Initially, $40\%$ of the students at the school dance are girls. then, $15$ more girls arrive, after which $52\%$ of the students at the dance are girls. how many students are now at the dance after the additional girls arrive?

This is the concept of algebra, to get the total number of dancers we proceed as follows. Let total number of dancers be x;
number of girls will be:
40/100*x
=0.4x
When the number of girls increase by 15, the new number of dances will be:
x+15 dancers
The new number of girls will be:
0.4x+15
since the new number of girls have increased to 52%, then the total number of dancers will be:
(0.4x+15)/(x+15)*100=52
(0.4x+15)/(x+15)=0.52
0.4x+15=0.52(x+15)
solving for x we get
0.4x+15=0.52x+7.8
putting like terms together and simplifying we get:
-0.12x=-7.2
thus
x=-7.2/(-0.12)
x=60
thus the total number of dancers on the floor was 60