The sum of their (father, mother and son) ages is 70. The father is 6 times as old as the son. When the sum of their ages is twice 70, the father will be twice as old as the son. How old is the mother?
The mother is 29 years and 2 months old.
Let's assume that son is X years old. Hence, father is 6X years old and mother is (70-7X) years old.
It is given that the sum of their ages is 70, which will total 140 after 70/3 years.
After 70/3 years, son will be (X + 70/3) years old and father will be (6X + 70/3) years old. Also, it is given that after 70/3 years, the father will be twice as old as the son. Thus,
(6X + 70/3) = 2 * (X + 70/3)
6X + 70/3 = 2X + 140/3
4X = 70/3
X = 35/6
Hence, their ages are
Son = X = 35/6 = 5 years and 10 months
Father = 6X = 6(35/6) = 35 years
Mother = (70 - 7X) = 70 - 7(35/6) = 29 years and 2 months