Inverse relationships
Some quantities have what is called an inverse relationship. As one of the quantities increases, the other decreases, and vice versa.
An example of this is the relationship between speed and time. Imagine a truck driver needs to drive 1000 km to deliver a load. The faster the driver travels, the less time it will take to cover this distance.
In the table below, the formula \text{Time} = \dfrac{\text{Distance}}{\text{Speed}} is used to find the time it would take to drive 1000 km at different average speeds (these times are rounded to one decimal place).
| Speed (km/h) | 60 | 70 | 80 | 90 | 100 | 110 | 120 |
|---|---|---|---|---|---|---|---|
| Time (hours) | 16.67 | 14.3 | 12.3 | 11.1 | 10 | 9.1 | 8.3 |
Plotting these values gives a hyperbola:
Notice that if the driver were to travel at an average speed of 40 km/h, it would take 25 hours (or just over a day.) to complete the journey.
Each point on the curve corresponds to a different combination of speed and time, and all the points on the curve satisfy the equation T = \dfrac{1000}{S} or TS = 1000, where T is the time taken and S is the speed travelled. From both equations, as average speed increases, the time taken will decrease.
An inverse relationship is a relationship between two quantities where when one increases, the other decreases, and vice versa.
Inverse variation
Two quantities x and y that are inversely proportional if {y}\propto{\frac{1}{x}}. That is, they have an equation of the form: y = k/x \text{ or } xy = k \text{ or } x = k/y,where k can be any number other than 0. Again, k is the constant of proportionality.
In the case of the speed and time taken by the truck driver in the example above, the constant of proportionality would be 1000 since TS = 1000.
The graph of an inverse relationship in the xy-plane is a hyperbola.
If two variables x and y vary inversely, their product xy will be constant.
Two quantities x and y that are inversely proportional if {y}\propto{\frac{1}{x}}, that is, y=\dfrac{k}{x} where k is the constant of proportionality.
Examples
Example 1
In the equation y=\dfrac{18}{x}, \, y varies inversely x. When x=6, \, y=3.
Solve for y when x=2.
For these two ordered pairs (x,y), what is the result when the y-value is multiplied by the x-value?
Example 2
The volume in litres, V, of a gas varies directly as the temperature in kelvins, T, and inversely as the pressure in pascals, p. If a certain gas occupies a volume of 2.1 L at 300 kelvins and a pressure of 11 pascals, solve for the volume at 360 kelvins and a pressure of 16 pascals.
You may use k to represent the constant of variation in your working if necessary.
Inverse Variation
If two variables x and y vary inversely, their product xy will be constant.
Two quantities x and y that are inversely proportional if {y}\propto{\frac{1}{x}}, that is,