This problem requires us to calculate how many additional staff members are needed to ensure that the lunch breaks do not disrupt the required staff-to-student ratio while all staff take their 30-minute lunch within 5 hours of clocking in.

Step 1: Analyze the staffing situation.

• The school has 96 students and 59 staff.
• 20 students require 20 staff.
• The remaining 76 students need 31 staff.

Thus, the total staffing requirements are $20 + 31 = 51$ staff members.

Step 2: Identify current staff clock-in times and breaks.

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

So, 39 staff members (16 + 16 + 7) are available, but they will need a 30-minute lunch break within 5 hours of their start time.

Step 3: Evaluate how lunch breaks affect staffing.

While staff take their lunch breaks, they are unavailable to supervise students. Given that 51 staff are required to supervise students at all times, we need to account for the periods when staff are on break.

If each staff member takes a 30-minute lunch break, this means for every 30-minute period when staff are on break, we need enough additional staff to cover the gap.

Number of staff needed during lunch breaks:

If 39 staff members will take lunch breaks (spread across different times), for each period when a group of staff is on break, we need enough additional staff to maintain the ratio. Since there are 51 staff required, and some will be taking breaks, we need at least enough extra staff to cover the gap.

Step 4: Determine how many staff are needed.

If we schedule lunch breaks in such a way that only a few staff go on break at a time, the number of additional staff needed depends on how many staff go on break at once. Ideally, the additional staff will cover the breaks in staggered shifts.

To maintain a consistent ratio without having all 39 staff taking breaks at the same time:

• Suppose breaks are staggered, with only 10 staff on break at a time.
• This would leave 41 staff available (since $51 - 10 = 41$).
• Thus, $51 - 41 = 10$ additional staff would be needed to ensure that the lunch breaks do not cause a shortfall in supervision.

Final Answer:

To ensure staff can take their lunch breaks without disrupting the required staff-to-student ratio, 10 additional staff members would be needed.

---

Would you like further details or have any questions?

Here are five related questions to expand on this topic:

1. How does the ratio of staff to students change if more students require one-on-one supervision?
2. How would the staffing requirement change if lunch breaks were extended to 45 minutes?
3. What would happen if fewer staff members were available for lunch breaks at the same time?
4. How can we optimize the schedule to reduce the need for additional staff?
5. How would the solution change if staff clocked in at different times?

Tip: When planning staff schedules, always stagger breaks to ensure that essential coverage is maintained during peak periods.