Problem 15 · 1993 AJHSME
Medium
Arithmetic & Operations
average-sum
The arithmetic mean (average) of four numbers is 85. If the largest of these numbers is 97, then the mean of the remaining three numbers is
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Answer: A — 81.0.
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Hint 1 of 2
An average hides a total: if four numbers average 85, you instantly know their sum. Recover the sum first — that's the number you can actually take things away from.
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Hint 2 of 2
You can't average the leftover three until you know their total. Take the full total, subtract the 97, then divide by 3.
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Approach: average → total → adjust → average again
- Mean 85 over 4 numbers means the sum is 4 × 85 = 340. (An average is the total shared equally, so total = average × count.)
- Drop the largest, 97: the other three sum to 340 − 97 = 243. Their mean is 243 ÷ 3 = 81.0.
- Why this transfers: almost every 'average changes when you add/remove a number' problem runs through the total. Convert average→sum, adjust the sum, then convert back. Sanity check: 97 was above the old average of 85, so pulling it out should drag the average down — and 81 < 85. ✓
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