IMC 2017 Q12
12. The combined age of Alice and Bob is 39. The combined age of Bob and Clare is 40. The combined age of Clare and Dan is 38. The combined age of Dan and Eve is 44. The total of all five ages is 105.
Which of the five is the youngest?
- A Alice
- B Bob
- C Clare
- D Dan
- E Eve
Intermediate Mathematical Challenge 2017, Question 12. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D Dan
Let the ages of Alice, Bob, Clare, Dan and Eve be \(a\), \(b\), \(c\), \(d\), \(e\) respectively.
So, for example, \(b + c = 40\) and \(d + e = 44\). Adding these gives \(b + c + d + e = 84\). We are also told that \(a + b + c + d + e = 105\). The difference between these equations shows that \(a = 105 - 84 = 21\). Hence \(b = 39 - 21 = 18\) and, similarly, \(c = 40 - 18 = 22\), \(d = 38 - 22 = 16\) and \(e = 44 - 16 = 28\). So Dan is the youngest.
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).