Lesson 2: Proportions

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.

Property

Two variables are said to be proportional if their ratio is constant, or always the same. This means one variable is a constant multiple of the other. To check if two variables are proportional, you can identify several pairs of corresponding values for the variables, and then compute their ratios to see if they are equal.

Examples

• A baker uses 3 cups of sugar for every 2 dozen muffins. The amount of sugar is proportional to the number of dozens of muffins because the ratio $\frac{3}{2}$ is constant.
• A taxi fare includes a 3 dollars flat fee plus 2 dollars per mile. The total cost is not proportional to the miles driven because the ratio of cost to miles changes. For 2 miles, the ratio is $\frac{2 \times 2 + 3}{2} = 3.5$, but for 5 miles it is $\frac{2 \times 5 + 3}{5} = 2.6$.
• The perimeter of a regular octagon is given by the formula $P = 8s$, where $s$ is the side length. The perimeter is proportional to the side length because the ratio $\frac{P}{s} = 8$ is always constant.

Explanation

Think of it like this: if two variables are proportional, they are partners that always move together at a steady pace. If you double one variable, the other one doubles too. Their relationship is perfectly predictable and consistent.