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, we need to find the value of \( c \) given the ratios \( (a+b):(b+c):(c+a) = 5:6:9 \) and the total \( a+b+c = 10 \).

### Step-by-step Solution:

1. **Express the Ratios in Terms of a Variable**:
We can express the ratios in terms of a variable \( x \):
\[ a + b = 5x, \quad b + c = 6x, \quad c + a = 9x \]

2. **Add the Three Equations**:
By adding these three equations, we can eliminate \( a \), \( b \), and \( c \):
\[ (a + b) + (b + c) + (c + a) = 5x + 6x + 9x \]
This simplifies to:
\[ 2(a + b + c) = 20x \]

3. **Substitute the Total Value**:
We know from the problem that \( a + b + c = 10 \). Substitute this into the equation:
\[ 2 \times 10 = 20x \]
This simplifies to:
\[ 20 = 20x \]

4. **Solve for \( x \)**:
Dividing both sides by 20 gives us:
\[ x = 1 \]

5. **Substitute \( x \) Back to Find \( a + b \), \( b + c \), and \( c + a \)**:
Now, we can find the values of \( a + b \), \( b + c \), and \( c + a \):
\[ a + b = 5x = 5 \times 1 = 5 \]
\[ b + c = 6x = 6 \times 1 = 6 \]
\[ c + a = 9x = 9 \times 1 = 9 \]

6. **Set Up Equations to Solve for \( c \)**:
We have:
\[ a + b = 5 \quad (1) \]
\[ b + c = 6 \quad (2) \]
\[ c + a = 9 \quad (3) \]
We also know \( a + b + c = 10 \).

7. **Use Equation (1) to Find \( c \)**:
From equation (1), we can express \( c \):
\[ c = 10 - (a + b) = 10 - 5 = 5 \]

8. **Final Value of \( c \)**:
Thus, the value of \( c \) is:
\[ c = 5 \]