The mean of the following frequency table is 50. But the . . .

Question : The mean of the following frequency table is 50. But the frequencies f1 and f2 in class 20-40 and 60-80 respectively are missing. Find the missing frequencies.

Classes	0-20	20-40	40-60	60-80	80-100	Total
Frequency	17	f1	32	f2	19	120

Doubt by Pushkar

Solution : 

Classes (CI)	Frequency (fi)	xi	di=xi-a	fidi
0-20	17	10	-40	-680
20-40	f1	30	-20	-20f1
40-60	32	50 = a	0	0
60-80	f2	70	20	20f2
80-100	19	90	40	760
	Σfi =120=f1+f2+68			Σfidi=-20f1+20f2+80

Now, 

f₁+f₂+68 = 120
f₁+f₂ = 52 — (1)

x̄=50 (Given)

By Assumed Mean Method

x̄ = a+[Σf_id_i/Σf_i]
50 = 50 + (-20f₁+20f₂+80)/ 120
50-50 = (-20f₁+20f₂+80)/ 120 
0 ×120 = -20f₁+20f₂+80 
0 = 20(-f₁+f₂+4)
0/20 = -f₁+f₂+4 
0 = -f₁+f₂+4
f₁-f₂ = 4 — (2)

Solving equations (1) and (2)
f₁+f₂ = 52 
f₁-f₂ = 4 
--------------
2f₁+0=56
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f₁ = 56/2
f₁ = 28
Putting the value of f1 in equation (1)

28+f₂ = 52
f₂ = 52 - 28
f₂ = 24

Hence, f₁= 28 & f₂=24