if a/(1-a)+b/(1-b)+c/(1-c)=1;then what is the value of 1/a+1/b+1/c=?

a / (1 - a) + b / (1 - b) + c / (1 - c) = 1

Step 1: Express in a Common Form
Rewriting each term:

Let x = a / (1 - a), y = b / (1 - b), and z = c / (1 - c). Then we have:

x + y + z = 1

From the definitions:

a = x / (1 + x),
b = y / (1 + y),
c = z / (1 + z).

We need to determine the value of:

1/a + 1/b + 1/c.

Step 2: Express in Terms of x, y, z
Using the reciprocal:

1/a = (1 + x) / x,
1/b = (1 + y) / y,
1/c = (1 + z) / z.

Thus:

1/a + 1/b + 1/c = (1 + x) / x + (1 + y) / y + (1 + z) / z.

Splitting each fraction:

1/a + 1/b + 1/c = (1/x + 1) + (1/y + 1) + (1/z + 1).

Rewriting:

1/a + 1/b + 1/c = 1/x + 1/y + 1/z + 3.

Step 3: Find 1/x + 1/y + 1/z
Since x + y + z = 1, applying the identity:

1/x + 1/y + 1/z ≥ 9 / (x + y + z) = 9 / 1 = 9.

Thus:

1/a + 1/b + 1/c = 9 + 3 = 12.

Final Answer:
1/a + 1/b + 1/c = 12.