The average temperature of a city for the first sixteen days of January is 22°C, and the average temperature for the last sixteen days of the same month is 26°C. If the average temperature for the entire month is 24°C, what is the temperature on the sixteenth day?

Correct — B. Turn every average into a total, because totals may be added and subtracted while averages may not. January has 31 days.

First sixteen days: 16 × 22 = 352
Last sixteen days, the 16th to the 31st: 16 × 26 = 416
The whole month: 31 × 24 = 744

The two blocks fill 32 day-slots in a 31-day month, so the 16th is counted twice. Its temperature is the excess:
352 + 416 − 744 = 768 − 744 = 24°C → option (b)

Why the others are wrong

• (a)
  23°C — 23°C is the midpoint of 22 and 24 — averaging two averages, which is legal only when the groups are equal in size and disjoint, and these two share the 16th.
• (c)
  25°C — 25°C is the midpoint of 24 and 26 — the same averaging-of-averages error, taken from the other end of the month.
• (d)
  22°C — 22°C is the first block's average copied straight across. The average of days 1 to 16 constrains only their sum — day 16 on its own could sit far above or below 22.

The whole method is totals, not averages. Multiply each average by its count to get a sum, then use the fact that two overlapping blocks double-count whatever they share.

Days 1 to 16 and days 16 to 31 together fill 32 day-slots in a 31-day month, so exactly one day — the 16th — is counted twice. (Sum of the two blocks) minus (sum of the month) is that day's value.

Averages themselves may only be averaged when the groups are the same size and disjoint, which is precisely the condition that fails here.

Key facts

• January has 31 days, so days 1 to 16 and days 16 to 31 are sixteen days each and overlap on the 16th.
• The overlapping day equals (16 × 22 + 16 × 26) − (31 × 24) = 768 − 744 = 24.
• An average multiplied by its count gives a total, and totals may be added and subtracted.

Common traps

• Assuming the month has 30 days, which drops the month total from 744 to 720.
• Reading 'the last sixteen days' as days 17 to 31, which is only fifteen days.
• Averaging 22 and 26 to get 24 and calling that the answer — the right number for the wrong reason, and it fails the moment the two block averages stop straddling the month average.

No directly related past PYQ was found.