A set of six distinct integers is split into two sets of three.
The first set of three integers has a mean of 10 and a median of 8.
The second set of three integers has a mean of 12 and a median of 9.
What is the smallest possible range of the set of all six integers?
- A8
- B10
- C11
- D12
- E14
- F15
Show the answer and worked solution
answer · E
- A8
- B10
- C11
- D12
- E14
- F15
Write the first set as {p, 8, q} with p < 8 < q and p + q = 22, and the second as {r, 9, s} with r < 9 < s and r + s = 27. To make the range small you want the two small values as large as possible and the two large values as small as possible, so push p and r up. The obstruction is distinctness: 8 is already used, so r ≠ 8, forcing r ≤ 7 and s ≥ 20; and if r = 7 then p ≤ 6, so q ≥ 16. That gives {6, 8, 16} and {7, 9, 20}, all six distinct, with range 20 − 6 = 14. Anything smaller is impossible: s ≥ 19 always, and the minimum of the six is at most 7 with equality only if r ≥ 7 and p = 7, which distinctness forbids.