What is the reasoning behind this inequality being displayed like this?

Question

The inequality for the following constraint→ "There cannot be more than 3 child tickets sold for every adult ticket sold."

Where child tickets are y

and adult tickets are x

is y ≤ 3x

Why can the answer not be 3y ≤ x instead?

1. I ask why not this format (3y ≤ x) and instead the other format (y ≤ 3x) because I assume it is 3 × (amount of child tickets, y) is less than ≤ (amount of adult tickets)

I see many questions with this type of format using words like "cannot be more", "up to" or "at least". Is there a technique or easier way to understand these constraints other than adding the coefficient (3 in this case) to the second-mentioned variable (adult tickets sold [y] in this case)?

Answer 1

It might help to first think of constraints like this as a ratio, and then convert to a linear equality later.

Here you are considering the number of child tickets (y) sold for/per adult ticket (x). The number of child tickets per adult ticket is the ratio y/x, and this ratio cannot be greater than 3. Hence y/x ≤ 3, and thus y ≤ 3x.

Unfortunately, the only general "method" for translating these kinds of problems into mathematical notation is "read the entire problem carefully and check your work."