Option 4 : \(13\frac13\)days

__Given:__

The efficiency of A = 2 × efficiency of B

Time taken by A = time taken by B - 20

__Concept used:__

Time = (Total work)/(Efficiency)

1.) When work is the same the ratio of time and efficiency is inversely proportional. eg - if time = 3 : 4 then, efficiency = 4 : 3

2.) The time required to complete the work is the reciprocal of the fraction of work done in one time period (one day).

__Calculation:__

Efficiency → A : B = 2 : 1

Time taken → A : B = 1 : 2

Difference in time = 2 - 1 = 1 ratio

∴ 1 ratio = 20 days

A's time = 20 days

B's time = 40 days

(A + B)'s one day work = \(\frac{1}{20}+\frac{1}{40}=\frac{3}{40}\)

Time taken to complete the work = \(\frac{40}{3}=13\frac13days\)

**∴ Time taken to complete the work is \(13\frac13days\) **