<s>[INST] Need help in figuring out what I am doing wrong when solving for n.. Here is the expression that I am trying to solve for n:
$$ \frac{4}{16+n} = \frac{10}{16+n} \frac{10}{16+n}$$
I am doing the following:
\begin{align}
\frac{4}{16+n} & = \frac{100}{(16 + n)^2} \\[8pt]
\frac{4}{16+n} & = \frac{100}{16^2 + 32n + n^2} \\[8pt]
\frac{4}{16+n} & = \frac{100}{256 + 32n + n^2} \\[8pt]
\frac{1}{16+n} & = \frac{100}{4(256 + 32n + n^2)} \\[8pt]
\frac{1}{16+n} & = \frac{20}{256 + 32n + n^2} \\[8pt]
16+n & = \frac{256 + 32n + n^2}{20} \\[8pt]
n & = \frac{256 + 32n + n^2 - 320}{20} \\[8pt]
n & = \frac{-64 + 32n + n^2}{20} \\[8pt]
20n & = -64 + 32n + n^2 \\[8pt]
20n -32n - n^2 & = -64 \\[8pt]
12n - n^2 & = -64 \\[8pt]
n(12 - n) & = -64
\end{align}
Not sure what to do next now... Book says that n = 9 
Cannot get 9...
Where am I wrong?
Thank you
 [/INST] You can cancel one of the $16 + n$ terms on each side, greatly simplifying the problem. Multiply both sides by $16 + n$. This gives
$$4 = \frac{10 \times 10}{16 + n}$$
Multiply both sides by $16 + n$ again. This gives
$$4(16 + n) = 100$$
Divide both sides by $4$:
$$16 + n = 25$$
Subtract $16$ from both sides:
$$n = 9$$
</s>