The average of 9 numbers is 56. When one number is removed, the average becomes 54. What number was removed?

Correct — B. Turn both averages into totals, then subtract.

Total of 9 numbers = 9 × 56 = 504
Total of the 8 left = 8 × 54 = 432
Removed number = 504 − 432 = 72 → option (b)

Shortcut: old average + (numbers left × fall in average) = 56 + 8 × 2 = 72

Why the others are wrong

• (a)
  52 — 52 is below the old average of 56, so removing it would raise the average, to 452 ÷ 8 = 56.5. The average fell.
• (c)
  82 — Removing 82 leaves 504 − 82 = 422, and 422 ÷ 8 = 52.75, not the stated new average of 54.
• (d)
  92 — Removing 92 leaves 504 − 92 = 412, and 412 ÷ 8 = 51.5, not the stated new average of 54.

An average is a total shared equally, so sum = count × average. To find a removed value, turn the old and new averages back into totals and subtract.

Direction check: when removing a value makes the average fall, that value was above the old average. The average here fell from 56 to 54, so the removed number is more than 56.

Key facts

• Sum = number of values × average.
• Removed value = old total − new total = 9 × 56 − 8 × 54 = 72.
• Removed value = old average + (values left × fall in average).
• Removing a value above the average lowers the average, and removing one below it raises it.

Common traps

• Multiplying the new average by 9 instead of 8: 504 − 9 × 54 = 18, forgetting that only 8 numbers remain.
• Pairing the old average with the old count: 56 + 9 × 2 = 74. The shortcut needs 56 + 8 × 2, or equally 54 + 9 × 2.

No directly related past PYQ was found.