1. Two candles of the same length but different thicknesses take different times to burn completely. The thicker candle takes 4 hours to burn completely, while the thinner candle takes 2 hours. The candles were lit at the same time, burned for a certain period, and then both were extinguished. After being extinguished, the remaining part of the thicker candle was three times longer than the remaining part of the thinner candle. How long did the candles burn?
Solution. Let $x$ be the length of the candles and let the candles burn for $t$ hours. From the thicker candle, a part of length $\frac{x}{4}$ will burn in one hour, and after time $t$, a part of length $x-\frac{x}{4} t$ will remain. From the thinner candle, a part of length $\frac{x}{2}$ will burn in one hour, and after time $t$, a part of length $x-\frac{x}{2} t$ will remain. Since the remaining part of the thicker candle is three times longer than the remaining part of the thinner candle, we get the equation

$$
x-\frac{x}{4} t=3\left(x-\frac{x}{2} t\right)
$$

whose solution is $t=\frac{8}{5} h$. The candles burned for 1 hour and 36 minutes, or 96 minutes.