Arun can do a piece of work in 8 hours. Mahesh can do it in 4 hours. With the assistance of Vijay, they completed the work in 2 hours. In how many hours can Vijay alone do it?

Correct option is D

Given:

Arun can complete the work in 8 hours

Mahesh can complete the work in 4 hours

Together, Arun, Mahesh, and Vijay complete the work in 2 hours

Formula Used:

Work done $$=\frac{1}{\text{Time taken}}$$​​

Solution:

Arun's work rate =$$\frac{1}{8}$$​ (since Arun can do it in 8 hours)

Mahesh's work rate $$= \frac{1}{4}$$​ (since Mahesh can do it in 4 hours)

Let Vijay's work rate =$$\frac{1}{x}$$​ (where x is the time Vijay takes to complete the work alone)

Together, their combined work rate is $$\frac{1}{2}$$​ (since they complete the work in 2 hours)

So,

$$\frac{1}{8} + \frac{1}{4} + \frac{1}{x} = \frac{1}{2} \\ \ \\\frac{3}{8} + \frac{1}{x} = \frac{1}{2} \\ \ \\\frac{1}{x} = \frac{1}{2} - \frac{3}{8} = \frac{4}{8} - \frac{3}{8} = \frac{1}{8}$$​

Therefore, x = 8.

Thus, Vijay can complete the work alone in 8 hours.

Alternate Method:

Assume total work (LCM of 8, 4, 2) = 8 units.

Work rate (units/hour):

Arun = 8 ÷ 8 = 1 unit/hour

Mahesh = 8 ÷ 4 = 2 units/hour

Total rate of all three = 8 ÷ 2 = 4 units/hour

Vijay’s rate = 4 - (Arun’s rate + Mahesh’s rate)

Vijay’s rate = 4 - (1 + 2) = 1 unit/hour

Time taken by Vijay alone = 8 ÷ 1 = 8 hours