If `a+ b- c : b+ c-a : c+ a- b= 8:7: 6`. Find `(1)/(a): (1)/(b): (1)/(c )`

Options

A. `195: 182: 210`
B. `182: 210: 195`
C. `210: 195: 182`
D. `15: 13: 14`

Answer

To solve the problem, we need to find the ratio \( \frac{1}{a} : \frac{1}{b} : \frac{1}{c} \) given the ratio \( a + b - c : b + c - a : c + a - b = 8 : 7 : 6 \).

### Step 1: Set up the equations based on the given ratio
From the given ratio, we can express:
- \( a + b - c = 8x \)
- \( b + c - a = 7x \)
- \( c + a - b = 6x \)

### Step 2: Add all three equations
Now, we will add all three equations:
\[ (a + b - c) + (b + c - a) + (c + a - b) = 8x + 7x + 6x \]
This simplifies to:
\[ (a + b - c + b + c - a + c + a - b) = 21x \]
The left-hand side simplifies to:
\[ a + b + c = 21x \]

### Step 3: Express \( c \) in terms of \( x \)
Now, we will use the first equation:
\[ a + b - c = 8x \]
Substituting \( a + b = 21x - c \):
\[ (21x - c) - c = 8x \]
This simplifies to:
\[ 21x - 2c = 8x \]
Rearranging gives:
\[ 2c = 21x - 8x = 13x \implies c = \frac{13x}{2} \]

### Step 4: Express \( a \) in terms of \( x \)
Now, we will use the second equation:
\[ b + c - a = 7x \]
Substituting \( c = \frac{13x}{2} \):
\[ b + \frac{13x}{2} - a = 7x \]
Rearranging gives:
\[ b - a = 7x - \frac{13x}{2} \]
Finding a common denominator:
\[ b - a = \frac{14x - 13x}{2} = \frac{x}{2} \implies b = a + \frac{x}{2} \]

### Step 5: Express \( b \) in terms of \( x \)
Now, we will use the third equation:
\[ c + a - b = 6x \]
Substituting \( c = \frac{13x}{2} \):
\[ \frac{13x}{2} + a - b = 6x \]
Rearranging gives:
\[ \frac{13x}{2} + a - (a + \frac{x}{2}) = 6x \]
This simplifies to:
\[ \frac{13x}{2} - \frac{x}{2} = 6x \implies \frac{12x}{2} = 6x \]
This is consistent, confirming our expressions.

### Step 6: Find \( a \), \( b \), and \( c \)
Now we can find \( a \), \( b \), and \( c \):
- From \( a + b + c = 21x \)
- Substitute \( b = a + \frac{x}{2} \) and \( c = \frac{13x}{2} \):
\[ a + (a + \frac{x}{2}) + \frac{13x}{2} = 21x \]
This simplifies to:
\[ 2a + 7x = 21x \implies 2a = 14x \implies a = 7x \]
Now substituting back:
\[ b = a + \frac{x}{2} = 7x + \frac{x}{2} = \frac{15x}{2} \]
And we already have:
\[ c = \frac{13x}{2} \]

### Step 7: Find \( \frac{1}{a} : \frac{1}{b} : \frac{1}{c} \)
Now we can find the ratios:
\[ \frac{1}{a} = \frac{1}{7x}, \quad \frac{1}{b} = \frac{2}{15x}, \quad \frac{1}{c} = \frac{2}{13x} \]
Thus, the ratio \( \frac{1}{a} : \frac{1}{b} : \frac{1}{c} \) becomes:
\[ \frac{1}{7} : \frac{2}{15} : \frac{2}{13} \]

### Step 8: Find a common denominator and simplify
The common denominator for \( 7, 15, 13 \) is \( 1365 \):
\[ \frac{195}{1365} : \frac{182}{1365} : \frac{210}{1365} \]
Thus, the ratio simplifies to:
\[ 195 : 182 : 210 \]

### Final Answer
The final answer is:
\[ \frac{1}{a} : \frac{1}{b} : \frac{1}{c} = 195 : 182 : 210 \]