A works 5.4 times as fast as B, and A takes 22 days less than B to complete the job when each works alone. Calculate the number of days taken to complete the same job if A and B work together.

Efficiency of A = 5.4 Efficiency of B 

A's time = B's time - 22 days

Concept Used: ​

Time taken to complete the work is inversely proportional to the efficiency

Time$$\propto \frac{1}{\text { Efficiency}}$$​  

if A complete the work in X days and B completes the work in Y days then

together they can  work in one day =$$\frac{1}{X}+\frac{1}{Y}$$ 

​$$\frac{\text{Efficiency of A}}{\text{Efficiency of B}} = \frac{54}{10} = \frac{27}{5}$$ 

Now , $$\frac{\text{Time taken by A}}{\text{Time taken by B}} = \frac{5}{27}$$ 

let time taken by A = 5x and time taken by b = 27x 

27x - 5x = 22

22x = 22

x = 1 

So, A's time to complete the work = 5 days, B's time to complete the work = 27 days

A's work rate = $$\frac{1}{5}$$ ,  B's work rate = $$\frac{1}{27}$$ 

One day work by A and B = $$\frac{1}{5} + \frac{1}{27} = \frac{27 + 5}{135} = \frac{32}{135}$$ 

​$$\therefore$$ time taken to complete the work by A and B = $$\frac{135}{32} = 4\frac{7}{32}days$$ 

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