Answer

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**Hint**: The total amount of work to be done remains constant irrespective of the factor of who is doing the work. Let the work capacities of the A and B be x and y, respectively. So, the total work to be finished comes to be $ 5x+5y $ . Then use the other point given in the question to form an equation in terms of x and y and express y in terms of x. Then find total work in terms of x and the coefficient of x in the expression is your answer.

**:**

__Complete step-by-step answer__To start with the solution, we let the work done by A per day be x, and the work done by B per day be y.

Total work done in n number days is given by:

$ n\times \sum{\left( \text{work capacity of each worker working} \right)\text{.}} $

Now, the total work is finished in five days if A and B work together. If we represent this mathematically, we get

$ \text{Total work that needs to be completed}=5x+5y.........(i) $

Also, it is given that if A works at twice his speed and B at half of his speed this works can be finished in 4 days, so for this case, the working capacity of A is 2x and that of B is $ \dfrac{y}{2} $ .

$ \text{Total work that needs to be completed}=4\times 2x+4\times \dfrac{1}{2}y=8x+2y.......(ii) $

Also, total work is independent of the number of workers and the type of worker doing the work. So, the total work in equation (i) and (ii) are the same.

$ 5x+5y=8x+2y $

$ \Rightarrow 3y=3x $

$ \Rightarrow x=y..........(iii) $

Now, we will substitute y from equation (iii) in equation (i), we get

$ \text{Total work that needs to be completed}=5x+5x=10x $

As we know that x is the working capacity of A, so its coefficient ‘10’ in the above result represents the number of days he would take to finish the total work. Hence, the answer to the above question is 10 days.

**Note**: Questions, including work, have two things to be wisely selected. One is the elements of the problem that you are treating as variables, and the other is the unit of work. You can either let work done per unit time of each worker as variables or the total work done by each worker to be a variable. The choice of unit and element for variable decides the complexity of solving.

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