Problem 5. Two candles have lengths that differ by 32 cm. If the longer one is lit at 15:00, and the shorter one at 19:00, then at 21:00 they will have the same length. The longer one will be completely burned out by 22:00, and the shorter one by midnight. The candles burn at a constant rate. What is the sum of the initial lengths of the candles?
Solution. Since the lengths of the candles at 21:00 are equal, the longer one burns out completely by 22:00, and the shorter one by midnight, it follows that the longer one takes 1 hour, and the shorter one 3 hours to burn the same length. Therefore, the longer candle burns 3 times faster than the shorter one. Let $x$ cm of the shorter candle burn per hour, then $3 x \mathrm{~cm}$ of the longer one will burn per hour. From 15:00 to 21:00, $6 \cdot 3 x \mathrm{~cm}$, i.e., $18 x \mathrm{~cm}$, will burn from the longer candle. From 19:00 to 21:00, $2 \cdot x \mathrm{~cm}$ will burn from the shorter candle. At 21:00, they have the same length, which means that $18 x-2 x=32$ and from this $x=2 \mathrm{~cm}$. Then, the length of the longer candle is $7 \cdot 3 x=42 \mathrm{~cm}$, since it burns out completely in 7 hours, and for the shorter one, which burns out in 5 hours, it is $5 x=10 \mathrm{~cm}$.

## IX Department