$12$ men can finish a job in $16$ days. $5$ men work at the start; and after 8 days, 3 men were added. How many days needed to finish the whole job? 
Twelve men can finish a job in 16 days. 5 men were working at the start and after 8 days, 3 men were added. How many days will it take to finish the whole job?

Solution:
So, the job will take $12 \times 16 = 192$ man days to finish.
In the first $8$ days we have done $8 \times 5 = 40$ man days.
Now we are doing $8 \times 8 = 64$ man days per day and need to do the remaining $192 - 40 = 152 $man days.


*

*Day $1 = 152 - 64 = 88$ man days left

*Day $2 = 88 - 64 = 24$
So, we’ll finish on day $3$ after the extra $3$ men are added.
So based on my solution, I came up with $11$ days but I feel like it's wrong. Can someone point out my mistakes if any? 

$12$ men can finish the job in $16$ days.
In $8$ days, $24$ men can finish the job.
So $5$ men would've finished $\frac 5{24}$ of the job within those $8$ days, leaving $\frac{19}{24}$ of the job remaining.
Add $3$ men, you get $8$ men now. $8$ men would take $24$ days to finish the job.
So the reinforced workforce would take an additional $\frac{19}{24} \times 24 = 19$ days to finish up.
The total time from the start is $8 + 19 = 27$.
