A couple has a son and a daughter. The age of the father is three times the daughter's age and the age of the mother is twice the son's age. The father is 9 years older than the mother and the daughter is 6 years younger than the son. What is the age of the mother?

Question: A couple has a son and a daughter. The age of the father is three times the daughter's age and the age of the mother is twice the son's age. The father is 9 years older than the mother and the daughter is 6 years younger than the son. What is the age of the mother?

Solution:
Let the ages of father, mother, son, and daughter be F, M, S, and D respectively.

According to the question (ATQ):
F = 3D .............(i)
M = 2S .............(ii)
F = M + 9 ...........(iii)
S = D + 6 ...........(iv)

From (ii) we get,
M = 2S
Or, M = 2(D + 6) [from (iv)]
Or, M = 2{(F/3) + 6} [from (i)]
Or, M = (2/3) × (F + 18)
Or, M = (2/3) × (M + 9 + 18) [from (iii)]
Or, M = (2/3) × (M + 27)
Or, 3M = 2M + 54
Or, M = 54

∴ The age of the mother is 54 years.