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A certain job was assigned to a group of men to do in $20$ days. But $12$men did not turn up for the job and the remaining men did the job in $32$ days. The original number of men in the group was:
A. $31$
B. $30$
C. $32$
D. $35$

Last updated date: 13th Jun 2024
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Hint:The problem can be simplified by converting the given statements and conditions into mathematical relations using variables. By considering the initial number of men as the variable $x$, we can derive simultaneous equations and approach to the solution.

Complete step-by-step answer:
It is given that initially, a job was assigned to a group of men to do in $20$ days. Since we don’t know the number of men assigned, let it be $ = x$
So it can be said that $x$men can do the job in $20$ days.
Now, we can use the elements of the unitary method to calculate the time taken by $1$ man to complete the job.
$x$ men take $20$ days to complete the job
So, $1$ man will take $20x$ days to complete the job. $ - - - - - - - - \left( 1 \right)$
Now it is stated that $12$men did not come to work and it took $32$ days to complete.
This can be converted into the mathematical form,
$\left( {x - 12} \right)$ men take $32$ days to complete.
So according to the statement,
$1$ man will take $32\left( {x - 12} \right)$ days to complete. $ - - - - - - - \left( 2 \right)$
From equation $\left( 1 \right)$ and $\left( 2 \right)$, the time taken can be equated:
Hence, $20x = 32\left( {x - 12} \right)$
               $20x = 32x - \left( {12 \times 32} \right)$
               $12 \times 32 = 32x - 20x$
               $12x = 12 \times 32$
                    $x = 32$
So, the original number of men in the group were $32$

So, the correct answer is “Option C”.

Note:The student should never forget the basic logic that more men will complete the work in the lesser time. : The student should always remember that we should aim to convert the given statements and relations into mathematical form.
The unitary method and other simple methods should be used to approach the solution.