Skill: Compare fractions with unlike denominators

Fractions must describe parts of the same-sized whole before you compare them. When the denominators are different, rename the fractions with a common denominator.

Compare:

$$\frac{3}{4}\quad\text{and}\quad\frac{5}{6}$$

1. Find a common denominator.
   A number that both $4$ and $6$ divide into is $12$.

2. Rename each fraction using denominator $12$:

$$\frac{3}{4}=\frac{3\times3}{4\times3}=\frac{9}{12}$$ $$\frac{5}{6}=\frac{5\times2}{6\times2}=\frac{10}{12}$$

3. Compare the numerators. Since $10>9$,

$$\frac{10}{12}>\frac{9}{12}$$

4. Write the comparison using the original fractions:

$$\boxed{\frac{5}{6}>\frac{3}{4}}$$

The denominator $6$ is larger than $4$, but that does not mean $\frac{5}{6}$ is smaller. The fractions must first be renamed so they show the same-sized pieces. Here, both fractions are measured in twelfths, and $10$ twelfths is greater than $9$ twelfths.