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 16 staff clocked in at 7:15AM and needed their lunch within 5 hours, 8 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?

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

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 16 staff clocked in at 7:15 AM and needed their lunch within 5 hours, 8 staff clocked in at 6:45 AM, and 15 clocked in at 7:45 AM 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?

This problem involves calculating the additional staff needed to cover lunch breaks for 59 school staff while maintaining a ratio of 51 staff required for 96 students. It includes analyzing clock-in times and ensuring staff get a 30-minute lunch within 5 hours of starting their shift. Ideal for students in Grades 9-12 to practice basic arithmetic and optimization.

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