Given that,

Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job.

If $M_{1}$ person can do $W_{1}$ work in $D_{1}$ days working $T_{1}$ hours in a day and $M_{2}$ Person can do $W_{2}$ work in $D_{2}$ days working $T_{2}$ hours in a day then the relationship between them is:

$$\frac{M_{1} \ast D_{1} \ast T_{1}}{W_{1}} = \frac{M_{2} \ast D_{2} \ast T_{2}}{W_{2}}$$

$(3\; \text{Men} + 8 \; \text{Machine})1 = (8 \; \text{Men} + 3 \; \text{Machine}) 2 $

$ \Rightarrow (3\; \text{Men} + 8 \; \text{Machine}) = (16 \; \text{Men} + 6 \; \text{Machine}) $

$ \Rightarrow 8 \; \text{Machine} – 6 \; \text{Machine} = 16 \; \text{Men} – 3\; \text{Men} $

$ \Rightarrow \boxed{2 \; \text{Machine} = 13 \; \text{Men}}$

If two machines can finish the job in $13$ days, then $13$ men can also finish that job in $13$ days.

$\therefore$ The number of men required to finish the job in $13$ days is $13$ men.

Correct Answer $:13$