Four years ago, the age of the mother was 13 times the age of her daughter. Two years hence, the mother's age will be 4 times the daughter's age. After how many years will the mother's age be 3 times that of her daughter?

Let the present age of the daughter be x years. Four years ago, the age of the daughter = (x - 4) years Four years ago, the age of the mother = 13 × (x - 4) = (13x - 52) years Two years hence, Age of the daughter = (x - 4) + 4 + 2 = (x + 2) years Age of the mother = (13x - 52) + 4 + 2 = (13x - 46) years According to the question, (x + 2) × 4 = (13x - 46)   =4x + 8 = 13x - 46 = 13x – 4x = 46 + 8   x = 6 Now, the present age of the mother = (13x - 52) + 4 = 13x - 48 = 13 × 6 - 48 = 30 Let after y years the age of the mother will be three times that of her daughter. According to the question, (6 + y) × 3 = (30 + y) = 18 + 3y = 30 + y = 2y = 30 - 18   y = 12/2   y = 6 ∴ After 6 years the mother's age will be 3 times that of her daughter.