`a+b : b+c : c+a = 4 : 7 : 9` given `a+b+c=70` find `c=?`

Answer

To solve the problem, we need to find the value of \( c \) given the ratios \( a+b : b+c : c+a = 4 : 7 : 9 \) and the total \( a+b+c = 70 \).

### Step-by-Step Solution:

1. **Set Up the Ratios**:
Given the ratios:
\[ a+b : b+c : c+a = 4 : 7 : 9 \]
We can express these in terms of a common variable \( k \):
\[ a+b = 4k, \quad b+c = 7k, \quad c+a = 9k \]

2. **Add the Equations**:
Now, we add all three equations:
\[ (a+b) + (b+c) + (c+a) = 4k + 7k + 9k \]
This simplifies to:
\[ 2a + 2b + 2c = 20k \]

3. **Express in Terms of \( a+b+c \)**:
Since \( a+b+c = 70 \), we can write:
\[ 2(a+b+c) = 2 \times 70 = 140 \]
So, we have:
\[ 20k = 140 \]

4. **Solve for \( k \)**:
Dividing both sides by 20 gives:
\[ k = \frac{140}{20} = 7 \]

5. **Find \( a+b \), \( b+c \), and \( c+a \)**:
Now we can find the values of \( a+b \), \( b+c \), and \( c+a \):
\[ a+b = 4k = 4 \times 7 = 28 \]
\[ b+c = 7k = 7 \times 7 = 49 \]
\[ c+a = 9k = 9 \times 7 = 63 \]

6. **Set Up the Equations**:
Now we have:
\[ a+b = 28 \quad (1) \]
\[ b+c = 49 \quad (2) \]
\[ c+a = 63 \quad (3) \]

7. **Use Equation (1) to Find \( c \)**:
From equation (1), we can express \( c \) in terms of \( a \) and \( b \):
\[ c = (b+c) - (a+b) = 49 - 28 = 21 \]

8. **Substituting Back**:
Now we can substitute \( c \) back into the equations to find \( a \) and \( b \):
From (2):
\[ b + 21 = 49 \implies b = 49 - 21 = 28 \]
From (1):
\[ a + 28 = 28 \implies a = 28 - 28 = 0 \]

9. **Final Calculation**:
Now we can find \( c \):
\[ c = 70 - (a + b) = 70 - 28 = 42 \]

Thus, the value of \( c \) is \( \boxed{42} \).