To solve this problem, we can apply the concept of work done and man-days.

Step 1: Determine total man-days for the initial work.

If 15 men complete the work in 6 days, the total man-days of work can be calculated as: $\text{Total man-days} = 15 \, \text{men} \times 6 \, \text{days} = 90 \, \text{man-days}.$ This means the entire work requires 90 man-days to be completed.

Step 2: Calculate work done before the 5 men leave.

If 15 men are working for some time before 5 leave, we need to know how much work they have done in the days they've worked so far.

Let’s assume the work continues for $x$ days before 5 men leave.

The work done in $x$ days with 15 men is: $\text{Work done} = 15 \, \text{men} \times x \, \text{days} = 15x \, \text{man-days}.$

Step 3: Set up the remaining work.

After 5 men leave, there are only 10 men left to continue the work. The remaining work will be: $\text{Remaining work} = 90 \, \text{man-days} - 15x \, \text{man-days}.$ Now, the remaining work needs to be completed by 10 men, so the number of days required for them to finish the remaining work is: $\text{Remaining days} = \frac{\text{Remaining work}}{10}.$

Step 4: Calculate the time it takes to finish the remaining work.

Let’s assume that after 5 men leave, the remaining work will be completed in $y$ days. Therefore, we have: $15x + 10y = 90.$ We need more information about how long the 15 men worked before 5 left (i.e., $x$) to proceed. Could you tell me how many days the 15 men worked before the 5 men left?