I'm learning about proportions in the book "the ART of PROBLEM SOLVING". In particular I don't understand what the author means when he says:
"two quantities are in a directly proportional relationship when their quotiënt is a constant when all else is held constant" and "two quantities are in inverse proportion if they have a constant product when all else is held constant"
I understand:
direct proportion: x/y = c if x goes up, y also goes up which keeps the fraction constant
inverse proportion: x*y= c if x goes up, y goes down which keeps the product constant
I just don't understand what the author means with "when all else is held constant". Even if other variables change, the proportions would still be related by the same constant. What is the "all else" he is referring to? Isn't the idea of a constant that it stays constant whatever is going on?
He quite literally means all other values need to be constant. Take the ideal gas law for instance. $$PV=nRT$$ (The meaning of the variables does not matter for this example.) $P$ and $V$ are inversely proportional if and only if $n$, $R$, and $T$ all constant because if, for example, $T$ changed, the product of the two number $P$ and $V$ would not change exactly proportional to each other.
The same holds true for direct proportion too. $$\frac{P}{T} = \frac{nR}{V}$$ $P$ and $T$ have a constant quotient if and only if none of the other values $n$, $R$, or $V$ change.