So, I was recently revisiting some old competition math problems and came to the jarring (and embarrassing) realization that I apparently don't understand joint variation as well as I thought. By way of example, if we have $4 apples = 3 oranges$ and $3 apples = 2 pears$, we could say that apples and oranges have a direct relationship and that apples and pears have a direct relationship, so: $apples/oranges = 3/4$ and $apples/pears = 2/3$.
My problem is that, we could also say that apples are jointly proportional to pears and oranges, which would yield: $4 apples = 6 oranges \times pears$ according to the joint variation formula. My question is how do we go from two direct variation equations to 1 joint variation equation?
This all came about when I was working through the following problem:
Given that x is directly proportional to y and to z and is inversely proportional to w and that x = 4 when (w,y,z) = (6,8,5), what is x when (w,y,z) = (4,10,9).
I know how to solve this problem, when I was doing this sort of thing, I was just taught to multiply when I saw inverse and divide for direct variation, so: $xw/yz=...$ and solve. I'm a little confused as to why this works.