Lesson 3: Understand Proportional Relationships: Equivalent Ratios

Property

Two quantities $x$ and $y$ are in a proportional relationship if the quotient $y/x$ is a fixed number $r$ whenever $x$ is not zero.
This may also be written $y = rx$ or $x = y/r$ (when $r$ is nonzero).
In a proportional relationship, $r$ is the unit rate of $y$ with respect to $x$.
This same unit rate, $r$, is also called the constant of proportionality.

Examples

• The cost of apples is proportional to their weight. If they cost 2 dollars per pound ($r=2$), then 5 pounds will cost $y = 2 \times 5 = 10$ dollars.
• The distance a car travels at a constant speed is proportional to time. At 60 mph ($r=60$), in 2.5 hours you travel $y = 60 \times 2.5 = 150$ miles.
• The number of pages read is proportional to the time spent reading. If you read 25 pages per hour ($r=25$), after 3 hours you will have read $y = 25 \times 3 = 75$ pages.

Explanation

This means two quantities are perfectly in sync. If you double one, the other doubles too. Their relationship is defined by a constant multiplier, called the constant of proportionality, which is just another name for the unit rate.

Property

In a proportional relationship, the ratio between two quantities, $y$ and $x$, is always constant. This constant value is called the constant of proportionality, represented by the letter $k$. It is also known as the unit rate.