The average weight of 12 people is 65 kg. If one person leaves and the new average becomes 64.5 kg, what is the weight of the person who left?

Correct — A. Compare the totals, not the averages.

Total of 12 people: 12 × 65 = 780 kg
Total of the 11 who stay: 11 × 64.5 = 709.5 kg
Subtract: 780 − 709.5 = 70.5 kg → option (a)

Why the others are wrong

• (b)
  80.5 — 80.5 leaves 780 − 80.5 = 699.5 kg for 11 people, an average of about 63.59 kg, not 64.5.
• (c)
  90.5 — 90.5 leaves 689.5 kg for 11 people, about 62.68 kg each. Losing someone that heavy pulls the average down by more than 2 kg, not by 0.5.
• (d)
  60.5 — 60.5 is below the old average of 65, so removing that person would raise the average, to about 65.41 kg. The average fell, so the leaver weighed more than 65.

An average hides a total: total = average × count. When someone leaves, write the total before and after, and the difference is the person who left.

Shortcut: the 11 who stay average 0.5 kg below 65, so together they fall 11 × 0.5 = 5.5 kg short of 65 each. For all 12 to average 65, the leaver must carry that 5.5: 65 + 5.5 = 70.5 kg.

Key facts

• Total = average × number of people: 12 × 65 = 780.
• If removing one value lowers the average, that value was above the old average.
• Leaver = old average + remaining count × fall in average = 65 + 11 × 0.5.

Common traps

• Multiplying the 0.5 fall by 12 instead of 11, which gives 71 kg.
• Treating the difference of the averages, 0.5, as a weight.

No directly related past PYQ was found.