Question 136

If a, b, c are non-zero, $$a+\frac{1}{b}=1$$ and $$b+\frac{1}{c}=1$$ then the value of abc is :

Solution

It is given that : $$b + \frac{1}{c} = 1$$

=> $$b = (1 - \frac{1}{c})$$ -------------Eqn(1)

Also, $$a + \frac{1}{b} = 1$$

=> $$a = 1 - \frac{1}{b}$$

=> $$a = 1 - \frac{1}{1 - \frac{1}{c}}$$ [Using Eqn(1)]

=> $$a = (1 - \frac{c}{c-1})$$ ---------------Eqn(2)

To find : $$abc$$

Using eqn(1) and (2)

= $$(1 - \frac{c}{c-1}) (1 - \frac{1}{c}) (c)$$

= $$(\frac{-1}{c-1}) (\frac{c-1}{c}) (c)$$

= $$-1$$

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