Math Core

Is $\dfrac{2}{3}$ of a pan of brownies more or less than $\dfrac{3}{4}$ of it? The numerators are different and the denominators are different, so you can't just count pieces. Equivalent fractions fix that: rename one or both fractions so they are easy to compare.

What you already know

Remember the two easy cases from third grade (always with the same size whole):

Same denominator: the pieces are the same size, so more pieces is more. $\dfrac{5}{8} > \dfrac{3}{8}$
Same numerator: you have the same number of pieces, so bigger pieces win. A smaller denominator means bigger pieces. $\dfrac{2}{3} > \dfrac{2}{5}$

The trick for harder comparisons is to turn them into one of these easy cases.

Strategy 1: Make the denominators the same

Compare $\dfrac{2}{3}$ and $\dfrac{3}{4}$ . Both 3 and 4 go into 12, so rename both fractions as twelfths:

\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \qquad\qquad \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}

Now the pieces are the same size. 9 twelfths is more than 8 twelfths, so $\dfrac{3}{4} > \dfrac{2}{3}$ .

Two equal bars cut into twelfths. Top: 2/3 = 8/12 is shaded. Bottom: 3/4 = 9/12 is shaded.

Often you only need to rename one fraction. To compare $\dfrac{1}{2}$ and $\dfrac{3}{8}$ , write $\dfrac{1}{2} = \dfrac{4}{8}$ . Then $\dfrac{4}{8} > \dfrac{3}{8}$ , so $\dfrac{1}{2} > \dfrac{3}{8}$ .

Strategy 2: Make the numerators the same

Sometimes it's quicker to match the tops. Compare $\dfrac{3}{4}$ and $\dfrac{6}{10}$ . Multiply $\dfrac{3}{4}$ by $\dfrac{2}{2}$ to get $\dfrac{6}{8}$ . Now both fractions are 6 pieces. Eighths are bigger than tenths, so $\dfrac{6}{8} > \dfrac{6}{10}$ , which means $\dfrac{3}{4} > \dfrac{6}{10}$ .

Strategy 3: Compare to a benchmark

A benchmark is an easy fraction you know well, like $\dfrac{1}{2}$ or $1$ . Compare $\dfrac{3}{8}$ and $\dfrac{4}{6}$ :

Half of 8 is 4, and 3 is less than 4, so $\dfrac{3}{8}$ is less than $\dfrac{1}{2}$ .
Half of 6 is 3, and 4 is more than 3, so $\dfrac{4}{6}$ is more than $\dfrac{1}{2}$ .

One is below one half and one is above, so $\dfrac{3}{8} < \dfrac{4}{6}$ . No renaming needed!

You can also think about how far a fraction is from 1 whole. $\dfrac{7}{8}$ is missing $\dfrac{1}{8}$ , and $\dfrac{5}{6}$ is missing $\dfrac{1}{6}$ . Eighths are smaller, so $\dfrac{7}{8}$ is missing less, and $\dfrac{7}{8} > \dfrac{5}{6}$ .

Ways to compare two fractions

Same denominator: rename so the pieces are the same size, then compare numerators.
Same numerator: rename so the number of pieces is the same, then the smaller denominator is greater.
Benchmark: check if each fraction is more or less than $\dfrac{1}{2}$ (or how close it is to $1$ ).

Write the result with $>$ , $<$ or $=$ . Comparisons only make sense when both fractions refer to the same whole.

Worked example: Rename one fraction

Compare $\dfrac{5}{12}$ and $\dfrac{1}{3}$ .

12 is $3 \times 4$ , so rename $\dfrac{1}{3}$ as twelfths: $\dfrac{1}{3} = \dfrac{1 \times 4}{3 \times 4} = \dfrac{4}{12}$ .

Now $\dfrac{5}{12} > \dfrac{4}{12}$ , so

\frac{5}{12} > \frac{1}{3}

Worked example: Tenths and fifths

Compare $\dfrac{7}{10}$ and $\dfrac{4}{5}$ .

Rename fifths as tenths: $\dfrac{4}{5} = \dfrac{8}{10}$ . Since 7 tenths is less than 8 tenths,

\frac{7}{10} < \frac{4}{5}

Worked example: Use one half

Compare $\dfrac{3}{5}$ and $\dfrac{5}{12}$ .

Half of 5 is $2\tfrac{1}{2}$ , and 3 is more than that, so $\dfrac{3}{5} > \dfrac{1}{2}$ . Half of 12 is 6, and 5 is less than 6, so $\dfrac{5}{12} < \dfrac{1}{2}$ .

\frac{3}{5} > \frac{5}{12}

Worked example: Which is longer?

A green ribbon is $\dfrac{5}{6}$ of a yard long. A red ribbon is $\dfrac{2}{3}$ of a yard long. Which is longer?

Rename thirds as sixths: $\dfrac{2}{3} = \dfrac{4}{6}$ . Then $\dfrac{5}{6} > \dfrac{4}{6}$ , so the green ribbon is longer.

Common mistake

Don't compare just the numerators or just the denominators when both are different. $\dfrac{2}{3}$ has smaller numbers than $\dfrac{5}{10}$ , but $\dfrac{2}{3}$ is greater: it is more than half, and $\dfrac{5}{10}$ is exactly half. Rename first, or use a benchmark.

Tip

If the fractions turn out equal after renaming, write $=$ . For example, $\dfrac{3}{4} = \dfrac{9}{12}$ , so $\dfrac{3}{4}$ ☐ $\dfrac{9}{12}$ gets the $=$ sign.

Practice

Practice 1

Which symbol makes this true? $\dfrac{1}{2}$ ☐ $\dfrac{5}{8}$

Practice 2

Which symbol makes this true? $\dfrac{3}{4}$ ☐ $\dfrac{7}{12}$

Practice 3

Which symbol makes this true? $\dfrac{6}{10}$ ☐ $\dfrac{3}{5}$

Practice 4

To compare $\dfrac{2}{3}$ and $\dfrac{5}{6}$ , Mia renames both as twelfths. $\dfrac{2}{3} = \dfrac{8}{12}$ . What is the missing numerator in $\dfrac{5}{6} = \dfrac{?}{12}$ ?

Type a number.

Practice 5

Which symbol makes this true? $\dfrac{4}{6}$ ☐ $\dfrac{2}{5}$

Practice 6

Which fraction is greater than $\dfrac{3}{4}$ ?

Practice 7

Sam ran $\dfrac{3}{5}$ of a mile. Ella ran $\dfrac{7}{10}$ of a mile. Who ran farther?

Practice 8

Put these fractions in order from least to greatest: $\dfrac{2}{3}$ , $\dfrac{1}{4}$ , $\dfrac{7}{12}$ .