Combined Variation

Last, but in no way least, is combined variation. Numbers are mixed, matched, computed, jumbled, and ultimately combined.

Combined variation mixes both direct and inverse variation. The general form looks like this:

Here, y varies directly with x, and y varies inversely with z. It's a double dose of variation. If x increases, y increases. If x decreases, y decreases. If z increases, y decreases. If z decreases, y increases.

Confused yet? Naw, take it one step at a time. Check out the sample problems below.

Sample Problem

If y varies directly with x and inversely with z, and y = 36 when x = 12 and z = 2, find y when x = 6 and z = 3.

First, write the general form for combined variation.

Plop in those given values and solve for k.

36 = 6k

k = 6

Now, plug our k-value into the general equation.

All that's left is to find y when x = 6 and z = 3.

y = 6(2) = 12

Owned.

Sample Problem

If y varies directly with x and inversely with z, and y = 5 when x = 100 and z = 5, find y when x = 8 and z = 4.

Where's our combined variation formula? We know we stuck it around here somewhere...

Ah, there it is. Plug in everything we know and solve for k.

5 = 20k

k = 0.25

That gives us our general equation.

Now, find y when x = 8 and z = 4.

y = 0.25(2) = 0.5 

Sample Problem

If y varies directly with x and inversely with z, and y = 20 when x = 10 and z = 6, find y when x = 15 and z = 3.

We can do this in our sleep by now. Kick things off with the general formula.

Plug in the given values and solve for k.

Multiply both sides by 6 to drop that fraction.

120 = 10k

k = 12

Throw that brand-spanking-new k into the general equation.

Hunt down y when x = 15 and z = 3.

y = 12(5) = 60

That's all, folks. If the world ever seems wacky and out of control, you can always fiddle around with math and find refuge in rational functions.