If (a + b) : (b + c) : (c + a) = 6 : 7 : 8 and (a + b + c) = 14, then the value of c is

Options

A. 6
B. 7
C. 8
D. 14

Answer

To solve the problem, we start with the given ratios and the total sum.

### Step-by-Step Solution:

1. **Set up the ratios**:
We are given that \((a + b) : (b + c) : (c + a) = 6 : 7 : 8\).
Let's express \(a + b\), \(b + c\), and \(c + a\) in terms of a common variable \(x\):
- \(a + b = 6x\)
- \(b + c = 7x\)
- \(c + a = 8x\)

2. **Express the total**:
We know from the problem that \(a + b + c = 14\).
We can also express \(a + b + c\) in terms of \(x\):
\[ a + b + c = (a + b) + (b + c) + (c + a) - (a + b + c) = 6x + 7x + 8x - (a + b + c) \]
This simplifies to:
\[ 2(a + b + c) = 21x \quad \Rightarrow \quad a + b + c = \frac{21x}{2} \]

3. **Set the equation**:
Since we know \(a + b + c = 14\), we can set up the equation:
\[ \frac{21x}{2} = 14 \]

4. **Solve for \(x\)**:
To solve for \(x\), multiply both sides by 2:
\[ 21x = 28 \]
Now, divide by 21:
\[ x = \frac{28}{21} = \frac{4}{3} \]

5. **Find \(a + b\)**:
Now we can find \(a + b\):
\[ a + b = 6x = 6 \times \frac{4}{3} = 8 \]

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

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

8. **Use the total 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 **6**.