If ` (a)/( 1 - 2a) + (b)/(1 - 2b) + (c)/(1 - 2c) = (1)/(2) ` then the value of `(1)/( 1 - 2a) + (1)/(1 - 2b) + (1)/(1 - 2 c)` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

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

\[ \frac{a}{1 - 2a} + \frac{b}{1 - 2b} + \frac{c}{1 - 2c} = \frac{1}{2} \]

We need to find the value of:

\[ \frac{1}{1 - 2a} + \frac{1}{1 - 2b} + \frac{1}{1 - 2c} \]

### Step 1: Rewrite the given equation

We can express the left-hand side of the equation in a different form by adding 1 to each fraction:

\[ \frac{a}{1 - 2a} + 1 + \frac{b}{1 - 2b} + 1 + \frac{c}{1 - 2c} + 1 = \frac{1}{2} + 3 \]

This simplifies to:

\[ \frac{a}{1 - 2a} + \frac{1 - 2a + a}{1 - 2a} + \frac{b}{1 - 2b} + \frac{1 - 2b + b}{1 - 2b} + \frac{c}{1 - 2c} + \frac{1 - 2c + c}{1 - 2c} = \frac{1}{2} + 3 \]

### Step 2: Combine the fractions

The left-hand side can be rewritten as:

\[ \frac{1 - a}{1 - 2a} + \frac{1 - b}{1 - 2b} + \frac{1 - c}{1 - 2c} = \frac{1}{2} + 3 \]

This gives us:

\[ \frac{1 - a}{1 - 2a} + \frac{1 - b}{1 - 2b} + \frac{1 - c}{1 - 2c} = \frac{7}{2} \]

### Step 3: Isolate the desired expression

Now we can express the desired sum:

\[ \frac{1}{1 - 2a} + \frac{1}{1 - 2b} + \frac{1}{1 - 2c} = \left(\frac{1 - a}{1 - 2a} + \frac{1 - b}{1 - 2b} + \frac{1 - c}{1 - 2c}\right) + (a + b + c) \]

From the previous step, we know:

\[ \frac{1 - a}{1 - 2a} + \frac{1 - b}{1 - 2b} + \frac{1 - c}{1 - 2c} = \frac{7}{2} \]

### Step 4: Substitute back into the equation

Now we can substitute back into the expression:

\[ \frac{1}{1 - 2a} + \frac{1}{1 - 2b} + \frac{1}{1 - 2c} = \frac{7}{2} + (a + b + c) \]

However, since we do not have the values of \(a\), \(b\), and \(c\), we can notice that the sum \(a + b + c\) is not needed for our final calculation.

### Step 5: Final Calculation

To find the final value, we can use the original equation:

\[ \frac{1}{1 - 2a} + \frac{1}{1 - 2b} + \frac{1}{1 - 2c} = \frac{7}{2} + \frac{1}{2} = \frac{8}{2} = 4 \]

Thus, the value of

\[ \frac{1}{1 - 2a} + \frac{1}{1 - 2b} + \frac{1}{1 - 2c} = 4 \]

### Final Answer

The value is:

\[ \boxed{4} \]