Problem 19 · 1991 AJHSME
Hard
Algebra & Patterns
minimize-rest
The average (arithmetic mean) of 10 different positive whole numbers is 10. The largest possible value of any of these numbers is
Show answer
Answer: C — 55.
Show hints
Hint 1 of 3
Average 10 over 10 numbers means a FIXED total — the ten numbers always add to 100, a fixed pie. To give one number the biggest possible slice, what should the other nine slices be?
Still stuck? Show hint 2 →
Hint 2 of 3
The total is locked at 100. One number is huge only if the other nine are as TINY as possible — and they must be different positive whole numbers. What are the nine smallest such numbers?
Still stuck? Show hint 3 →
Hint 3 of 3
The nine smallest different positive whole numbers are 1, 2, 3, …, 9. Take their sum away from 100.
Show solution
Approach: fix the total, then starve the other nine numbers
- Average 10 across 10 numbers means the total is fixed: 10 × 10 = 100. With a fixed total, one number is biggest exactly when the rest are smallest.
- The nine others must be different positive whole numbers, so the smallest they can be is 1, 2, 3, …, 9. That sum is (9 × 10) ÷ 2 = 45.
- The largest number takes whatever's left: 100 − 45 = 55.
- Why this transfers: a fixed average is a fixed total — a budget. To maximize one part of a fixed budget, push every other part to its allowed minimum (and use the "different" rule to make them 1, 2, 3, …). The same "minimize the rest" move solves countless extremal problems.
- Trap to dodge: 91 (= 100 − 9, leaving 1's) ignores "different"; the values must be distinct, forcing 1 through 9, not nine copies of 1.
Mark:
· log in to save