Question

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

Answer 1

`z = 135/4`

Explanation:

Based on the given information, we can write:

`z = k(y/x)`

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, `y/x` may not be equal to `z`, so we need to put a constant `k` in there in order to scale `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(y/x)`
`5 = k(2/6)`
`15 = k`

Since `k=15`, we can now say that `z = 15(y/x)`.

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

`z = 15(y/x)`
`z = 15(9/4)`
`z = 135/4`