If (a+b): (b+c): (c+a)= 5: 6: 9 and a+b+c=10. What is the value of c :

Options

A. 2
B. 3
C. 5
D. 7

Answer

To solve the problem step by step, we will use the given ratios and the total sum of \( a + b + c \).

### Step-by-Step Solution:

1. **Understanding the Ratios**:
We are given the ratios:
\[ (a+b) : (b+c) : (c+a) = 5 : 6 : 9 \]
This means we can express \( a+b \), \( b+c \), and \( c+a \) in terms of a common variable \( k \):
\[ a+b = 5k, \quad b+c = 6k, \quad c+a = 9k \]

2. **Adding the Equations**:
Now, we add all three equations:
\[ (a+b) + (b+c) + (c+a) = 5k + 6k + 9k \]
This simplifies to:
\[ 2a + 2b + 2c = 20k \]

3. **Using the Total Sum**:
We know from the problem that:
\[ a + b + c = 10 \]
Therefore, we can substitute \( a + b + c \) into our equation:
\[ 2(a + b + c) = 20k \implies 2 \times 10 = 20k \implies 20 = 20k \]
From this, we can solve for \( k \):
\[ k = 1 \]

4. **Finding the Values of \( a+b \), \( b+c \), and \( c+a \)**:
Now that we have \( k \), we can find the values of \( a+b \), \( b+c \), and \( c+a \):
\[ a+b = 5k = 5 \times 1 = 5 \]
\[ b+c = 6k = 6 \times 1 = 6 \]
\[ c+a = 9k = 9 \times 1 = 9 \]

5. **Setting Up the Equations**:
We now have:
\[ a + b = 5 \quad (1) \]
\[ b + c = 6 \quad (2) \]
\[ c + a = 9 \quad (3) \]

6. **Solving for \( c \)**:
To find \( c \), we can use equations (1) and (2):
- From equation (1): \( a + b = 5 \)
- From equation (2): \( b + c = 6 \)

We can express \( a \) from equation (1):
\[ a = 5 - b \quad (4) \]
Substitute equation (4) into equation (3):
\[ c + (5 - b) = 9 \]
Rearranging gives:
\[ c - b = 4 \quad (5) \]

Now we can substitute equation (2) into equation (5):
From equation (2), we have \( c = 6 - b \). Substitute this into equation (5):
\[ (6 - b) - b = 4 \]
Simplifying gives:
\[ 6 - 2b = 4 \implies 2b = 2 \implies b = 1 \]

7. **Finding \( c \)**:
Now substitute \( b = 1 \) back into equation (2) to find \( c \):
\[ 1 + c = 6 \implies c = 6 - 1 = 5 \]

### Final Answer:
Thus, the value of \( c \) is:
\[ \boxed{5} \]