Lesson 2: Proportions

Property

Two quantities are in a proportional relationship if the ratio between them is constant. This can be verified in two main ways:

1. Using a table: Test for equivalent ratios. For any pair of corresponding quantities $(x, y)$, the ratio $\frac{y}{x}$ must be the same for all non-zero pairs.
2. Using a graph: The graph of the relationship must be a straight line that passes through the origin $(0, 0)$.

Examples

• A table shows hours worked and earnings. If 2 hours earns 30 dollars, 3 hours earns 45 dollars, and 5 hours earns 75 dollars, the relationship is proportional because the rate is always 15 dollars per hour.
• A recipe calls for 2 cups of flour for every 1 cup of sugar. A graph of flour vs. sugar would be a straight line through $(0,0)$ and $(1,2)$, showing it's proportional.
• A cell phone plan costs 10 dollars per month plus 1 dollar per gigabyte. A graph of the cost would be a line starting at $(0,10)$, not the origin, so it is not proportional.

Explanation

Think of it like a recipe. If you double the flour, you must double the sugar. A proportional relationship means two quantities scale up or down together at a steady rate. The graph is a straight line starting from zero because zero input means zero output.

Property

A proportion is an equation stating that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$

Cross Products Property: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$ where $b \neq 0$ and $d \neq 0$

Property

In a proportional relationship, the unit rate represents the constant of proportionality. When comparing two proportional relationships, you can use their unit rates to determine which relationship has a greater or lesser rate of change. The unit rate is found by simplifying the ratio to have a denominator of 1.

Examples