Lesson 2: Identifying Proportional Relationships

Property

A relationship between two quantities, $x$ and $y$, is proportional if their ratio $\frac{y}{x}$ is constant for all corresponding non-zero values. This constant ratio is called the constant of proportionality, $k$. The relationship can be described by the equation:

Examples

The cost of buying apples at $3 per apple is a proportional relationship.

• 2 apples cost $6$: $\frac{6}{2} = 3$
• 5 apples cost $15$: $\frac{15}{5} = 3$
• The ratio is constant, so the relationship is proportional with the equation $y = 3x$.

A car traveling at a constant speed of 50 miles per hour represents a proportional relationship between time ($x$) and distance ($y$).

• In 2 hours, the car travels 100 miles: $\frac{100}{2} = 50$
• In 3.5 hours, the car travels 175 miles: $\frac{175}{3.5} = 50$
• The ratio is constant, so the relationship is proportional with the equation $y = 50x$.

Explanation

To determine if a relationship is proportional, check if the ratio of the dependent variable ($y$) to the independent variable ($x$) is the same for every pair of values. If this ratio is constant, the relationship is proportional. This constant value, often denoted by $k$, is the constant of proportionality. A key feature of proportional relationships is that when one quantity is zero, the other must also be zero.

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.