# Same Bottom, Compare the Top

> **How do you compare fractions with the same denominator?** Compare the numerators. When the denominators match, both fractions are made of the same-size pieces, so the one with more pieces is bigger. For example, 5/7 is bigger than 2/7, because five sevenths is more than two sevenths of the same whole.

- Big idea: When the bottom numbers match, the pieces are the same size, so you can just count them. That’s why 5/7 is more than 2/7.
- Common confusion: Fractions with the same bottom number must be the same size, because both are “sevenths”. Or the smaller top number makes the bigger fraction, the way the smaller bottom number did for 1/6 and 1/8.
- It often sounds like this: “2/7 and 5/7 are both sevenths, so they’re the same size.”
- Family: [Same Value, Different Forms](https://slices.org/explore/equivalence-and-comparison) (slice 13 of 20)
- [Try the lesson yourself](https://slices.org/slices/same-bottom-compare-the-top): about 3 minutes, free, with no sign-in.

## In plain words

Take two bars of exactly the same size and cut each into 7 equal pieces. Every piece on both bars is one seventh, and all the sevenths are the same size.

Shade 2 pieces on one bar and 5 on the other. Because the pieces match, comparing the fractions comes down to comparing the counts, and 5 pieces is more than 2 pieces.

So 5/7 is bigger than 2/7. When two fractions have the same bottom number, the one with the bigger top number is bigger.

The bottom number tells you the size of the pieces and the top number tells you how many there are. When the size is the same, only the count matters. This only works when both fractions are of the same whole, because 3/8 of a large pizza can be more than 5/8 of a small one.

On a number line, 2/7 is 2 equal steps from 0 and 5/7 is 5 of the same steps. The point further to the right is the bigger number.

## A worked example

Two friends have identical chocolate bars, each made of 8 equal squares. One eats 3 squares, which is 3/8 of a bar, and the other eats 5 squares, which is 5/8 of a bar. The squares are all the same size, so 5 squares is more chocolate and 5/8 is bigger than 3/8.

## Why this confusion is common

People who have just learned that 1/8 is smaller than 1/6 sometimes carry away the rule “smaller number, bigger fraction” and apply it to the top number too. Others notice that both fractions are “sevenths” and decide they must be the same size. Both come from looking at the digits instead of the pieces. Once the same-size pieces are visible, the comparison becomes a matter of counting.

## Try this away from the screen

Take two identical egg cartons with 12 cups each, and put a small object in 5 cups of one carton and in 8 cups of the other. Each cup is 1/12 of a carton. Ask whether the cups are the same size, then which carton is fuller, 5/12 or 8/12, and how they know.

## Helpful ways to talk about it

| Instead of saying | Try saying |
| --- | --- |
| “Just look at the top numbers.” | “Are the pieces the same size? If they are, what do we need to compare?” |
| “The bigger number always wins.” | “What does the 7 tell us about the pieces? What does the 5 tell us?” |

## Practice questions

[Try these questions one at a time](https://slices.org/set?t=Practice%3A+Same+Bottom%2C+Compare+the+Top&q=s13-transfer-1%2Cs13-transfer-2%2Cs13-review-1%2Cs13-review-2), with pictures you can change and a reply to every answer.

1. A pizza is cut into 8 equal slices. Which is more pizza: 3/8 or 5/8? **Answer:** 5/8. Every slice is one eighth of the same pizza, so you only need to compare how many there are. 5 slices is more than 3, so 5/8 is more pizza than 3/8.
2. Which is the largest: 4/9, 7/9 or 2/9? **Answer:** 7/9. Every step on this line is one ninth. 7/9 is 7 steps from 0, which is further right than 4 steps or 2 steps, so it’s the largest.
3. Which is smaller: 7/10 or 3/10? **Answer:** 3/10. Both fractions count tenths of the same bar, and 3 tenths is fewer pieces than 7 tenths, so 3/10 is smaller.
4. Which comparison is true? The choices are 9/12 < 5/12, 9/12 = 5/12 and 9/12 > 5/12. **Answer:** 9/12 > 5/12. 9/12 and 5/12 count pieces of the same size, so the one with more pieces is greater, and that’s 9/12.

## Questions people ask

### Why is 5/7 bigger than 2/7?

Both fractions are made of sevenths, so the pieces are the same size. Five pieces of the same size is more than two of them.

### Does comparing the top numbers always work?

No, only when the bottom numbers are the same and both fractions are of the same whole. If the bottom numbers are different, the pieces are different sizes, so you need another way, such as comparing piece sizes when the top numbers match, or comparing each fraction with one half.

### How do you write the comparison with < or >?

The open side of the sign faces the bigger number. Write 5/7 > 2/7 and read it as “five sevenths is greater than two sevenths”, or write 2/7 < 5/7, which reads “two sevenths is less than five sevenths”.

## Curriculum connections (US Common Core)

- [3.NF.A.3.d](https://www.thecorestandards.org/Math/Content/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.

## Words to know

- [Numerator](https://slices.org/glossary/numerator): The numerator is the top number of a fraction, and it counts how many of the equal parts you have.
- [Denominator](https://slices.org/glossary/denominator): The denominator is the bottom number of a fraction, and it tells you how many equal parts make one whole.
- [Equal parts](https://slices.org/glossary/equal-parts): Equal parts are exactly the same size as each other. A fraction only makes sense when the whole has been split into parts like that.
- [Whole](https://slices.org/glossary/whole): The whole is the complete thing that a fraction is part of. It might be a pizza, a length on a number line or a group of objects.

## Related slices

- [Same Top, Compare the Pieces](https://slices.org/slices/same-top-compare-the-pieces): Both 3/5 and 3/8 have 3 pieces, but fifths are bigger than eighths, so 3/5 is the bigger fraction.
- [The Top Number](https://slices.org/slices/the-top-number): The top number counts the parts you have, so 3/5 means 3 of the 5 equal parts.
- [The Bigger Numerator Trap](https://slices.org/slices/bigger-numerator-trap): Even though 3/8 has the bigger top number, 2/3 is the bigger fraction, because thirds are much bigger pieces than eighths.
- [Compare Without Guessing](https://slices.org/slices/compare-without-guessing): When the numbers are all different, you can compare each fraction with one half instead of guessing.