Variations

Types of variation

Statement	Equation
$y$ varies directly as $x$	$y = kx$
$y$ varies directly as $x^2$	$y = kx^2$
$y$ varies inversely as $x$	$y = \dfrac{k}{x}$
$z$ varies jointly as $x$ and $y$	$z = kxy$
$z$ varies directly as $x$ and inversely as $y$	$z = \dfrac{kx}{y}$
$y$ is partly constant and partly varies as $x$	$y = k_1 + k_2 x$

$k$ is a non-zero constant, the variation constant.

Method

1. Write the equation with $k$.
2. Substitute the given pair of values to find $k$.
3. Use the equation to find the unknown.

For partial variation there are two constants, so you need two pairs of values and simultaneous equations.

Graphs

• $y = kx$: a straight line through the origin.
• $y = \dfrac{k}{x}$: $xy$ is constant.
• $y = k_1 + k_2 x$: a straight line that does not pass through the origin (unless $k_1 = 0$).

Percentage changes

Use multipliers. If $y = kx^2$ and $x$ increases by 10%:

$$\text{new } y = k(1.1x)^2 = 1.21kx^2$$

so $y$ increases by 21%. If $z = kxy$, $x$ increases by 20% and $y$ decreases by 20%, then $z$ is multiplied by $1.2 \times 0.8 = 0.96$: a 4% decrease.

Common mistake: "increases by 20% and decreases by 20%" does not cancel out. Always multiply the factors.