Guess their ages?
A middle-aged couple, Mr and Mrs Tan have four children - Benedict, Bob, Brenda and Brian. Soon after their three sons rode their motorbikes for the first time, Mrs Tan became pregnant with Billy. Then, Mr Tan tried to calculate the average age of everyone in the Tan family and found it to be an integer.
When Billy was just born, Mr Tan did the same again and interestingly ended up with another integer. What is the most likely sum of the ages of everyone in the Tan family?
Assumptions:
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No one's birthdays passed between the two times Mr Tan calculated the average age.
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The four siblings are listed in order of their birth (i.e. Benedict is the oldest followed by Bob, Brenda and Brian).
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When a person is x years and y months old where y is between 0 and 11 inclusive, that person is taken to be x years old even if he/she is due to turn (x+1) years old within the same year.
The answer is 168.
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1 solution
When Mr and Mrs Tan had four children and their average age is an integer, it tells us that the total age must be divisible by 6. When Billy was born, the average age was still an integer. Since no one's birthdays has passed, the total age remains the same. This total must also be divisible by 7. In other words, the total age is divisible by LCM of 6 and 7= 42.
But how do we determine which multiple of 42? Look out for clues in the question. Since Brenda's 3 brothers are able to ride motorcycles, including even Brian, the youngest, the four children must be at least of a certain age, perhaps assume that Brian must be at least 18. This would mean that the total ages of the 4 children is probably more than 1 8 × 4 = 7 2 . Round this up to 80 since we also need to factor in the age difference among the siblings.
Now the multiples of 42 greater than 80 include 84, 126, 168, 210 and so on and so forth. 84 is an absurd answer as it means that Mr and Mrs Tan's total age is 4. Try 126 now: 126 - 80 = 46 which means the average age of Mr and Mrs Tan is 2 4 6 = 2 3 which also does not make sense considering the ages of their children (probably at least 18). Moving on with 168, 168 - 80 = 88. Average age of Mr and Mrs Tan is 2 8 8 = 4 4 which is more reasonable. This also satisfies the condition that both are middle-aged.
If you were to try 210, you would realise that the average age of Mr and Mrs Tan would be about 2 2 1 0 − 8 0 = 2 1 3 0 = 6 5 which would violate the condition that the couple is middle aged. Also I doubt whether Mrs Tan is able to conceive at that age...
Hence the most likely total is 1 6 8 . An example would be Mr Tan - 44, Mrs Tan - 44, Benedict - 22, Bob -21 , Brenda - 19, Brian - 18 which satisfies all conditions stipulated and implied by the question.