The proportion of students in three classes is 3 : 4 : 5. If the number of students is increased by 20 in each class, then the proportion changes to 4 : 5 : 6. What is the initial number of students in the three classes taken together?

The proportion of students in three classes is 3 : 4 : 5.

let the number of students be 3x, 4x, 5x

now 20 students increased in each class

so 3x+20: 4x+20 : 5x+20 is 4:5:6 

now $$\frac{3x+20}{4x+20}=\frac{4}{5}$$ 

we get 15x+100 = 16x +80

x= 20

initial number of students in 3 class taken together is 3x+4x+5x = 12x = $$12\times 20$$= 240

alternatively 

initial proportion of the three class is 3: 4: 5

final proportion of the three class is  4: 5: 6 

observing the change of 1 unit from initial to final that is ( 4-3 =1 , 5-4= 1, 6-5 = 1) the change has come due to increase in number of students in each section which is equal to 20 

hence 1 unit = 20

now initial number of students taken from 3 class together = 3+4+5 = 12 units 

we know 1 unit = 20 

therefore 12 units = $$12\times 20$$ = 240