GK 2019 Q17
17. Julio has two cylindrical candles with different heights and diameters. The two candles burn wax at the same uniform rate. The first candle lasts 6 hours, while the second candle lasts 8 hours. He lights both candles at the same time and three hours later both candles are the same height. What is the ratio of their original heights?
- A 4 : 3
- B 8 : 5
- C 5 : 4
- D 3 : 5
- E 7 : 3
Grey Kangaroo 2019, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 5 : 4
Let the original heights of the first and second candles be \(x\) and \(y\) respectively. Since the first candle lasts six hours, after three hours its height is \(\tfrac{1}{2}x\). Similarly, since the second candle lasts eight hours, after three hours its height is \(\tfrac{5}{8}y\). After three hours the two candles have the same height and hence \(\tfrac{1}{2}x = \tfrac{5}{8}y\). Therefore \(\tfrac{x}{y} = \tfrac{5}{4}\) and hence the ratio of their original heights is 5 : 4.
Solution © UK Mathematics Trust, from the GK 2019 solutions.