PK 2017 Q14
14. Johannes told me that it rained seven times during his holiday. When it rained in the morning, it was sunny in the afternoon; when it rained in the afternoon, it was sunny in the morning. There were 5 sunny mornings and 6 sunny afternoons. Without more information, what is the least number of days that I can conclude that the holiday lasted?
- A 7
- B 8
- C 9
- D 10
- E 11
Pink Kangaroo 2017, Question 14. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 9
Let \(m\) be the number of days with sunny mornings and wet afternoons. Let \(n\) be the number of days with sunny mornings and sunny afternoons. There were 5 sunny mornings so \(m + n = 5 \ldots\) (1). Since there are seven wet days, the number of days with wet mornings and sunny afternoons must be \(7 - m\). There are 6 sunny afternoons so \(n + (7 - m) = 6\), which rearranges to \(m = n + 1 \ldots\) (2).
Equations (1) and (2) together give \(m = 3\), \(n = 2\), so Johannes had 3 days with sunny mornings and wet afternoons, 2 days sunny all day, and 4 days with wet mornings and sunny afternoons, a total of 9 days (not counting any cloudy days he may have had!).
Solution © UK Mathematics Trust, from the PK 2017 solutions.