In a family, a couple has a son and daughter. The age of the father is three times that of his daughter and the age of the son is half of his mother. The wife is nine years younger to her husband and the brother is seven years older than his sister. What is the age of the mother?

Options

A. 40 years
B. 45 years
C. 50 years
D. 60 years

Explanation

To solve the problem step by step, we will define variables for the ages of the family members and set up equations based on the information given.

### Step 1: Define Variables
Let:
- \( x \) = age of the mother
- \( x + 9 \) = age of the father (since the mother is 9 years younger than the father)
- \( \frac{x}{2} \) = age of the son (since the son's age is half of the mother's age)
- \( \frac{x}{2} + 7 \) = age of the daughter (since the son is 7 years older than the daughter)

### Step 2: Set Up the Equation
According to the problem, the father's age is three times that of his daughter. We can express this relationship as:
\[ x + 9 = 3 \left( \frac{x}{2} + 7 \right) \]

### Step 3: Solve the Equation
Now, let's simplify and solve the equation:
1. Expand the right side:
\[ x + 9 = 3 \left( \frac{x}{2} + 7 \right) = \frac{3x}{2} + 21 \]
2. Rearranging gives us:
\[ x + 9 - 21 = \frac{3x}{2} \]
\[ x - 12 = \frac{3x}{2} \]
3. To eliminate the fraction, multiply the entire equation by 2:
\[ 2(x - 12) = 3x \]
\[ 2x - 24 = 3x \]
4. Rearranging gives:
\[ 2x - 3x = 24 \]
\[ -x = 24 \]
\[ x = 60 \]

### Conclusion
The age of the mother is \( 60 \) years.