A class of 30 students appeared in a test. The average score of 12 students is 60, and that of the rest is 80. What is the average score of the class?

Correct — B. The two groups are unequal, so each average must carry the weight of its group.

The rest = 30 − 12 = 18 students.
Marks of the 12 = 12 × 60 = 720
Marks of the 18 = 18 × 80 = 1440

Class total = 720 + 1440 = 2160
Class average = 2160 ÷ 30 = 72 → option (b).

Why the others are wrong

• (a)
  71 — 71 would need the 60-scoring group to number 13.5 students, since a class of 30 with m students at 60 averages 80 − 2m⁄3. The paper fixes m = 12.
• (c)
  70 — 70 is the plain average of 60 and 80, and that is correct only for an equal split, m = 15. Here 12 score 60 and 18 score 80, so the mean is pulled toward 80.
• (d)
  73 — 73 needs m = 10.5 by the same formula, and no whole-number split of 30 reaches it. With m = 12 the average is 72.

This is a weighted mean. Averaging two group averages only works when the groups are the same size; otherwise each average must be multiplied by its own count.

Total marks = 12 × 60 + 18 × 80, and the divisor is the whole class, 30.

Read as a shift from the larger group, the class average is 80 − 8: the 12 low scorers sit 20 marks below 80, and 12 × 20 ÷ 30 = 8.

Key facts

• A weighted mean is the sum of all values divided by the total count, not the mean of the means.
• 12 students at 60 contribute 720 marks and 18 at 80 contribute 1440, for 2160 in all.
• For a class of 30 split m students at 60 and the rest at 80, the average is 80 − 2m⁄3.
• The weighted mean always lies between the two group averages, nearer the larger group.

Common traps

• Averaging the two averages to 70, which is right only for an equal split.
• Taking the rest as 30 instead of 18 — the rest is the class minus the 12.
• Dividing by 2 groups rather than by 30 students.

No directly related past PYQ was found.