GK 2016 Q12
12. Two kangaroos Bo and Ing start to jump at the same time, from the same point, in the same direction. After that, they each make one jump per second. Each of Bo's jumps is 6 m in length. Ing's first jump is 1 m in length, his second is 2 m, his third is 3 m, and so on. After how many jumps does Ing catch Bo?
- A 10
- B 11
- C 12
- D 13
- E 14
Grey Kangaroo 2016, Question 12. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 11
Since the two kangaroos jump at the same time and in the same direction, Ing will catch Bo when they have jumped the same distance. After \(n\) jumps, Bo has jumped \(6n\) metres whereas Ing has jumped \((1 + 2 + 3 + \ldots + n)\) metres. The sum of the whole numbers from 1 to \(n\) is given by the formula \(\tfrac{1}{2}n(n + 1)\) (it is left as an exercise for the reader to prove this) and hence \(\tfrac{1}{2}n(n + 1) = 6n\). Hence \(n + 1 = 12\), which has solution \(n = 11\). Therefore Ing will catch Bo after 11 jumps.
Solution © UK Mathematics Trust, from the GK 2016 solutions.