Lesson 3: Comparing Proportional and Nonproportional 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

A relationship between two quantities, $x$ and $y$, is non-proportional if the ratio $\frac{y}{x}$ is not constant for all related pairs of $(x, y)$. This means there is no single constant of proportionality.

Examples