If $ a \propto b $ and $ b \propto \sqrt{c} $, then is $ a \propto \sqrt{c} $?

Question

I'm sorry for asking a really simple question, but I just wanted to make sure..

If $ a \propto b $ and $ b \propto \sqrt{c} $, then is $ a \propto \sqrt{c} $ ?

I think this is true, because if I increase $c$ by factor of $9$, $b$ will increase by a factor of $3$, and if $b$ increases by a factor of $3$, then $a$ will increase by a factor of $3$. Therefore, this is the same thing as saying $a \propto \sqrt{c}$.

Thank you for answering my question..

Answer 1

If $a \propto b$ then $a=K_1 b$ for some constant $K_1$ and so if $b \propto \sqrt{c}$ then $b = K_2 \sqrt{c}$ for some constant $K_2$.

Hence, $a=K_1 b = K_1 (K_2 \sqrt{c}) = (K_1K_2) \sqrt{c} \Rightarrow a \propto \sqrt{c}$

---

Footnote: Proportionality (Mathematics)

Answer 2

Both $\dfrac ab$ and $\dfrac b{\sqrt c}$ are constants. $\dfrac a{\sqrt c}=\left(\dfrac ab\right)\left(\dfrac b{\sqrt c}\right)$ is also a constant.