Lesson 6: Direct Variation

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

Proportional relationships can be represented by an equation of the form $y = kx$ or $y = rx$. In this equation:

• $x$ is the independent variable (input).
• $y$ is the dependent variable (output).
• $r$ (or $k$) is the constant of proportionality (the unit rate).

This equation shows that the output is always a constant multiple of the input.

Examples

• A machine prints 80 pages in 5 minutes. The unit rate is $r = \frac{80}{5} = 16$ pages per minute. The equation is $p = 16m$, where $p$ is pages and $m$ is minutes.
• The cost for apples is 2.50 dollars per pound. If $C$ is the total cost and $p$ is the number of pounds, the equation is $C = 2.5p$.
• A graph of a proportional relationship passes through $(4, 32)$. The unit rate is $\frac{32}{4}=8$. The equation representing this graph is $y = 8x$.

Explanation

An equation is like a powerful calculator for a proportional relationship. Once you find the constant rate ($r$), you can plug in any amount for $x$ to instantly find its corresponding amount $y$, without having to fill out a huge table.