Four years ago, a mother was four times as old as her daughter. Six years later, the mother will be two and a half times as old as her daughter at that time. Find the present ages of mother and her daughter.

Let present ages of mother and her daughter be x and y years respectively.

Given,

Four years ago, a mother was four times as old as her daughter.

⇒ (x - 4) = 4(y - 4)

⇒ x - 4 = 4y - 16

⇒ x = 4y - 16 + 4

⇒ x = 4y - 12 …….(1)

Given,

Six years later, the mother will be two and a half times as old as her daughter at that time.

⇒ (x + 6) = 2.5(y + 6)

⇒ x + 6 = 2.5y + 15

⇒ x = 2.5y + 15 - 6

⇒ x = 2.5y + 9 ……..(2)

From (1) and (2), we get :

⇒ 4y - 12 = 2.5y + 9

⇒ 4y - 2.5y = 9 + 12

⇒ 1.5y = 21

⇒ y = $\dfrac{21}{1.5}$ = 14.

Substituting value of y in equation (1), we get :

⇒ x = 4 × 14 - 12 = 56 - 12 = 44.

Hence, age of mother is 44 years and age of daughter is 14 years.