Thirds are bigger, so 2/3 wins.” | “Is each fraction more or less than half of the whole? How do you know?” |
Practice questions
For grown-ups helping out: the answers are tucked away so learners can try first.
Which is bigger: 4/9 or 3/4?
Show the answer
Answer: 3/4. Four ninths stops before the middle of the line, because 4 ninths is less than half. Three fourths reaches well past the middle. So 3/4 > 4/9.
Sam eats 5/12 of a pizza. Ana eats 3/5 of a pizza the same size. Who eats more pizza?
Show the answer
Answer: Ana. Half of 12 slices is 6, so 5/12 is less than half a pizza. Half of 5 slices is 2 and a half, so 3/5 is more than half. Ana eats more.
Which is more: four 1/10 pieces, or three 1/6 pieces?
Show the answer
Answer: Three 1/6 pieces. Three sixths make exactly half a whole. Five tenths would make half, so four tenths is a little less. Three 1/6 pieces are more.
Which of these is the largest?
Show the answer
Answer: 4/5. 4 of 5 parts is most of a bar. 7 of 16 is less than half, because 8 of 16 would be half, and 2 of 9 is even less. So 4/5 is the largest.
Questions people ask
Why is 2/3 bigger than 3/8 when 3 is bigger than 2?
Because the pieces are different sizes. A third of a whole is more than twice as big as an eighth, so two thirds is more than half of the whole, while three eighths is less than half.
When does a bigger numerator mean a bigger fraction?
When the denominators are the same, so every part is the same size: 5/8 is bigger than 3/8. When the denominators are different, you need to think about the size of the parts as well as how many there are.
How do you compare fractions with different numerators and denominators?
Draw both on bars of the same length or on one number line, compare each with one half, or rewrite both with a common denominator. For example, 3/8 = 9/24 and 2/3 = 16/24, so 2/3 is bigger.
So does a smaller numerator mean a bigger fraction?
No. Neither number decides on its own. 2/3 is bigger than 3/8, but 2/9 is smaller than 3/8. A fraction’s size comes from both numbers together: how many parts, and how big they are.
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.
- Benchmark fraction
- A familiar fraction, such as 1/2, used as a reference point for comparing other fractions.
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:
- 4.NF.A.2
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Related slices
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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
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Why 3/5 is bigger than 3/8: the same number of pieces, but fifths are bigger pieces than eighths. · About 3 minutes
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How to compare fractions like 3/8 and 4/7 by asking which side of one half each one is on. · About 4 minutes
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