Answer

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**Hint:**It is given that 2 men can do a certain amount of work in one day and for doing the same amount of work in one day 3 boys are required. Let x be the amount of work done by a boy and y be the amount of work by a man. Therefore the relation between the work done by the boys and the men in one day is given by the equation $3x=2y$. Using this relation, let us calculate the required data.

**Complete Step by Step Solution:**

Let us express the given situation as an algebraic equation.

Seventy-five men do a piece of work in 24 days. Let us assume that the amount of work done here is ${{W}_{1}}$.

The amount of work done by a boy in one day is equal to x. Therefore the work done ${{W}_{1}}$ by 75 boys in 24 days can be written as shown below.

$\Rightarrow {{W}_{1}}=75\left( x \right)\times 24=1800x$

$\Rightarrow {{W}_{1}}=1800x$ ……(1)

We have to find out the number of men who can do double the amount of work done by 75 boys in 24 days. Hence the final work ${{W}_{2}}$ that has to be done can be assumed as ${{W}_{2}}=2{{W}_{1}}$. Therefore multiply equation (1) by 2.

$\Rightarrow {{W}_{2}}=2{{W}_{1}}=2\times 1800x$

$\Rightarrow {{W}_{2}}=3600x$ ……(2)

Work done by three boys in one day is equal to the work done by two men in one day.

$\Rightarrow 3x=2y$

$\Rightarrow x=\dfrac{2}{3}y$

Let us substitute the above relation in (2)

$\Rightarrow {{W}_{2}}=3600x=3600\times \dfrac{2}{3}y$

$\Rightarrow {{W}_{2}}=2400y$

We know that the amount of work a man can do in one day is equal to y. For 20 days, the amount of work done is 20y. Therefore the number of men required to complete the work in 20 days can be calculated as follows.

$\Rightarrow \dfrac{2400y}{20y}=120$ men.

**The 120 men will be required to do the double amount of work in 20 days. Hence option C is the right choice.**

**Note:**

In day-work problems like these, establishing a relationship between the variable is a crucial step. These relations can be substituted in suitable equations to get the solution. And we should be able to write down the given condition into an algebraic equation with the given variables.

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