In \(n\) hours, \(1^{\text {st }}\) candle burns \(\frac{n}{8}\) th of its height and in \(n\)h.

\(2^{\text {nd }}\) candle burns \(\frac{n}{10}\) th of its height.

Now, the ratios of their respective heights after burning will be \(1-\frac{n}{8}: 1-\frac{n}{10}\).

The ratio of height wanted \(\rightarrow \frac{1}{2}: 1\).

So, the time taken,

\[

\begin{aligned}

& \frac{\frac{8-n}{84}}{\frac{10-n}{10^{5}}}=\frac{1}{2} \Rightarrow \frac{5(8-n)}{4(10-n)}=\frac{1}{2} \\

\Rightarrow & 40-5 n=2(10-n) \Rightarrow 40-5 n=20-2 n .

\end{aligned}

\]

\(\Rightarrow 3 n=20 \Rightarrow n=\frac{20}{3}=6 \frac{2}{3}\) hours.

Thus, it will take \(6 \frac{2}{3}\) hours for the first candle to reach half the height of second.