Amit can do a piece of work in 8 hours. Gopal can do it in 20 hours. With the help of Akshay, they complete the work in 4 hours. In how many hours can Akshay alone do the same work three times?

Correct option is B

Given:
Amit can do the work in 8 hours.
Gopal can do the work in 20 hours.
Amit, Gopal, and Akshay together can do the work in 4 hours.

Concept used:

The rate of work of each person is the reciprocal of the time they take to complete the work. example,

A complete work in x day,
Rate of work in one day = $$\frac{1}{x}$$​ day
Solution:

Rate of work of Amit = $$\frac{1}{8}$$​(since Amit can do the work in 8 hours)

Rate of work of Gopal = $$\frac{1}{20}$$​ (since Gopal can do the work in 20 hours)
Let the rate of work of Akshay be $$\frac{1}{x}$$​, where x is the time Akshay takes to complete the work alone.

The equation for their combined work rate is,

​$$\frac{1}{8} + \frac{1}{20} + \frac{1}{x} = \frac{1}{4}$$

​​$$\frac{1}{x} = \frac{1}{4} -\frac{1}{8} - \frac{1}{20}$$

$$\frac{1}{x} = \frac{10-5-2}{40}$$

$$\frac{1}{x} = \frac{3}{40}$$

x = $$\frac{40} {3}$$

Akshay alone can complete the work in $$\frac{40}{3}$$​ hours.

Now, the work three times.

Time for 3 times = $$3 \times \frac{40}{3}$$​​

Time for 3 times = 40 hours

Thus, the correct answer is (b).

Alternative Method:

Akshay of  efficiency = (Amit + Gopal+ Akshay) - (Amit + Gopal)

Akshay of  efficiency = 10 - 7

Akshay of  efficiency =  3

Work = 40 

Time for 3 times = $$\frac{3\times 40}{3}$$

Time for 3 times = $$\frac{120}{3}$$

Time for 3 times = 40​​