Amit can do a piece of work in 6 hours. Gopal can do it in 42 hours. With the assistance of Akshay, they completed the work in 3 hours. In how many hours can Akshay alone do it?

Correct option is C

Given:

Amit = 6 hours, Gopal = 42 hours, Together with Akshay = 3 hours

Solution:

Let, Time for Akshay alone = x hours

$$\dfrac{1}{6} + \dfrac{1}{42} + \dfrac{1}{x} = \dfrac{1}{3} \\ \ \\\dfrac{4}{21} + \dfrac{1}{x} = \dfrac{1}{3} \\ \ \\\dfrac{1}{x} = \dfrac{1}{3} - \dfrac{4}{21} = \dfrac{3}{21} = \dfrac{1}{7}$$ 

x = 7 hours.

​Alternate Solution:

Assume total work = LCM(6, 42) = 42 units.
Amit’s rate =$$\frac {42}6=$$​7 units/hour; Gopal’s rate =$$\frac{42}{42}$$​= 1 unit/hour.
In 3 hours, Amit + Gopal do 3 × (7 + 1) = 24 units.
Remaining work = 42 - 24 = 18 units,

done by Akshay in 3 hours

Akshay’s rate = $$\frac{18}3$$​= 6 units/hour.
Time for Akshay alone = $$\frac {42}6$$​= 7 hours.