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 than her brother. The mother's age is:

Correct option is C

Given:

1. The father’s age is four times that of the son.
2. The daughter’s age is one-third of her mother’s age.
3. The wife (mother) is 6 years younger than her husband (father).
4. The daughter (sister) is 3 years older than her brother (son).

Formula Used:

1. Relationships between ages are given as:
Father's age} = 4 $$\times$$​ Son's age
Daughter's age = $$\frac{1}{3} \times Mother's age$$​
Mother's age = Father's age - 6.
Daughter's age = Son's age + 3.

Solution:

1. Let the son’s age be x .
Then:
Father's age = 4x.
Mother's age = 4x - 6.
Daughter's age = x + 3.

2. From the relationship between the daughter's and mother's ages:
Daughter's age = $$\frac{1}{3} \times$$​ Mother's age
Substituting:
​$$x + 3 = \frac{1}{3} \times (4x - 6)$$​

3. Multiply through by 3 to eliminate the fraction:
3(x + 3) = 4x - 6.
Expand:
3x + 9 = 4x - 6.
Simplify:
9 + 6 = 4x - 3x.
x = 15.

4. Substituting x = 15 (son’s age):
Father's age = 4 $$\times$$​ 15 = 60.
Mother's age = 60 - 6 = 54.
Daughter's age = 15 + 3 = 18.

Final Answer:

The mother's age is 54 years.
**Option C: 54 years**