Answer

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**Hint:**In this problem we have to calculate the number of extra workers required to complete the building, from the given data. In this problem we are going to use the formula $\dfrac{{{M}_{1}}{{D}_{1}}}{{{W}_{1}}}=\dfrac{{{M}_{2}}{{D}_{2}}}{{{W}_{2}}}$, where ${{M}_{1}}$ is the number of men for painting a building in first case, ${{D}_{1}}$ is the number of days required to complete the painting in first case, ${{W}_{1}}$ is the work done in first case. Similarly, ${{M}_{2}}$, ${{D}_{2}}$, ${{W}_{2}}$ are the corresponding values for the second case. In this problem in both the cases the work is same which is painting a same school building, so we will get ${{W}_{1}}={{W}_{2}}$, and we will assume the number of men required in second case as $x$. Now we will substitute all the values we have in the formula and simplify the equation to get the required result.

**Complete step by step solution:**

Given that, $17$ men can complete painting a school building in $120$ days.

In this case

Men required ${{M}_{1}}=17$,

Number of days required ${{D}_{1}}=120$,

Work done is ${{W}_{1}}$.

In the problem we also have if the work has to be completed in $102$ days.

In this case

Men required, let’s say ${{M}_{2}}=x$,

Number of days required ${{D}_{2}}=102$,

Work done is ${{W}_{2}}$.

In both cases they will paint for the same school building, so the work done in both cases is the same. Hence ${{W}_{1}}={{W}_{2}}$.

We have the formula $\dfrac{{{M}_{1}}{{D}_{1}}}{{{W}_{1}}}=\dfrac{{{M}_{2}}{{D}_{2}}}{{{W}_{2}}}$.

Substituting all the value we have in the above formula and simplifying it, then we will get

$\begin{align}

& \Rightarrow \dfrac{17\times 120}{{{W}_{1}}}=\dfrac{x\times 102}{{{W}_{1}}} \\

& \Rightarrow x=\dfrac{17\times 120}{102} \\

& \Rightarrow x=20 \\

\end{align}$

**From the above value we can say that the number of men required for painting the school in $102$ days is $20$. That means we need $20-17=3$ extra men for completing this job.**

**Note:**We can also use another method to solve the above equation. First, we will calculate the ratio of the number of days in the first case to the number of days in the second case which is given by $\dfrac{120}{102}$. Now we will multiply the above value with $17$ , then we will get $17\times \dfrac{120}{102}=20$. From this method also we have the same result.

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