In a family, the sum of the ages of father, mother and son is 90 years today. The father's age is 6 times the son's age. The son's age will be half the father's age when the sum of the ages of all three becomes exactly twice their present total. Find the present age of the mother.

The mother's present age is 37.5 years.

Father = 6 times son; the sum grows by 3 per year.

Sum doubles from 90 to 180 after (180 - 90)/3 = 30 years.

Then son = (s + 30) and father = (6s + 30) with son = half of father.

s + 30 = (6s + 30)/2 gives 4s = 30, so s = 7.5 and father = 45.

Mother = 90 - 45 - 7.5 = 37.5.