Math Core

Would you rather have $\dfrac{3}{8}$ of a pizza or $\dfrac{5}{8}$ ? What about $\dfrac{1}{3}$ or $\dfrac{1}{6}$ ? You can decide which fraction is bigger without cutting a single slice, as long as you know two simple rules.

The same whole

Before you compare, make sure both fractions are parts of the same size whole. Half of a watermelon is more fruit than half of a grape! In this lesson, every comparison uses the same whole.

We use three symbols to compare:

Symbol	Meaning
$>$	is greater than
$<$	is less than
$=$	is equal to

Rule 1: Same denominator

When the denominators are the same, the pieces are the same size. Then just count the pieces. More pieces means more.

🟩🟩🟩⬜⬜⬜⬜⬜ is $\dfrac{3}{8}$

🟩🟩🟩🟩🟩⬜⬜⬜ is $\dfrac{5}{8}$

5 eighths is more than 3 eighths, so $\dfrac{5}{8} > \dfrac{3}{8}$ .

You can see this on a number line too. The fraction farther to the right is greater.

Rule 2: Same numerator

When the numerators are the same, you have the same number of pieces. Now check the piece size. Remember: a bigger denominator means the whole is cut into more parts, so each piece is smaller.

$\dfrac{1}{3}$ is 1 piece from a whole cut into 3. $\dfrac{1}{6}$ is 1 piece from a whole cut into 6. Thirds are bigger pieces, so $\dfrac{1}{3} > \dfrac{1}{6}$ .

The dot at $\dfrac{2}{6}$ is the same point as $\dfrac{1}{3}$ , and it is to the right of $\dfrac{1}{6}$ .

Two ways to compare

Same denominator: the fraction with the bigger numerator is greater. $\dfrac{4}{6} > \dfrac{1}{6}$
Same numerator: the fraction with the smaller denominator is greater. $\dfrac{2}{3} > \dfrac{2}{8}$

Common mistake

With the same numerator, don't pick the bigger denominator! $\dfrac{1}{8}$ is less than $\dfrac{1}{4}$ , even though 8 is more than 4. Eighths are smaller pieces.

Worked example: Same denominator

Compare $\dfrac{2}{4}$ and $\dfrac{3}{4}$ .

The denominators are both 4, so the pieces are the same size. 3 pieces is more than 2 pieces.

\frac{2}{4} < \frac{3}{4}

Worked example: Same numerator

Compare $\dfrac{3}{8}$ and $\dfrac{3}{6}$ .

Both have 3 pieces. Sixths are bigger than eighths, because 6 parts per whole is fewer than 8. So 3 sixths is more.

\frac{3}{8} < \frac{3}{6}

Worked example: A word problem

Eli walked $\dfrac{2}{3}$ of a mile. Ana walked $\dfrac{2}{4}$ of a mile. Who walked farther?

Same numerator (2), so compare piece size. Thirds are bigger than fourths, so $\dfrac{2}{3} > \dfrac{2}{4}$ . Eli walked farther.

Tip

Compare to one half as a check. $\dfrac{3}{8}$ is less than $\dfrac{4}{8}$ , which is half. $\dfrac{3}{6}$ is exactly half. So $\dfrac{3}{8} < \dfrac{3}{6}$ , just like we found.

Practice

Practice 1

Which symbol makes this true? $\dfrac{5}{6}$ ☐ $\dfrac{2}{6}$

Practice 2

Which symbol makes this true? $\dfrac{1}{8}$ ☐ $\dfrac{1}{2}$

Practice 3

Which fraction is the greatest?

Practice 4

Which symbol makes this true? $\dfrac{2}{3}$ ☐ $\dfrac{2}{6}$

Practice 5

Jada ate $\dfrac{1}{6}$ of a pie. Kai ate $\dfrac{1}{3}$ of the same pie. Who ate more?

Practice 6

Which symbol makes this true? $\dfrac{2}{4}$ ☐ $\dfrac{4}{8}$

Practice 7

Put these in order from least to greatest: $\dfrac{3}{4}$ , $\dfrac{3}{8}$ , $\dfrac{3}{6}$ .