If `( a + b) : ( b + c ) : ( c + a) = 5 : 12 : 11 and a + b + c =28`, then `(1)/( a) : (1)/( b) : ( 1)/( c )` is equal to:

Options

A. `2: 3: 9`
B. `2 : 9 : 3`
C. `9 : 6 : 2`
D. `6 : 9 : 2`

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) based on the given ratios and then calculate the ratio of \( \frac{1}{a} : \frac{1}{b} : \frac{1}{c} \).

### Step 1: Set up the equations based on the given ratios
Given the ratios:
\[ (a + b) : (b + c) : (c + a) = 5 : 12 : 11 \]
We can express this in terms of a variable \( x \):
\[ a + b = 5x, \quad b + c = 12x, \quad c + a = 11x \]

### Step 2: Add the equations
Adding all three equations:
\[ (a + b) + (b + c) + (c + a) = 5x + 12x + 11x \]
This simplifies to:
\[ 2a + 2b + 2c = 28x \]

### Step 3: Relate to the total sum
We know from the problem statement that:
\[ a + b + c = 28 \]
Thus, we can set:
\[ 2(a + b + c) = 28x \implies 2 \cdot 28 = 28x \implies 56 = 28x \]
From this, we can solve for \( x \):
\[ x = 2 \]

### Step 4: Calculate \( a + b \), \( b + c \), and \( c + a \)
Now substituting \( x \) back into the equations:
\[ a + b = 5x = 5 \cdot 2 = 10 \]
\[ b + c = 12x = 12 \cdot 2 = 24 \]
\[ c + a = 11x = 11 \cdot 2 = 22 \]

### Step 5: Solve for \( a \), \( b \), and \( c \)
We have the following system of equations:
1. \( a + b = 10 \)
2. \( b + c = 24 \)
3. \( c + a = 22 \)

From equation 1, we can express \( b \):
\[ b = 10 - a \]

Substituting \( b \) into equation 2:
\[ (10 - a) + c = 24 \implies c = 24 - 10 + a \implies c = 14 + a \]

Now substituting \( c \) into equation 3:
\[ (14 + a) + a = 22 \implies 14 + 2a = 22 \implies 2a = 8 \implies a = 4 \]

Now substituting \( a \) back to find \( b \) and \( c \):
\[ b = 10 - 4 = 6 \]
\[ c = 14 + 4 = 18 \]

### Step 6: Find the ratio \( \frac{1}{a} : \frac{1}{b} : \frac{1}{c} \)
Now we have \( a = 4 \), \( b = 6 \), and \( c = 18 \). We can find:
\[ \frac{1}{a} = \frac{1}{4}, \quad \frac{1}{b} = \frac{1}{6}, \quad \frac{1}{c} = \frac{1}{18} \]

### Step 7: Simplify the ratios
To find the ratio \( \frac{1}{4} : \frac{1}{6} : \frac{1}{18} \), we can find a common denominator, which is 36:
\[ \frac{1}{4} = \frac{9}{36}, \quad \frac{1}{6} = \frac{6}{36}, \quad \frac{1}{18} = \frac{2}{36} \]
Thus, the ratio becomes:
\[ 9 : 6 : 2 \]

### Final Answer
The final ratio \( \frac{1}{a} : \frac{1}{b} : \frac{1}{c} \) is:
\[ \boxed{9 : 6 : 2} \]