PK 2016 Q20
20. It takes 4 hours for a motorboat to travel downstream from X to Y. To return upstream from Y to X it takes the motorboat 6 hours. How many hours would it take a wooden log to be carried from X to Y by the current, assuming it is unhindered by any obstacles? [Assume that the current flows at a constant rate, and that the motorboat moves at a constant speed relative to the water.]
- A 5
- B 10
- C 12
- D 20
- E 24
Pink Kangaroo 2016, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 24
Let \(D\) be the distance from \(X\) to \(Y\), and let \(u\) be the speed of the boat and \(v\) be the speed of the current. When travelling downstream the overall speed of the boat in the current is \(u + v\), and travelling upstream against the current it is \(u - v\). Then using time \(= \dfrac{\text{distance}}{\text{speed}}\), we get \(\dfrac{D}{u + v} = 4\) for the journey downstream and \(\dfrac{D}{u - v} = 6\) for the journey upstream. Inverting the equations gives \(\dfrac{u + v}{D} = \dfrac{1}{4}\) or \(4u + 4v = D \ldots\) (1) and \(\dfrac{u - v}{D} = \dfrac{1}{6}\) or \(6u - 6v = D \ldots\) (2). Multiplying (1) by 3 and (2) by 2, we get \(12u + 12v = 3D \ldots\) (3) and \(12u - 12v = 2D \ldots\) (4).
Subtracting, (3) − (4) gives \(24v = D\) which rearranges to \(\dfrac{D}{v} = 24\) (hours), which is the time taken for the log to float downstream at the speed of the current alone.
Solution © UK Mathematics Trust, from the PK 2016 solutions.