GK 2015 Q11
11. A bush has 10 branches. Each branch has either 5 leaves only or 2 leaves and 1 flower. Which of the following could be the total number of leaves the bush has?
- A 45
- B 39
- C 37
- D 31
- E None of A to D
Grey Kangaroo 2015, Question 11. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E None of A to D
The maximum number of leaves the bush could have is 10 × 5 = 50. Each branch that has two leaves and a flower instead of five leaves reduces the number of leaves the bush has by three. Therefore the total number of leaves the bush has is of the form \(50 - 3n\) where \(n\) is the number of branches with two leaves and a flower. It is straightforward to check that none of options A to D has that form and so the answer is E.
Solution © UK Mathematics Trust, from the GK 2015 solutions.