Direct, and Inverse Proportion; Invariants.

[Faizan Sheikh]

If two quantities are directly proportional then their quotient is an invariant(it does not change, it is constant). Further, if we have another pair of two quantities that are directly proportional then their quotient is also an invariant. Moreover, if you multiple these two quotients, you end up with a third invariant.
But, consider the following example.
Given that x is directly proportional to y and to z and is inversely propotional to w, and that x=4 when (w,y,z)=(6,8,5), what is x when (w,y,z)=(4,10,9)?

Correct Solution:
xw is a contant, x/z is a constant, and x/y is a constant.
Thus, xw/yz is constant. (They just combined the constant terms)

so, xw/yz=(4)(6)/(8)(5)=3/5.

Thus, when (w,y,z)=(4,10,9), we find

x=3yz/5w=27/2.


Wrong Solution a.k.a My Solution:
Since xw is a contant, x/z is a constant, and x/y is a constant. Therefore if we multiply all these constant terms we will get a constant.

Thus, (x^3)w/yz is a constant. The rest is immaterial since I do not end up with 27/2.

I guess there is something wrong with (x^3)w/yz being a constant. Can anybody please explain to me why (x^3)w/yz is not a constant? Hence explain why I do not get 27/2 if my method is followed?

 

[HallsofIvy]

I would like to see how the "combined the constants" in the first solution!

The problem with your solution is that "x is directly proportional to z" means
x/z= k₁ (constant) for fixed y and w. "x is directly proportional to y" means x/y= k₂ (constant) for fixed z and w. k₁ may depend on y and w, k₂ may depend on z and w. That's why you cannont just multiply the two equations and say
x²/yz= constant.