GK 2018 Q16
16. Eleven points are marked from left to right on a straight line. The sum of all the distances between the first point and the other points is 2018 cm. The sum of all the distances between the second point and the other points, including the first one, is 2000 cm. What is the distance between the first and second points?
- A 1 cm
- B 2 cm
- C 3 cm
- D 4 cm
- E 5 cm
Grey Kangaroo 2018, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 2 cm
Let the distance between the first two points be \(x\) cm. Consider the other nine points. The sum of their distances from the first point is \((2018 - x)\) cm and the sum of their distances from the second point is \((2000 - x)\) cm. Also, for each of these nine points, its distance from the second point is \(x\) cm less than its distance from the first point. Therefore, when we total the distances over the nine points, we obtain \(9x = (2018 - x) - (2000 - x)\) and hence \(9x = 18\). Therefore \(x = 2\) and so the distance between the first and second points is 2 cm.
Solution © UK Mathematics Trust, from the GK 2018 solutions.