The question is the following: YOur school has contracted with a professional magician to perform at the school. The school has guaranteed an attendance of at least 1000 and total ticket receipts of at least $4800. The tickets are $4 for students and $6 for non students, of which the magician receives $2.50 and $4.50 respectively. What is the minimum amount of money the magician could receive?

I am unsure how to set this up. So far I have 1000 is greater than or equal to 4s+6n (S=student n=nonstudent).

I think i would look at this from another direction. the first part of the question says there is a guaranteed attendance and a guaranteed total ticket receipt. i would take this to mean that the magician is guaranteed a minimum of either 1000 attendees or $4800 which ever is greatest.

so... if 1000 students showed up, that would only be $4000 (the minimum for 1000 attendees), which would mean we should work with the greater which would be minimum receipts of $4800.

then... i know that of every $4 taken in, the magician gets $2.50, or 62.5%

thus... 0.625*4800=$3000

but, you have to check the assumptions we made in the beginning.

To find the minimum amount of money the magician could receive, we need to set up an inequality based on the given conditions.

Let's assume that "s" represents the number of student tickets sold, and "n" represents the number of non-student tickets sold.

The total number of tickets sold can be expressed as: s + n.

The total attendance should be at least 1000, so we can write the first equation: s + n ≥ 1000.

The total ticket receipts should be at least $4800, which can be expressed as: 4s + 6n ≥ 4800.

Now, let's consider the amount of money the magician receives from each ticket sold.

For each student ticket sold, the magician receives $2.50. Therefore, the magician's earnings from student tickets can be calculated as: 2.50s.

For each non-student ticket sold, the magician receives $4.50. So, the magician's earnings from non-student tickets can be calculated as: 4.50n.

To find the minimum amount the magician could receive, we need to minimize the magician's earnings, which is the objective function. So, we want to find the minimum value of: 2.50s + 4.50n.

To summarize, we have the following system of inequalities:
s + n ≥ 1000
4s + 6n ≥ 4800

And the objective function to minimize:
Minimize: 2.50s + 4.50n

Solving this system of inequalities will give us the values of "s" and "n" that satisfy the conditions and minimize the magician's earnings.

To solve this problem, you will need to consider two conditions imposed by the school:

1. Attendance: The school has guaranteed an attendance of at least 1000 people. Let's represent the number of students as "s" and the number of non-students as "n". Since the attendance must be at least 1000, you can set up the inequality: s + n ≥ 1000.

2. Total ticket receipts: The school has guaranteed total ticket receipts of at least $4800. To calculate the total amount of money received, you'll need to multiply the number of students by the student ticket price ($4) and the number of non-students by the non-student ticket price ($6). Additionally, you'll need to subtract the amounts that the magician receives per ticket. With this information, the equation can be set up as follows:

4s + 6n - (2.50s + 4.50n) ≥ 4800.

To solve for the minimum amount of money the magician could receive, you should find the minimum value of the expression (2.50s + 4.50n). Since the goal is to minimize the magician's earnings, this value should be as small as possible while still meeting the conditions set by the school.

Now you can solve this system of inequalities (s + n ≥ 1000 and 4s + 6n - (2.50s + 4.50n) ≥ 4800) to find the minimum value of (2.50s + 4.50n) and determine the minimum amount the magician could receive.

This is a "linear programming" question

first condition:
S + n > 1000

second condition:
4S + 6n > 4800 or
2S + 3n > 2400

now graph these in the first quadrant on a S n grid, use the intercepts and it will be easy
it is easy to solve S+n=1000 with 2S+3n=2400 to get
S=600 and n=400

The profit equation for the magician would be
Prof = 2.5S + 4.5n

check:
if S=600, n=400
Prof = 600(2.5) + 400(4.5) = 3300