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The first candle burns off in 8 hours and the second in 10 hours. How long until the first candle will be half the height of the second?

candles

in Mathematics by Diamond (58,497 points) | 41 views

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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.
by Diamond (58,497 points)

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