The average score of 25 students in a test is 68. The top 5 scorers had an average of 80, and the lowest 5 had an average of 50. What is the average of the remaining 15 students?

Correct — A. Work in totals: take the two known groups out of the class total, then divide by the 15 left.

Class total = 25 × 68 = 1700
Top 5 = 5 × 80 = 400
Lowest 5 = 5 × 50 = 250
Remaining 15 = 1700 − 400 − 250 = 1050

Average of the 15 = 1050 ÷ 15 = 70 → option (a)

Why the others are wrong

• (b)
  68 — 68 is the class average, which assumes the top and bottom groups cancel. They do not: the top 5 sit 60 marks above 68 in total, the lowest 5 sit 90 below.
• (c)
  72 — 72 × 15 = 1080. With 400 and 250 that makes 1730, a class average of 69.2, not 68.
• (d)
  66 — 66 × 15 = 990. With 400 and 250 that makes 1640, a class average of 65.6, not 68.

Average × count = total lets you add and subtract groups. Averages cannot be subtracted directly, because the groups differ in size.

The faster route measures each group against the class average of 68:
Top 5: +12 each = +60
Lowest 5: −18 each = −90

The deviations must sum to zero, so the middle 15 supply +30, which is +2 each: 68 + 2 = 70.

Key facts

• Group total = group average × group size.
• Deviations from the overall average, summed over every member, come to zero.
• Average of the rest = (overall total − known group totals) ÷ members left.

Common traps

• Assuming the middle group matches the class average because the top and bottom groups are the same size: +12 and −18 pull by different amounts.
• Dividing the remaining 1050 by 25 instead of 15: divide by the size of the group you are asked about.

No directly related past PYQ was found.