The mean of the following frequency distribution is 57.6 and the sum of all the frequencies is 50. Find the values of p and q :

Classes	Frequency
0 - 20	7
20 - 40	p
40 - 60	12
60 - 80	q
80 - 100	8
100 - 120	5

We construct the following table :

Given,

The sum of frequencies = 50.

∴ 32 + p + q = 50
⇒ p + q = 18
⇒ p = 18 - q      …..(i)

$\therefore \text{Mean} = \dfrac{∑f$iyi}{∑f_i} \\[1em] \Rightarrow 57.6 = \dfrac{1940 + 30p + 70q}{50} \\[1em] \Rightarrow 57.6 \times 50 = 1940 + 30p + 70q \\[1em] \Rightarrow 2880 = 1940 + 30p + 70q \\[1em] \Rightarrow 30p + 70q = 2880 - 1940 \\[1em] \Rightarrow 30p + 70q = 940

Putting value of p from (i) in above equation,

$\Rightarrow 30(18 - q) + 70q = 940 \\[1em] \Rightarrow 540 - 30q + 70q = 940 \\[1em] 540 + 40q = 940 \\[1em] 40q = 940 - 540 \\[1em] 40q = 400 \\[1em] q = 10$

Using (i),

⇒ p = 18 - q = 18 - 10 = 8.

Hence, the value of p = 8 and q = 10.