Joint Proportionality I don't understand simple proportions! (Although i'm already in calculus and doing good) 
If $x/y = c_1$ and $xz = c_2$, why can we combine this to say $xz = c_3y$? Shouldn't $xz$ stay the same regardless of what $y$ does?
In particular consider this standard problem: 
 $x$ is directly proportional with $y$ and inversely proportional with $z$.
If 
1) $x = 20, y = 10, z = 5$
Then 
2) $x = ? , y = 100, z = 1$8  
The answer to this problem:
$x = 500/9$ in the second situation
BUT $xz = 100$ in the first situation and $xz = 1000$ in the second situation. But $xz$ is constant so how can this be?
Also, is there a place where I can ask such elementary questions because I feel a bit weird asking it alongside these advanced math questions.
 A: It rather depends on what you mean when you say "$x$ is directly proportional with $y$ and inversely proportional with $z$".  I suspect it may be intended to mean something like "$x$ is directly proportional to $y$ when $z$ is held constant and $x$ is inversely proportional to $z$ when $y$ is held constant"
Take the example of ideal gases which produce similar results to the problem:


*

*Charles's law says "when the pressure on a sample of a dry gas is held constant, the Kelvin temperature and the volume will be directly related" i.e. $\frac{V}{T}=k_c$

*Boyle's law says "the absolute pressure exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies if the temperature and amount of gas remain unchanged within a closed system" i.e. $PV=k_b$

*Gay-Lussac's law of pressure–temperature which says "the pressure of a gas of fixed mass and fixed volume is directly proportional to the gas's absolute temperature" i.e. $\frac{P}{T}=k_g$
When all three are combined, allowing all of pressure, temperature and volume to vary together in a closed system, you get $$\dfrac{PV}{T}=k$$ and the suggested result to the problem comes from solving $\dfrac{x \times 18}{ 100} = \dfrac{20 \times 5}{10}$
The alternative approach would be to say that since $x$ is directly proportional to $y$ and inversely proportional to $z$, for consistency you would have to have $y$ inversely proportional to $z$, in which case it would be impossible to have a question which goes from $y=10,z=5$ to $y=100,z=18$.  That does not seem to be what was intended  
