Constant of Variation

The constant of variation in a direct variation is the constant (unchanged) ratio of two variable quantities.

The formula for direct variation is

$y = k x$ (or $y = k x$ )

where $k$ is the constant of variation .

Example 1:

If $y$ varies directly as $x$ and $y = 15$ when $x = 24$ , find $x$ when $y = 25$ .

Find the constant of variation.

\begin{array}{l} k = \frac{y}{x} = \frac{15}{24} = \frac{5}{8} \\ y = \frac{5}{8} x \end{array}

To find $x$ , substitute $25$ for $y$ .

$\begin{array}{l} 25 = \frac{5}{8} x \\ x = 40 \end{array}$

The constant of variation in an indirect variation is the constant (unchanged) product between two variable quantities.

The formula for indirect variation is

$x y = k$ (or $y = \frac{k}{x}$ )

where $k$ is the constant of variation .

Example 2:

If it takes $4$ hours at an average speed of $90$ km/h to do a certain journey, how long would it take at $120$ km/h?

Find the constant of variation.

$k$ = speed · time

$k = 90 \cdot 4 = 360$

Then,

$\begin{array}{l} time = \frac{k}{speed} \\ = \frac{360}{120} \\ = 3 hours . \end{array}$