Skill: Compare ratios using equivalent forms

To compare two ratios, rewrite them in an equivalent form that makes the comparison easy. You may multiply or divide both terms of a ratio by the same nonzero number. This keeps the ratio’s value unchanged.

Example: Which is greater, $3:5$ or $4:7$?

Rewrite each ratio as a fraction:

$$3:5=\frac{3}{5} \qquad\text{and}\qquad 4:7=\frac{4}{7}$$

Find a common denominator. Both $5$ and $7$ divide into $35$.

• Multiply $\frac{3}{5}$ by $\frac{7}{7}$:

$$\frac{3}{5}=\frac{3\cdot7}{5\cdot7}=\frac{21}{35}$$

• Multiply $\frac{4}{7}$ by $\frac{5}{5}$:

$$\frac{4}{7}=\frac{4\cdot5}{7\cdot5}=\frac{20}{35}$$

Now compare the numerators:

$$\frac{21}{35}>\frac{20}{35}$$

Therefore,

$$\boxed{3:5>4:7}$$

Remember to multiply or divide both parts of each ratio by the same number. Comparing the terms separately—for example, saying $4:7$ is greater because $4>3$—can lead to an incorrect conclusion.