In a family, a couple has a son and a daughter. The age of the father is three times 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. 50 years
C. 45 years
D. 60 years

Answer

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 provided.

### Step 1: Define Variables
Let:
- \( D \) = Age of the daughter
- \( F \) = Age of the father
- \( M \) = Age of the mother
- \( S \) = Age of the son

### Step 2: Set Up Equations
From the problem, we have the following relationships:
1. The age of the father is three times the age of his daughter:
\[ F = 3D \]
2. The age of the son is half of his mother:
\[ S = \frac{1}{2}M \]
3. The wife (mother) is nine years younger than her husband (father):
\[ M = F - 9 \]
4. The brother (son) is seven years older than his sister (daughter):
\[ S = D + 7 \]

### Step 3: Substitute Equations
We can substitute \( F \) from the first equation into the third equation:
\[ M = 3D - 9 \]

Now we have:
1. \( F = 3D \)
2. \( S = \frac{1}{2}(3D - 9) \)
3. \( S = D + 7 \)

### Step 4: Solve for D
Now we can set the two expressions for \( S \) equal to each other:
\[ \frac{1}{2}(3D - 9) = D + 7 \]

Multiply both sides by 2 to eliminate the fraction:
\[ 3D - 9 = 2D + 14 \]

Now, subtract \( 2D \) from both sides:
\[ D - 9 = 14 \]

Add 9 to both sides:
\[ D = 23 \]

### Step 5: Find Other Ages
Now that we have the age of the daughter, we can find the other ages:
1. **Father's Age**:
\[ F = 3D = 3 \times 23 = 69 \]
2. **Mother's Age**:
\[ M = F - 9 = 69 - 9 = 60 \]
3. **Son's Age**:
\[ S = D + 7 = 23 + 7 = 30 \]

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