A can finish a piece of work in 16 days and B can finish it in 12 days. They worked together for 4 days and then A left. B finished the remaining work. For how many total number of days did B works to finish the work completely?

To find the total number of days B worked to finish the work completely, we can break down the problem into simple, logical steps.

Step 1: Determine the rate of work for A and B.
Let the total work be 1 unit.
A can finish the work in 16 days, so A's work rate per day is:
$\frac{1}{16}$
B can finish the work in 12 days, so B's work rate per day is:
$\frac{1}{12}$

Step 2: Find their combined rate of work when working together.
When working together, their combined work rate per day is:
$\frac{1}{16}+\frac{1}{12}$
To add these fractions, find the least common multiple (LCM) of 16 and 12, which is 48:
$\frac{3}{48}+\frac{4}{48}=\frac{7}{48}$
So, together they complete $\frac{7}{48}$ of the work per day.

Step 3: Calculate the work completed in the first 4 days.
They worked together for 4 days, so the amount of work completed is:
$4\times \frac{7}{48}=\frac{7}{12}$

Step 4: Find the remaining work.
The remaining work after A leaves is:
$1-\frac{7}{12}=\frac{5}{12}$

Step 5: Calculate the time taken by B to finish the remaining work alone.
B finishes the remaining work $\frac{5}{12}$ at a rate of $\frac{1}{12}$ per day:
Number of days B worked alone = $\frac{\text{Remaining Work}}{\text{B's Rate}}$
$\text{Days}=\frac{5/12}{1/12}=5\text{days}$

Step 6: Calculate the total number of days B worked.
B worked for the first 4 days with A, and then worked for another 5 days alone to finish the remaining work.
Total days B worked = $4+5=9\text{days}$.

Therefore, the total number of days B worked is 9.