If a+b:b+c:c+a = 6:7:8 anda + b+ c =14, then find c.

Options

A. 6
B. 7
C. 8
D. 10

Answer

To solve the problem, we need to find the value of \( c \) given the ratios and the sum of \( a + b + c \).

### Step-by-Step Solution:

1. **Understanding the Ratios**:
We are given the ratios:
\[ a + b : b + c : c + a = 6 : 7 : 8 \]
We can express these in terms of a common variable \( k \):
\[ a + b = 6k, \quad b + c = 7k, \quad c + a = 8k \]

2. **Adding the Equations**:
Now, let's add all three equations:
\[ (a + b) + (b + c) + (c + a) = 6k + 7k + 8k \]
This simplifies to:
\[ 2a + 2b + 2c = 21k \]
Dividing the entire equation by 2 gives:
\[ a + b + c = \frac{21k}{2} \]

3. **Using the Given Sum**:
We know from the problem statement that:
\[ a + b + c = 14 \]
Setting the two expressions for \( a + b + c \) equal to each other:
\[ \frac{21k}{2} = 14 \]

4. **Solving for \( k \)**:
To find \( k \), we multiply both sides by 2:
\[ 21k = 28 \]
Now, divide by 21:
\[ k = \frac{28}{21} = \frac{4}{3} \]

5. **Finding \( a + b \)**:
Now that we have \( k \), we can find \( a + b \):
\[ a + b = 6k = 6 \times \frac{4}{3} = 8 \]

6. **Finding \( b + c \)**:
Next, we find \( b + c \):
\[ b + c = 7k = 7 \times \frac{4}{3} = \frac{28}{3} \]

7. **Finding \( c + a \)**:
Finally, we find \( c + a \):
\[ c + a = 8k = 8 \times \frac{4}{3} = \frac{32}{3} \]

8. **Finding \( c \)**:
We can use the equations we have to find \( c \). We know:
\[ a + b + c = 14 \]
We can express \( c \) as:
\[ c = 14 - (a + b) = 14 - 8 = 6 \]

Thus, the value of \( c \) is:
\[ \boxed{6} \]