Two candles of identical lengths and different thicknesses were lit. Half of the thin candle and one third of the thick candle could be seen to burn down in 15 minutes. How long (in minutes) after the candles had been lit was the ratio of their lengths 4.

The thin candle burns out in 30 minutes and the thick one in 45 minutes, so after $$t$$ minutes their lengths are $$L\left(1 - \frac{t}{30}\right)$$ and $$L\left(1 - \frac{t}{45}\right)$$. Setting the thick one four times the thin, $$1 - \frac{t}{45} = 4 - \frac{4t}{30}$$, so $$\frac{t}{9} = 3$$ and $$t = 27$$ minutes. Then the lengths are $$0.1L$$ and $$0.4L$$.