I know the title isn’t that specific but honestly, I can’t think of a specific title that isn’t the whole problem.
Anyways. Force is directly proportional to mass. Force is directly proportional to acceleration.
Therefore, $\frac{force}{mass*acceleration} = k$ Ans this is true.
But…… $$\frac{force}{mass} = constant_1$$ $$\frac{acceleration}{force} = constant_2$$ Multiplying these equation gives us: $$\frac{acceleration}{mass} = constant_3$$
And surely this isn’t true. Acceleration and mass are not directly proportional. Am I making a mistake?
When you say two variables, $x$ and $y$, are directly proportional, this means that when $x$ experiences some change, $y$ experiences that same change times a constant. However, this constant is dependent on other variables that may not be identified.
Your claim that $\frac{force}{mass}=constant_1$ is dependent on the fact that any other variable is not modified. Specifically, the $constant_1$ is only known to be constant wrt variations in $force$ or $mass$. We cannot claim that $constant_1$ is independent of some other variable, say $acceleration$
In fact, we know that $constant_1$ is indeed also directly proportional to $acceleration$.