Given: Mean = 28, $\Sigma f_i$ = 100
Step 1: Find p and q using $\Sigma f_i = 100$
$$12 + p + 27 + 20 + q + 6 = 100$$
$$65 + p + q = 100$$
$$p + q = 35 \quad \text{...(1)}$$
Step 2: Calculate $\Sigma f_i x_i$ using class marks
| Class | $x_i$ | $f_i$ | $f_i x_i$ |
|-------|--------|--------|------------|
| 0–10 | 5 | 12 | 60 |
| 10–20 | 15 | p | 15p |
| 20–30 | 25 | 27 | 675 |
| 30–40 | 35 | 20 | 700 |
| 40–50 | 45 | q | 45q |
| 50–60 | 55 | 6 | 330 |
$$\Sigma f_i x_i = 1765 + 15p + 45q$$
Step 3: Apply mean formula
$$\bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i} \Rightarrow 28 = \frac{1765 + 15p + 45q}{100}$$
$$2800 = 1765 + 15p + 45q$$
$$15p + 45q = 1035$$
$$p + 3q = 69 \quad \text{...(2)}$$
Step 4: Solve equations (1) and (2)
Subtract (1) from (2):
$$2q = 34 \Rightarrow q = 17$$
$$p = 35 - 17 = 18$$
$$\boxed{p = 18, \quad q = 17}$$
Source: Chapter 13, Section 13.2 (Mean of Grouped Data)
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