| Class Interval | Midpoint \((x_i)\) | Frequency \((f_i)\) | \((x_i f_i)\) |
| 0–20 | 10 | 17 | 170 |
| 20–40 | 30 | \(f_1\) | \(30f_1\) |
| 40–60 | 50 | 32 | 1600 |
| 60–80 | 70 | \(f_2\) | \(70f_2\) |
| 80–100 | 90 | 19 | 1710 |
| Total | | \(\sum f_i = 68 + f_1 + f_2\) | \(\sum x_i f_i = 3480 + 30f_1 + 70f_2\) |
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∴ According to the question:
\[
68 + f_1 + f_2 = 120
\Rightarrow f_1 + f_2 = 120 - 68 = 52 \quad \text{(i)}
\]
Also,
\[
\frac{3480 + 30f_1 + 70f_2}{120} = 50
\Rightarrow 3480 + 30f_1 + 70f_2 = 6000
\Rightarrow 30f_1 + 70f_2 = 2520
\Rightarrow 3f_1 + 7f_2 = 252 \quad \text{(ii)}
\]
Subtracting \(3 \times \text{(i)}\) from (ii):
\[
3f_1 + 7f_2 - 3f_1 - 3f_2 = 252 - 156
\Rightarrow 4f_2 = 96
\Rightarrow f_2 = 24
\]
Substitute \(f_2 = 24\) into equation (i):
\[
f_1 = 52 - 24 = 28
\] Therefore, the required solution is:
\(f_1 = 28\) and \(f_2 = 24\)
