PK 2017 Q23
23. On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are of 101, 102, 103, 104, 105 and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?
- A 75%
- B 80%
- C 90%
- D 95%
- E 100%
Pink Kangaroo 2017, Question 23. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 80%
The total mass is 621g so any three masses with total mass over 310.5g could be in the heavier pan. There are eight of these triples that include the 106g mass: (106, 105, 104), (106, 105, 103), (106, 105, 102), (106, 105, 101), (106, 104, 103), (106, 104, 102), (106, 104, 101), and (106, 103, 102).
Without 106, there are 2 ways to make a set over 310.5g: (105, 104, 103) and (105, 104, 102).
Hence the probabililty that the 106g mass is included in the heavier pan is \(\tfrac{8}{8 + 2} = \tfrac{8}{10}\) or 80%.
Solution © UK Mathematics Trust, from the PK 2017 solutions.