Task 4 - 320814 Alexander observed two candles, a white one and a red one that was half as long. Both were lit at the same time; after 2 hours, the white candle had burned down, while the red one (being wider) took 5 hours to burn completely.

How long after being lit had it been until the two candles were exactly the same length?
In every hour, the length $L$ of the white candle, as long as it burned, decreased by $\frac{1}{2} L$, while the length $\frac{L}{2}$ of the red candle decreased by $\frac{1}{5} \cdot \frac{L}{2}=\frac{1}{10} L$.

After $x$ hours, the length of the white candle had thus decreased by $\frac{x}{2} L$ to $L-\frac{x}{2} L=\left(1-\frac{x}{2}\right) L$, and the length of the red candle by $\frac{x}{10} L$ to $\frac{L}{2}-\frac{x}{10} L=\left(\frac{1}{2}-\frac{x}{10}\right) L$.

If the candles were of equal length after $x$ hours, then

$$
1-\frac{x}{2}=\frac{1}{2}-\frac{x}{10} \quad ; \quad x=\frac{5}{4}
$$

Thus, it had taken $\frac{5}{4}$ hours ( $=75$ minutes) for the candles to be of equal length.

Solutions of the 1st Round 1991 adopted from [5]

### 5.34.2 2nd Round 1992, Class 8