z varies inversely with x and directly with y. When x = 6 and y = 2, z = 5. What is the value of z when x = 4 and y = 9?

20/9 5/27 135/4 60

Mar 20, 2017

$z = \frac{135}{4}$

Explanation:

Based on the given information, we can write:

$z = k \left(\frac{y}{x}\right)$

Where $k$ is some constant we don't know that will make this equation true. Since we know that $y$ and $z$ vary directly, $y$ needs to go on the top of the fraction, and since $x$ and $z$ vary inversely, $x$ needs to go on the bottom of the fraction. However, $\frac{y}{x}$ may not be equal to $z$, so we need to put a constant $k$ in there in order to scale $\frac{y}{x}$ so that it matches up with $z$.

Now, we plug in the three values for $x , y ,$and $z$ which we know, in order to find out what $k$ is:

$z = k \left(\frac{y}{x}\right)$
$5 = k \left(\frac{2}{6}\right)$
$15 = k$

Since $k = 15$, we can now say that $z = 15 \left(\frac{y}{x}\right)$.

To get the final answer, we now plug $x$ and $y$ into this equation.

$z = 15 \left(\frac{y}{x}\right)$
$z = 15 \left(\frac{9}{4}\right)$
$z = \frac{135}{4}$