# Linear Programming Word Problem: Theater

I'm having trouble understanding the second constraint.

A theater is presenting a program on drinking and driving for students and their parents[...] admission is 2.00 dollars for parents and 1.00 dollar for students. However, the situation has two constraints: 1) The theater can hold no more than 150 people and 2) every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?

Let $$x$$ = number of students

$$y$$ = number of parents

Since the question prompt wants the maximum amount of revenue, then the objective function is $$z = x + 2y$$

The theater can hold no more than 150 people so the first constraint is simply: $$x + y \leq 150$$

The second constraint is, "every two parents must bring at least one student," but I don't understand how to model this.

I've looked up a solution and it said $$y \leq 2x$$ is how to model this constraint, but I don't understand why. If there must be at least one student for every two parents then why wouldn't the inequality be $$2y \geq x$$?

Yes

y <= 2x

The easiest way to test this is with some data points and sketch a graph:

y_______x

2_______1,2,3,4,5…

4_______2,3,4,5,6…

6_______3,4,5,6,7…

Hence:

y <= 2x

2 <= 2x1 True

and

2y >= x

2x2 >= 5 False

Graph of y <= 2x