at the science school, there is one microscope for every 5 students. The principal figures that in order to have one for every 4 students, she needs 6 additional microscopes. How many students are at the school?

120 students

Explanation:

Let x be the number of scopes
Let a be the number of student at school

1x = 5 students
But 5x = a

1x = 4 students
But the principal needs 6 additional scope so that all the students can use it.

=> (x + 6) = 4 students
=> 4(x + 6) = a

Thus,
5x = 4(x + 6)
5x = 4x + 24
5x - 4x = 24
x = 24

So, there are 24 scopes.

Plug x in both equation and compare:

5x = a
5(24) = a
120 = a

4(x + 6) = a
4(24 + 6) = a
4(30) = a
120 = a

Therefore, the students at the school are 120