If `(a + b - c) : ( b + c - a) : (a + c - b) = 6 : 5 : 7`, then `(1)/(a) : (1)/(b) : (1)/( c)` is equal to:

Options

A. `156 : 132 : 143`
B. `132 : 156 : 143`
C. `156 : 143 : 132`
D. `143 : 156 : 132`

Answer

To solve the problem, we start with the given ratio:

\[ (a + b - c) : (b + c - a) : (a + c - b) = 6 : 5 : 7 \]

### Step 1: Set up equations based on the ratio
Let:
\[ a + b - c = 6x \]
\[ b + c - a = 5x \]
\[ a + c - b = 7x \]

### Step 2: Add all three equations
Now, we add these three equations together:
\[ (a + b - c) + (b + c - a) + (a + c - b) = 6x + 5x + 7x \]

This simplifies to:
\[ (a + b - c + b + c - a + a + c - b) = 18x \]

The left-hand side simplifies to:
\[ a + b + c = 18x \]

### Step 3: Express \(c\) in terms of \(x\)
Using the first equation:
\[ a + b - c = 6x \implies c = a + b - 6x \]

Substituting \(c\) into \(a + b + c = 18x\):
\[ a + b + (a + b - 6x) = 18x \]
\[ 2a + 2b - 6x = 18x \]
\[ 2a + 2b = 24x \implies a + b = 12x \]

### Step 4: Express \(a\) and \(b\) in terms of \(x\)
Now, substituting \(a + b = 12x\) into the equations:
1. From \(a + b - c = 6x\):
\[ c = 12x - 6x = 6x \]

2. From \(b + c - a = 5x\):
\[ b + 6x - a = 5x \implies b - a = -x \implies b = a - x \]

3. From \(a + c - b = 7x\):
\[ a + 6x - (a - x) = 7x \implies 6x + x = 7x \]
This is consistent.

### Step 5: Substitute to find \(a\) and \(b\)
Now substituting \(b = a - x\) into \(a + b = 12x\):
\[ a + (a - x) = 12x \implies 2a - x = 12x \implies 2a = 13x \implies a = \frac{13x}{2} \]
\[ b = a - x = \frac{13x}{2} - x = \frac{11x}{2} \]
\[ c = 6x \]

### Step 6: Find \(\frac{1}{a} : \frac{1}{b} : \frac{1}{c}\)
Now we need to find:
\[ \frac{1}{a} : \frac{1}{b} : \frac{1}{c} = \frac{2}{13x} : \frac{2}{11x} : \frac{1}{6x} \]

### Step 7: Simplify the ratios
To simplify:
\[ \frac{2}{13x} : \frac{2}{11x} : \frac{1}{6x} = \frac{2}{13} : \frac{2}{11} : \frac{1}{6} \]
Multiplying through by \(78\) (the LCM of \(13\), \(11\), and \(6\)):
\[ \frac{2 \cdot 6}{13} : \frac{2 \cdot 6}{11} : \frac{78}{6} = \frac{12}{13} : \frac{12}{11} : 13 \]

### Final Step: Convert to integer ratios
The final ratio is:
\[ \frac{12 \cdot 11 \cdot 6}{13 \cdot 11 \cdot 6} : \frac{12 \cdot 13 \cdot 6}{13 \cdot 11 \cdot 6} : \frac{12 \cdot 13 \cdot 11}{13 \cdot 11 \cdot 6} \]
This gives us:
\[ \frac{156}{132} : \frac{132}{156} : \frac{143}{156} \]

Thus, the answer is:
\[ \frac{156}{132} : \frac{132}{156} : \frac{143}{156} \]

### Final Result
The answer is \(156 : 132 : 143\).