PK 2018 Q8
8. Two buildings are located on one street at a distance of 250 metres from each other. There are 100 students living in the first building. There are 150 students living in the second building. Where should a bus stop be built so that the total distance that all residents of both buildings have to walk from their buildings to this bus stop would be the least possible?
- A In front of the first building
- B 100 metres from the first building
- C 100 metres from the second building
- D In front of the second building
- E Anywhere between the buildings
Pink Kangaroo 2018, Question 8. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D In front of the second building
All distances in this solution are in metres. Let \(x\) be the distance of the bus stop from the second building. Then the total distance walked by the 150 students in this building is \(150x\). The distance travelled by each of the students in the first building is \(250 - x\), so the total of their distances is \(100(250 - x)\). Adding these to get the total distance of all students gives \(150x + 100(250 - x) = 50x + 25000\). This is minimised by setting \(x = 0\), so the bus stop should be in front of the second building!
Solution © UK Mathematics Trust, from the PK 2018 solutions.