Many quantities vary in response to the variation of the input of some other quantity. For example, the speed of a car varies with the amount of fuel flowing into the engine.
When a quantity y varies with respect to another quantity x you say that y is directly proportional to x. In symbols, \[ y \propto x \] To get an equation you need to provide a constant, so that \[ y = kx \] and the constant k is called the constant of proportionality.
From the work on linear relationships you can see that the constant k is the gradient of the line passing through the origin.
Example Given a table of y and x values.
x | 1 | 2 | 3 | 4 | 5 | 6 |
y | 3 | 6 | 9 | 12 | 15 | 18 |
To find the constant of proportionality for the relation between y and x:
Form the proportion \[ \frac{y}{x} = \frac{a}{b} \] where a and b are a pair of x, y values from the table. Choose x = 2, y = 6. Then \[ \begin{align} \frac{y}{x} &= \frac{6}{2} \\ y &= \frac{6}{2}x \\ &= 3x \ \end{align}\] so the constant of proportionality is k = 3.
You can confirm the constant of proportionality is also the gradient of the line passing through the points.
Using the extreme points in the table above \[ \begin{align} \frac{\text{rise}}{\text{run}} &= \frac{18-3}{6-1} \\ &= \frac{15}{5} \\ &= 3 \ \end{align}\] Since the relationship is a straight line the gradient is constant.
Prediction
You can use proportional variation to predict a value outside the range of your data.
Example If y is proportional to x, and y = 6 when x = 3, to find the value of y when x = 9:
Form the proportion using the data \[ \begin{align} \frac{y}{x} &= \frac{6}{3} \\ \ \end{align}\]
Next, find the constant of proportionality \[ \begin{align} y &= \frac{6}{3}x \\ \ \end{align}\] and the constant of proportionality is k = 2.
To find the value of y when x = 9, substitute the given x value into the proportionality equation: \[ \begin{align} y &= 2x \\ &= 2 \times 9 \\ &= 18 \ \end{align}\]
Guided Examples
Conversion Graphs
You can use proportionality graphs to convert from one set of units to another.
Example Given a graph of metres to feet and inches.
To convert 16 feet to metres:
To convert 3 metres to feet:
Word Problems
Many word problems are about using proportional variation.
Example 8 workers can erect 72 metres of fence in 1 day. If the length of fence is proportional to the number of workers, how many metres of fence can 12 workers erect in the same time period.
Form the proportion. If f is the number of metres of fence and w is the number of workers then \[ \begin{align} \frac{f}{w} &= \frac{72}{8} \\ \ \end{align}\]
Next, find the constant of proportionality \[ \begin{align} \frac{f}{w} &= \frac{72}{8} \\ f &= \frac{72}{8}w \\ &= 9w \ \end{align}\] and the constant of proportionality is 9.
To find the value of f when w = 12, substitute the given w value into the proportionality equation: \[ \begin{align} f &= 9 \times w \\ &= 9 \times 12 \\ &= 108 \ \end{align}\] so 12 workers can erect 108 metres of fence in one day.