Amar, Akram and Vaibhav together can complete a work in 20 days. If Amar can work thrice as faster than Akram and Akram can work twice as faster than Vaibhav, then in how many days Vaibhav alone can complete the same work?

Let the daily work rate of Vaibhav be $$v$$ (units of work per day).

Akram is twice as fast as Vaibhav, so Akram’s rate is $$2v$$.

Amar is thrice as fast as Akram, hence Amar’s rate is $$3 \times 2v = 6v$$.

Total combined rate of work = $$v + 2v + 6v = 9v$$ (units per day).

They finish the whole work together in 20 days, so

$$\text{Work} = (\text{rate})\times(\text{time}) \implies 1 = 9v \times 20 \; \Rightarrow \; 9v \times 20 = 1$$

Solving for $$v$$:

$$v = \frac{1}{9 \times 20} = \frac{1}{180}$$ (work per day).

The time Vaibhav alone would take is the reciprocal of his rate:

$$\text{Time} = \frac{1}{v} = 180 \text{ days}$$

Hence Vaibhav alone can complete the work in 180 days.

Option B which is: 180 days