In the Jolly Quizmaster one of the teams has come together for a very unusual reason; Alice, Bob, Carol, Dave and Eve all have birthdays on the same day, although they were born in different years.
Before one of the Quiz Nights the were chatting away, and we overheard the following statements:
- Dave said to Bob: "I'm nine years older than Eve."
- Eve said to Bob: "I'm seven years older than Alice."
- Alice said to Bob: "Your age is exactly 70% greater than mine."
- Bob said to Carol: "Eve is younger than you."
- Carol said to Dave: "The difference between our ages is six years."
- Carol said to Alice: "I'm ten years older than you."
- Carol said to Alice: "Bob is younger than Dave."
- Bob said to Carol: "The difference between your age and Dave's is the same as the difference between Dave's and Eve's."
Now, we know all these people, as they are regulars in the pub and at the Quiz Nights that we run, and some of the statements were untrue. In fact it dawned on us that when someone spoke to a person older than them they told the truth, but they lied whenever they spoke to someone younger.
So, how old are the members of this unusual Quiz Team?
In statement 6, Carol tells Alice she's older than her by 10 years. If Carol is younger then she's lying, but that is not possible, so Carol must be older than Alice, just not by 10 years.
In statement 7, Carol again lies to Alice that Bob is younger than Dave. So Dave is older than Bob.
Knowing this, we know that Dave tells the truth to Bob in statement 1, and that he is 9 years older than Eve.
And in statement 2 Eve is telling the truth to Bob that she is 7 years older than Alice.
From this we know Alice is telling the truth to Bob that his age is 70% greater than her own. For Bob's age to be a whole number, Alice's age must be a multiple of 10. Since Bob is older than Dave, and Dave is 7 + 9 = 16 years older than Alice, that means Bob has to be more than 16 years older than Alice. The lowest multiple of 7 greater than 16 is 21.
So Alice's age is a multiple of 10 and she is at least 30 years old.
In that case, Carol is lying to Dave that the difference in their ages is 6 years, but Bob tells the truth to Carol that the difference between Carol's age and Dave's is the same as the difference between Dave's and Eve's, which is 9 years.
And so we get Alice is 30, Bob is 51, Carol is 55, Dave is 46 and Eve is 37.