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$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 $\text{Time }=\frac{\text{Distance }}{\text{Speed }}$Time =Distance Speed is used to find the time it would take to drive $1000$1000 km at different average speeds (these times are rounded to one decimal place).
| Speed (km/h) | $60$60 | $70$70 | $80$80 | $90$90 | $100$100 | $110$110 | $120$120 |
| Time (hours) | $16.67$16.67 | $14.3$14.3 | $12.5$12.5 | $11.1$11.1 | $10$10 | $9.1$9.1 | $8.3$8.3 |
Plotting these values gives a hyperbola:
Notice that if the driver were to travel at an average speed of $40$40 km/h, it would take $25$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 $T=\frac{1000}{S}$T=1000S or $TS=1000$TS=1000, where $T$T is the time taken and $S$S is the speed travelled. From both equations, as average speed increases, the time taken will decrease.
Inverse variation
Two quantities $x$x and $y$y that are inversely proportional if $y\propto\frac{1}{x}$y∝1x. That is, they have an equation of the form:
$y=\frac{k}{x}$y=kx or $xy=k$xy=k or $x=\frac{k}{y}$x=ky,
where $k$k can be any number other than $0$0. Again, $k$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$1000 since $TS=1000$TS=1000.
The graph of an inverse relationship in the $xy$xy-plane is a hyperbola.
If two variables $x$x and $y$y vary inversely, their product $xy$xy will be constant.
Two quantities $x$x and $y$y that are inversely proportional if $y\propto\frac{1}{x}$y∝1x, that is,
$y=\frac{k}{x}$y=kx
where $k$k is the constant of proportionality.
Practice questions
Question 1
In the equation $y=\frac{18}{x}$y=18x, $y$y varies inversely as $x$x. When $x=6$x=6, $y=3$y=3.
Solve for $y$y when $x=2$x=2.
For these two ordered pairs $\left(x,y\right)$(x,y), what is the result when the $y$y-value is multiplied by the $x$x-value?
question 2
The volume in litres, $V$V, of a gas varies directly as the temperature in kelvins, $T$T, and inversely as the pressure in pascals, $p$p. If a certain gas occupies a volume of $2.1$2.1 L at $300$300 kelvins and a pressure of $11$11 pascals, solve for the volume at $360$360 kelvins and a pressure of $16$16 pascals.
You may use $k$k to represent the constant of variation in your working if necessary.