In psychology 101, the ratio of the number of girls to the number of boys is $3:1$. after one girl drops the class and three more boys sign up, the ratio of the number of girls to the number of boys becomes $5:2$. how many students are in the class after these changes?

Initially,

g = number of girls;

b = number of boys.

Because the ratio of girls to boys is 3:1, therefore

g = 3b (1)

Finally, we have

g-1 girls because one girl dropped the class;

b+3 boys because three more boys signed up.

Because the new ratio of girls to boys is 5:2, therefore

(g - 1) / (b + 3) = 5/2

Cross multiply.

2 (g - 1) = 5 (b + 3)

2g - 2 = 5b + 15

2g - 5b = 17 (2)

Substitute (1) into (2) to obtain

2 (3b) - 5b = 17

b = 17

g = 3b = 3*17 = 51

Final number of students in the class = 17 + 51 = 68

Answer: 68