The oldest child in the given family described is; Adam
We are told that;
Adam's age = Beth's age + Carol's age
David's age = Beth's age + Carol's age (4 years ago)
Adam's age = 2 * David's age (8 years ago)
Thus, we can say that Beth and Carol are younger than both Adam and David since Adam's age was twice that of David's age 8 years ago and as such Adam must be the oldest child.
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