Find the missing frequency f1,f2 and f3 in frequency . . .

Question : Find the missing frequency f₁,f₂ and f₃ in frequency distribution, when it is given that f₂:f₃=4:3 and mean is 50.

Class Interval	Frequency
0-20	17
20-40	f1
40-60	f2
60-80	f3
80-100	19
Total	120

Doubt by Gauri
Solution : 

Mean (x̄) = 50 (Given)

f₂:f₃=4:3 (Given)
Let f₁=x
Let f₂=4y and f₃=3y

Class Interval (CI)	Frequency (fi)	Class Marks (xi)	fixi
0-20	17	10	170
20-40	x	30	30x
40-60	4y	50	200y
60-80	3y	70	210y
80-100	19	90	1710
Total	120		30x+410y+1880

Now, 
17+x+4y+3y+19=120
x+7y+36=120

x+7y=120-36
x+7y=84 — (1)

By Direct Method

Mean (x̄)= Σf_ix_i/Σf_i
50 = (30x+410y+1880)/120
50×120 = 30x+410y+1880

6000 = 30x+410y+1880
6000-1880 = 30x+410y
4120 = 30x+410y
4120 =10(3x+41y)
412 = 3x+41y
3x+41y=412 — (2)

Solving equation (1) and (2)

 (x+7y=84)×3
 3x+41y=412

 3x + 21y = 252
 3x + 41y = 412
-     -           - 
____________
0    -20y = -160
____________

20y=160
y=160/20
y=8
putting in eq (1)

x+7(8)=84
x+56=84
x=84-56
x=28

f₁=x=28

f₂=4y=4(8) = 32
f₃=3y=3(8) = 24

Hence, f₁=28, f₂=32, f₃=24