Question

If $A$ varies as $C$, and $B$ varies as $C$, then $A\pm B$ and $\sqrt{AB}$ will each vary as $C$.

Answer 1

$A$ varies as $C$ then let $A=kC$ where $k$ is a constant

$B$ varies as $C$ then let $B=mC$ where $m$ is a constant

Then

$A\pm B=kC\pm mC=C(k\pm m)$

$\sqrt{AB}=\sqrt{kC.mC}=\sqrt{C^2.km}=C\sqrt{km}$

We see that both varies with $C$.