(I'm Overwhelmed) Module-3-Arithmetic Progress Quiz #11 last question

In Solution, both sides were divided by 4 but 15 wasn’t divided. I thought all numbers on both sides had to be divided.

What’s the question?

In a university, there are 12 biology teachers,15 English teachers, 32 engineers, and n administrators. If the ratio of biology teachers to English teachers is the same as that of Engineers to administrators. How many administrators are there?

The Solution part seemed easier to me because I did it in a different way that took me longer, and I want to understand in which conditions we divide all sides and when we need to divide some numbers.
Thank you!

Sorry, I somehow forgot about this one. Let’s see your method - there are different ways to do such a problem after all.