A man and his wife have three children, John, Ben and Mary.
The difference between their parents' ages is the same as
between John and Ben and between Ben and Mary.
Let d be the difference between the father's and the mother's ages,
which is also the difference between John's and Ben's ages, and
also the difference between Ben's and and Mary's ages.
We cannot tell which parent is older.
We can only tell that Bill is the middle child, but we cannot tell
which of John and Mary is the oldest and youngest.
Let Bill's age be x
Then the younger child's age is x-d
And the older child's age is x+d
The ages of John and Ben, multiplied together, equals the age of the father.
The ages of Ben and Mary multiplied together equals the age of the mother.
The product of Bill's age and the older child's age = the older parent's age
The product of Bill's age and the younger child's age = the younger parent's age
x(x-d) = younger parent's age
x(x+d) = older parent's age
The younger parent's age + d = the older parent's age.
x(x-d) + d = x(x+d)
x²-dx+d = x²+dx
d = 2dx
0 = 2dx - d
0 = d(2x-1)
d=0, or 2x-1 = 0
2x = 1
x = 
Certainly Bill is not half a year old (a 6 month-old baby), so d=0
That means all three children are the same age, and the two parents are
the same age.
John, Bill, and Mary are the same age, x. They are either triplets or
adopted children.
x(x-d) = younger parent's age
x(x+d) = older parent's age
And since d=0, they are both x² years old.
So each child is x years old and each parent is x² years old.
The combined ages of the family amounts to 90 years.
Since the sum of their ages is 90,
x + x + x + x² + x² = 90
2x² + 3x - 90 = 0
That factors as
(x - 6)(2x + 15) = 0
x - 6 = 0; 2x + 15 = 0
x = 6 2x = -15
x = -15/2, which we ignore.
So the children are 6 years old each,
and the parents are 6² or 36 years old each.
Edwin
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