PK 2016 Q17
17. There are 2016 kangaroos in a zoo. Each of them is either grey or pink, and at least one of them is grey and at least one is pink. For every kangaroo, we calculate this fraction: the number of kangaroos of the other colour divided by the number of kangaroos of the same colour as this kangaroo (including himself). Find the sum of all the 2016 fractions calculated.
- A 2016
- B 1344
- C 1008
- D 672
- E more information required
Pink Kangaroo 2016, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 2016
Let \(G\) be the number of grey kangaroos, and \(P\) the number of pink kangaroos, so \(G + P = 2016\). For each grey kangaroo we calculate the fraction as \(\dfrac{P}{G}\); and there are \(G\) of these, so the total of the fractions for grey kangaroos is \(G \times \dfrac{P}{G} = P\).
Similarly the total of the fractions calculated for the pink kangaroos is \(G\). Thus the total of all the fractions is \(P + G = 2016\).
Solution © UK Mathematics Trust, from the PK 2016 solutions.