One night two cylindrical wax candles of different heights and different diameters were lit. One of the candles was 20 cm taller than the other. They were both lit at the same time and each burned at a steady rate. Five hours after they were lit they were both the same height. The taller one burned all of its wax six hours after it was lit, and the shorter one burned all of its wax 10 hours after it was lit. What was the ratio of the original height of the shorter candle to the original height of the taller candle? Express your answer as a common fraction.
My thinking: The more volume the candle has, the faster it will burn. We have that it took $10$ hours for the bigger candle to burn out. We can use the equation $\frac{h\cdot r^2}{10^3}=\frac{r(h+20)}{6^3}$. I'm not sure if this is correct though. I don't know how to proceed. Help is greatly appreciated.
The radii do not matter. The whole problem just refers to the heights of the two candles and how long it took them to burn down. You should define $x$ as the height of the tall candle and $y$ as the height of the shorter. Write equations that express what you know. You are given the time each burned which gives the fraction that burn per hour.