Let's break this down step by step:

Step 1: Staff Required for Students

• 20 students require 20 staff.
• The remaining 76 students need 31 staff.

So, a total of $20 + 31 = 51$ staff are required at all times.

Step 2: Analyze Staff Lunch Schedule

We need to make sure there are always enough staff to cover for those on lunch breaks. The lunch breaks are 30 minutes long, and the staff must take their lunch within 5 hours of clocking in.

Here’s the breakdown of the staff clock-in times:

• 16 staff clocked in at 7:15 AM, requiring lunch by 12:15 PM.
• 8 staff clocked in at 6:45 AM, requiring lunch by 11:45 AM.
• 15 staff clocked in at 7:45 AM, requiring lunch by 12:45 PM.

Step 3: Plan for Lunch Coverage

Lunch coverage needs to be provided while maintaining the required staffing levels for students. Let's focus on the lunch window for each group of staff.

Group 1: 8 Staff (Clock-in at 6:45 AM)

• Lunch between 11:45 AM - 12:15 PM.
• During this period, 8 staff members will be unavailable, leaving 51 - 8 = 43 staff on duty. Since 51 staff are required, we are short by 8 staff during this time.

Group 2: 16 Staff (Clock-in at 7:15 AM)

• Lunch between 12:15 PM - 12:45 PM.
• During this period, 16 staff members will be unavailable, leaving 51 - 16 = 35 staff on duty. In this case, we are short by 16 staff.

Group 3: 15 Staff (Clock-in at 7:45 AM)

• Lunch between 12:45 PM - 1:15 PM.
• During this period, 15 staff members will be unavailable, leaving 51 - 15 = 36 staff on duty. We are short by 15 staff in this case.

Step 4: Calculate Total Additional Staff Needed

The period when we are short by the most staff is during Group 2's lunch break, where 16 staff members are taking lunch, and we are left with only 35 staff on duty (short by 16).

Therefore, to ensure all staff members get their 30-minute lunch and the current ratios are maintained