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**Hint:**We start solving the problem by assigning the variables for total work that need to be done and total men required to complete the job in 7 months. We then find the total work by multiplying 420 months with 9 months. We then equate this with the multiplication of 7 months with the total people required. Once we find the total no. of people required we subtract it with 420 to get the number of extra men that need to be employed.

__Complete step-by-step answer__:According to the problem, we have given that a certain piece of work can be completed in 9 months by a workforce of 420 men. We need to find the number of extra men to be employed in order to complete the job in 7 months.

Let us assume the total work to be done as x and the total men required to complete the job in 7 months be y. So, 420 men takes 9 months to complete work x.

We get the total work done by all 420 men in 9 months.

So, we get $x=420\times 9$.

$\Rightarrow x=3780$ ---(1).

Now, the amount of work obtained in equation (1) has to be completed in 7 months by y men.

So, the total amount of work done by y men in 7 months is $7\times y$ which is equal to x.

So, we get $7y=x$.

From equation (1) we get,

$\Rightarrow 7y=3780$.

$\Rightarrow y=\dfrac{3780}{7}$.

$\Rightarrow y=540$.

We have found a total of 540 men required to complete the job in 7 months. But we already have 420 men working for us. In order to find the extra men employed, we subtract 420 from the 540 people.

The extra men employed is $\left( 540-420 \right)=120$.

We have found the extra men that need to be employed in order to complete the work in 7 months as 120.

**∴ The extra men that need to be employed in order to complete the work in 7 months is 120 people.**

**Note**: We can see that there is an inverse relation between the total number of men employed and time required to complete the work done. We can also find the work done by each man in 9 months by dividing x with the total man power. Similarly, we can expect problems to find the work done by each man in a day or in a month using the obtained values.

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