A school has 96 students and 59 staff. 20 students require 20 staff at all times while the remaining 76 students need 31 staff at all times. If 8 staff clocked in at 7:15AM and needed their lunch within 5 hours, 16 staff clocked in at 6:45AM, and 15 clocked in at 7:45AM and all need their lunch within 5 hours of when they clocked in, how many additional staff would it take to ensure staff received their 30-minute lunch and maintain current ratios? Show a schedule

### 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?

A school has 96 students and 59 staff. 20 students require 20 staff at all times while the remaining 76 students need 31 staff at all times. If 8 staff clocked in at 7:15AM and needed their lunch within 5 hours, 16 staff clocked in at 6:45AM, and 15 clocked in at 7:45AM and all need their lunch within 5 hours of when they clocked in, how many additional staff would it take to ensure staff received their 30-minute lunch and maintain current ratios? Show a schedule.

This scheduling problem involves managing 96 students with 59 staff members, balancing lunch breaks while maintaining required student-to-staff ratios. Learn how to calculate staff shortages and determine additional hires, using ratios and time management concepts. Suitable for Grades 9-12.

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