Lesson 2-1: Connect Ratios, Rates, and Proportions

Property

A ratio compares two quantities, often of the same kind.

A rate is a special type of ratio that compares quantities with different units (like miles and gallons, or dollars and hours).

Property

A proportion is an equation of the form $\frac{a}{b} = \frac{c}{d}$, where $b \neq 0, d \neq 0$. The proportion states two ratios or rates are equal. For any proportion of this form, its cross products are equal: $a \cdot d = b \cdot c$. Cross products can be used to test whether a proportion is true.

Examples

• The sentence "4 is to 9 as 20 is to 45" is written as the proportion $\frac{4}{9} = \frac{20}{45}$.
• To determine if $\frac{6}{11} = \frac{30}{55}$ is a proportion, we check the cross products. Since $6 \cdot 55 = 330$ and $11 \cdot 30 = 330$, the equation is a proportion.
• To check if $\frac{8}{10} = \frac{30}{40}$ is a proportion, we find the cross products. $8 \cdot 40 = 320$ and $10 \cdot 30 = 300$. Since the products are not equal, it is not a proportion.

Explanation

A proportion is a statement that two ratios are equal, like a balanced scale. The cross-product rule is a quick check: if the products of the numbers on the diagonal are equal, the ratios form a true proportion.