What am I doing wrong on this question about "varies directly" and "varies inversely"? I don't get this problem:
$R$ varies directly as $S$ and inversely as $T$. When $R = \frac{4}{3}, T = \frac{9}{14}$, $S = \frac{3}{7}$. Find $S$ when $R = \sqrt{48}$ and $T = \sqrt{75}$.
My attempt:
We know
$$ R = c_1 S$$
and
$$ R = \frac{c_2}{T}$$
Plugging things in, we get $c_1 = \frac{28}{9}$ and $c_2 = \frac{6}{7}$. Now, we just solve for $S$.
$$ \sqrt{48} = \frac{6}{7} S$$
$$ 4 \sqrt{3} = \frac{6}{7} S$$
$$ 2 \sqrt{3} = \frac{3}{7} S$$
$$S = \frac{14}{3} \sqrt{3} $$
But, as far as I know, the correct answer appears to be $30$. What am I doing wrong? Am I getting wrong the definition of "directly" and "inversely"?

You can’t treat the relationship between $R$ and $S$ and the relationship between $R$ and $T$ as if they were independent: the hypothesis actually means that
$$R=c\cdot\frac{S}T$$
for some constant $c$.
Thus, the relationship $R=c_1T$ holds only when $T$ is held constant, and the relationship $R=\frac{c_2}T$ holds only when $S$ is held constant.
You know that
$$\frac43=c\cdot\frac{3/7}{9/14}=c\cdot\frac37\cdot\frac{14}9=\frac23\,,$$
so
$$c=\frac{4/3}{2/3}=2\,.$$
When $R=\sqrt{48}$ and $T=\sqrt{75}$ we have
$$\sqrt{48}=\frac{2S}{\sqrt{75}}\,,$$
so
$$S=\frac12\sqrt{48\cdot75}=30\,.$$
