how many faculty of each rank should be hired at each college to exactly meet the demand for sections and completely exhaust the salary budget

Rank
table. Lecture. Instrutor
Section taught. 3. 4
annual salary. 20. 25
(Thousands $)

Table. Community college
1. 2. 3
demand for sec. 30. 33. 35
Salary budget. 200. 210. 220
(Thousand$)

I don't understand how to set up to solve

Lecturer: teaches 3 sections for $20K

Instructor: teaches 4 sections for $25K
For each college, assume we hire
t=number of teachers,
i=number of instructors,
then if we were to satisfy requirements and spend all the budget, then
20t/3+25i/4 = budget
or
80t+75i=12*budget ...(1)
where t and i are integers.
For simple cases like these, the resulting diophantine equation (1) can be solved by trial and error or by more rigorous means, depending on the requirements of your teacher.

can someone explain what to do next?

To solve this problem, we need to determine the number of faculty members of each rank (lecture and instructor) that should be hired at each college, considering the demand for sections and the salary budget.

First, let's start by setting up some variables:
Let L1, L2, L3 represent the number of lecturers hired at colleges 1, 2, and 3, respectively.
Let I1, I2, I3 represent the number of instructors hired at colleges 1, 2, and 3, respectively.

Now, let's calculate the total cost of hiring faculty at each college:

For Community College 1:
The demand for sections is 30, and the number of sections taught by a lecturer is 3. So the number of lecturers needed is 30 / 3 = 10.
The demand for sections is 30, and the number of sections taught by an instructor is 4. So the number of instructors needed is 30 / 4 = 7.

The total annual salary cost for lecturers at Community College 1 is 10 * $20,000 = $200,000.
The total annual salary cost for instructors at Community College 1 is 7 * $25,000 = $175,000.

The total cost of hiring faculty at Community College 1 is $200,000 + $175,000 = $375,000.

Similarly, we can calculate the total cost of hiring faculty at colleges 2 and 3:

For Community College 2:
The demand for sections is 33, and the number of sections taught by a lecturer is 3. So the number of lecturers needed is 33 / 3 ≈ 11.
The demand for sections is 33, and the number of sections taught by an instructor is 4. So the number of instructors needed is 33 / 4 ≈ 8.

The total annual salary cost for lecturers at Community College 2 is 11 * $20,000 = $220,000.
The total annual salary cost for instructors at Community College 2 is 8 * $25,000 = $200,000.

The total cost of hiring faculty at Community College 2 is $220,000 + $200,000 = $420,000.

For Community College 3:
The demand for sections is 35, and the number of sections taught by a lecturer is 3. So the number of lecturers needed is 35 / 3 ≈ 12.
The demand for sections is 35, and the number of sections taught by an instructor is 4. So the number of instructors needed is 35 / 4 ≈ 9.

The total annual salary cost for lecturers at Community College 3 is 12 * $20,000 = $240,000.
The total annual salary cost for instructors at Community College 3 is 9 * $25,000 = $225,000.

The total cost of hiring faculty at Community College 3 is $240,000 + $225,000 = $465,000.

Now, we need to check if the total cost for each college matches the salary budget given:

For Community College 1:
The salary budget is $200,000. The total cost of hiring faculty at Community College 1 is $375,000, which exceeds the budget. Thus, it is not possible to meet the demand and exhaust the budget at this college.

For Community College 2:
The salary budget is $210,000. The total cost of hiring faculty at Community College 2 is $420,000, which exceeds the budget. Thus, it is not possible to meet the demand and exhaust the budget at this college as well.

For Community College 3:
The salary budget is $220,000. The total cost of hiring faculty at Community College 3 is $465,000, which also exceeds the budget. Therefore, it is not possible to meet the demand and exhaust the budget at this college either.

Based on these calculations, it seems that it is not possible to hire the required number of faculty members at each college while completely exhausting the salary budget. Adjustments will need to be made to meet the requirements and stay within the budget.