Problem Breakdown:

1. Total students: 96 students.

   • 20 students require 20 staff (1-to-1 ratio).
   • 76 students require 31 staff (approx. 1 staff for every 2.45 students).

2. Staff required to supervise all students:

   • For the 20 students, 20 staff are needed.
   • For the 76 students, 31 staff are needed.
   • So, total staff required at all times: $20 + 31 = 51$ staff.

3. Staff clock-in times and lunch requirements:

   • 8 staff clocked in at 7:15 AM, lunch needed by 12:15 PM (within 5 hours).
   • 16 staff clocked in at 6:45 AM, lunch needed by 11:45 AM.
   • 15 staff clocked in at 7:45 AM, lunch needed by 12:45 PM.

4. Lunch breaks: All staff need a 30-minute lunch break within 5 hours of clocking in.

Solution:

Step 1: Assess the staff availability during lunch periods

We must ensure 51 staff are always available, even when staff are on their 30-minute lunch break. Let's determine the staff available during the lunch windows and then calculate how many additional staff are needed.

Step 2: Calculate lunch windows

1. Staff clocking in at 6:45 AM:

   • Lunch window: 11:45 AM to 12:15 PM.
   • 16 staff need lunch during this window.

2. Staff clocking in at 7:15 AM:

   • Lunch window: 12:15 PM to 12:45 PM.
   • 8 staff need lunch during this window.

3. Staff clocking in at 7:45 AM:

   • Lunch window: 12:45 PM to 1:15 PM.
   • 15 staff need lunch during this window.

Step 3: Schedule and calculate shortages during lunch windows

We will analyze each 30-minute lunch window and check how many staff are available, then determine if we need additional staff to maintain the required 51 staff on duty.

1. 11:45 AM - 12:15 PM (16 staff on lunch):

   • Available staff: $8 + 15 = 23$ staff (since 16 are on lunch).
   • Staff required: 51.
   • Staff shortage: $51 - 23 = 28$ additional staff needed.

2. 12:15 PM - 12:45 PM (8 staff on lunch):

   • Available staff: $16 + 15 = 31$ staff (since 8 are on lunch).
   • Staff required: 51.
   • Staff shortage: $51 - 31 = 20$ additional staff needed.

3. 12:45 PM - 1:15 PM (15 staff on lunch):

   • Available staff: $8 + 16 = 24$ staff (since 15 are on lunch).
   • Staff required: 51.
   • Staff shortage: $51 - 24 = 27$ additional staff needed.

Step 4: Minimum additional staff needed

The largest shortage occurs during the 11:45 AM - 12:15 PM window, where 28 additional staff are required.

Thus, the school would need to hire 28 additional staff to ensure all staff receive their 30-minute lunch break and the student-to-staff ratios are maintained at all times.

Schedule Summary:

• 11:45 AM - 12:15 PM: 28 additional staff required.
• 12:15 PM - 12:45 PM: 20 additional staff required.
• 12:45 PM - 1:15 PM: 27 additional staff required.

Tip:

Staggering lunch breaks more efficiently, or introducing floating staff who only work during peak times, could reduce the number of additional hires needed.

Follow-up Questions:

1. How would reducing the lunch break time impact the staff requirement?
2. What if additional staff clocked in at 8:00 AM—how would that affect the schedule?
3. Could shifts be adjusted to accommodate lunch without hiring more staff?
4. What is the cost implication of hiring 28 additional staff for the school?
5. How could technology or automation reduce the staff requirements in such a scenario?