GK 2017 Q9
9. Four cousins Alan, Bob, Carl and Dan are 3, 8, 12 and 14 years old, although not necessarily in that order. Alan is younger than Carl. The sum of the ages of Alan and Dan is divisible by 5. The sum of the ages of Carl and Dan is divisible by 5. What is the sum of the ages of Alan and Bob?
- A 26
- B 22
- C 17
- D 15
- E 11
Grey Kangaroo 2017, Question 9. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 17
The ages of the four cousins are 3, 8, 12 and 14. When these are added in pairs, we obtain 3 + 8 = 11, 3 + 12 = 15, 3 + 14 = 17, 8 + 12 = 20, 8 + 14 = 22 and 12 + 14 = 26. Only two of these, 15 and 20, are divisible by 5. However, we are told that the sum of the ages of Alan and Dan and the sum of the ages of Carl and Dan are both divisible by 5. Hence, since Dan's age appears in both sums that are divisible by 5, his age is 12. Since Alan is younger than Carl, Alan's age is 3 and Carl's age is 8. Hence Bob's age is 14. Therefore the sum of the ages of Alan and Bob is 3 + 14 = 17.
Solution © UK Mathematics Trust, from the GK 2017 solutions.