# If z varies directly as x and inversely as y, and z = 8 when x = 12 and y = 3, the value of the constant of proportionality is what number?

Nov 13, 2017

#### Answer:

$k = 2$

#### Explanation:

when z varies directly proportional to x means a division

$c = \frac{z}{x}$ where c is a constant of proportionality

when z varies inversely proportional to x it means a product

$z y = c$

now with both

$k = \frac{z y}{x}$

z=8; x=12 and y=3

$k = \frac{8.3}{12}$

$k = 2$

Nov 13, 2017

#### Answer:

$k = 2$

#### Explanation:

$\text{the initial statement is } z \propto \frac{x}{y}$

$\text{to convert to an equation multiply by k the constant}$
$\text{of variation}$

$\Rightarrow z = k \times \frac{x}{y} = \frac{k x}{y}$

$\text{to find k use the given condition}$

$z = 8 \text{ when "x=12" and } y = 3$

$z = \frac{k x}{y} \Rightarrow k = \frac{y z}{x} = \frac{3 \times 8}{12} = \frac{24}{12} = 2$