A group of 210 students appeared in some test. The mean of 1/3rd of students…

Concept — the combined (weighted) mean. When a set is split into sub-groups, the mean of the whole set is the total of all the values divided by the total count. Each sub-group mean therefore enters weighted by that sub-group’s own size: x̄ = (n1 x̄1 + n2 x̄2) ÷ (n1 + n2). Only when the two sub-groups are equal in size does this reduce to the plain average of the two means.

Application.

1. Size of the one-third block: n1 = 210 ÷ 3 = 70 students; its mean is x̄1 = 60.

2. Size of the remaining block: n2 = 210 − 70 = 140 students; its mean is x̄2 = 78.

3. Total marks of the one-third block: 70 × 60 = 4200.

4. Total marks of the remaining block: 140 × 78 = 10920.

5. Total marks of the whole group: 4200 + 10920 = 15120.

6. Mean of the whole group: 15120 ÷ 210 = 72.

Cross-check. The two blocks stand in the size ratio 1 : 2, so the combined mean must lie twice as far from 60 as it does from 78. The 18-mark gap between the two sub-group means splits as 12 and 6, giving 60 + 12 = 72 from below and 78 − 6 = 72 from above — the same value, so the arithmetic is consistent.

Note also that the plain average of 60 and 78 is 69. The combined mean must be larger than that, because the bigger block is the one carrying the higher mean — which is exactly why the sizes, not just the two means, decide the answer.