Anil is twice as good a student as Bharat and is able to finish a work in 30 minutes less than Bharat’s time. Find the time in which both of them can finish the same work together?

Given Anil is twice as good a student as Bharat

$$\Rightarrow$$ Efficiency  of  Anil : Efficiency  of  Bharat = 2:1

and also, Efficiency is inversely proportional to Time taken,

$$\Rightarrow$$ Time  taken  by  Anil : Time  taken  by  Bharat = 1:2...............................(1)

Given that Time taken by Anil is 30 min less than Bharat's time.

Let say, Time taken by Bharat be 't' minutes.

Then the time taken by Anil = t-30 minutes

Substituting these in equation (1), we get

$$\frac{t-30}{t} = \frac{1}{2}$$

$$\Rightarrow$$ t = 60  minutes.

Therefore the time taken by Anil and Bharat are 30 minutes and 60 minutes respectively.

Let Efficiency of Bharat be 'x', then Efficiency of Anil will be '2x'

$$\Rightarrow$$ Total Work = Efficiency\times Time taken = $$(2x)\times 30 (or) (x)\times 60 = 60x$$  units.

Efficiency when Anil and Bharat are working together = x+2x = 3x

Total Work = 60x units

Time taken(T) by Anil and Bharat together to complete the work = Total work/ Efficiency when Anil and Bharat work together

$$\Rightarrow T = \frac{60x}{3x}$$

$$\Rightarrow$$ T = 20  minutes.