The combined age of a man and his wife is six times the combined ages of their children. Two years ago their united ages were ten times the combined ages of their children. Six years hence their combined age will be three times the combined age of the children. The number of children they have is________

Answer

To solve the problem step by step, we will define variables and set up equations based on the information given in the question.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the age of the man be \( m \) years.
- Let the age of the wife be \( w \) years.
- Let the combined age of their children be \( x \).
- Let the number of children be \( n \).

2. **Set Up the First Equation:**
- According to the problem, the combined age of the man and his wife is six times the combined ages of their children.
- This gives us the equation:
\[ m + w = 6x \quad \text{(Equation 1)} \]

3. **Set Up the Second Equation (Two Years Ago):**
- Two years ago, the ages of the man and wife were \( m - 2 \) and \( w - 2 \) respectively.
- The combined age of their children two years ago would be \( x - 2n \) (since each child was 2 years younger).
- The problem states that their united ages were ten times the combined ages of their children two years ago:
\[ (m - 2) + (w - 2) = 10(x - 2n) \]
- Simplifying this, we have:
\[ m + w - 4 = 10x - 20n \]
- Substituting \( m + w = 6x \) from Equation 1 into this equation:
\[ 6x - 4 = 10x - 20n \]
- Rearranging gives us:
\[ 20n - 4 = 4x \]
- Dividing the entire equation by 4:
\[ 5n - 1 = x \quad \text{(Equation 2)} \]

4. **Set Up the Third Equation (Six Years Hence):**
- Six years hence, the ages of the man and wife will be \( m + 6 \) and \( w + 6 \) respectively.
- The combined age of their children will be \( x + 6n \).
- The problem states that their combined age will be three times the combined age of the children:
\[ (m + 6) + (w + 6) = 3(x + 6n) \]
- Simplifying this, we have:
\[ m + w + 12 = 3x + 18n \]
- Substituting \( m + w = 6x \) from Equation 1:
\[ 6x + 12 = 3x + 18n \]
- Rearranging gives us:
\[ 3x = 18n - 12 \]
- Dividing the entire equation by 3:
\[ x = 6n - 4 \quad \text{(Equation 3)} \]

5. **Substituting Equations:**
- Now we have two expressions for \( x \):
- From Equation 2: \( x = 5n - 1 \)
- From Equation 3: \( x = 6n - 4 \)
- Setting these equal to each other:
\[ 5n - 1 = 6n - 4 \]
- Rearranging gives:
\[ 4 - 1 = 6n - 5n \]
\[ 3 = n \]

6. **Conclusion:**
- The number of children \( n \) is \( 3 \).

### Final Answer:
The number of children they have is **3**.