A couple has a son and a daughter. The age of the father is four times that of the son and the age of the daughter is one-third of that of her mother. The wife is 6 years younger to her husband and the sister is 3 years older then her brother. The mother's age is = ?

Explanation:-

$$\eqalign{ & M \to {\text{Mother, }}F \to {\text{Father, }}S \to {\text{Son and }}D \to {\text{Daughter}} \cr & F = {\text{ }}4S,{\text{ }}D{\text{ }} = \frac{1}{3}M,{\text{ }}M = {\text{ }}F - 6{\text{ }}and{\text{ }}S = {\text{ }}D - 3 \cr & \therefore M = 3D = 3\left( {S + 3} \right) \cr & = 3S + 9 = \frac{3}{4}F + 9 = \frac{3}{4}\left( {M + 6} \right) + 9 \cr & = \frac{3}{4}M + \frac{3}{4} \times 6 + 9 \cr & \Rightarrow \left( {M - \frac{3}{4}M} \right) = \left( {\frac{9}{2} + 9} \right) \cr & \Rightarrow \frac{1}{4}M = \frac{{27}}{2} \cr & \Rightarrow M = \left( {\frac{{27}}{2} \times 4} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 54\,{\text{years}} \cr}$$
∴ The mother is 54 years old.