If the average of the first four of five numbers in decreasing order is 25 and the average of the last four numbers is 20, then what is the difference between the first and the last number?

Correct — D, 20. Call the five numbers a₁ to a₅ in decreasing order. The first four average 25, so a₁ + a₂ + a₃ + a₄ = 100. The last four average 20, so a₂ + a₃ + a₄ + a₅ = 80. Subtract the second total from the first — the three middle numbers cancel completely, leaving a₁ − a₅ = 100 − 80 = 20. The middle values never have to be known.

• (a)5 — Five is the difference between the two averages, 25 − 20. The difference of the numbers is that gap multiplied by four, because each average covers four terms.
• (b)10 — Ten would follow if each average covered two terms. Here each covers four, so the multiplier is four.
• (c)15 — Fifteen matches no step in the working; the totals differ by exactly 20 and that difference is the answer.

Two overlapping blocks of the same list differ only in their end members. Converting both averages into totals and subtracting makes the shared middle terms vanish, so the answer emerges without solving for any individual value.

The 'in decreasing order' phrase matters only for the sign — it tells you the first number is the larger, so the difference is +20 rather than −20. Generalise the structure: if two blocks of k terms overlap in k − 1 terms and their averages differ by d, the two end members differ by kd. Here k = 4 and d = 5, giving 20.

• Sum of the first four = 4 × 25 = 100; sum of the last four = 4 × 20 = 80.
• Subtracting the two sums cancels a₂, a₃ and a₄, leaving a₁ − a₅ = 20.
• General rule: overlapping blocks of k terms whose averages differ by d have end members differing by kd.
• The individual values are not determined by the data — only their difference is, which is why the question asks only for that.

• Answering 5 by subtracting the averages instead of the totals.
• Trying to pin down the individual numbers, which the data do not fix.
• Losing the sign by ignoring that the list is in decreasing order.

Asked with two overlapping sub-groups of a list and their averages, the target being the difference between the two end members, or the value of one of them when the other is given.

• practice — not a real PYQ

  The average of the first three of four numbers is 16 and the average of the last three is 20. The difference between the fourth and the first number is

  • (a)4
  • (b)8
  • (c)12
  • (d)16

  Answer(c) 12 — the totals are 48 and 60; subtracting cancels the two shared numbers, leaving the difference of the end members as 12.

• practice — not a real PYQ

  The average weight of 10 students falls by 200 g when one of them, weighing 60 kg, is replaced by a new student. The new student weighs

  • (a)56 kg
  • (b)58 kg
  • (c)59 kg
  • (d)62 kg

  Answer(b) 58 kg — a drop of 0.2 kg in the average of 10 means the total fell by 2 kg, so the replacement is 60 − 2 = 58 kg.