Problem 16 · 2022 AMC 8
Medium
Arithmetic & Operations
sum-constraint
Four numbers are written in a row. The average of the first two is 21, the average of the middle two is 26, and the average of the last two is 30. What is the average of the first and last of the numbers?
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Answer: B — 25.
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Hint 1 of 2
You can't pin down the four numbers from three clues — but you don't have to. The question asks only about the combination a + d, and that one combination is findable. Turn each average into a sum first.
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Hint 2 of 2
The pairs are a+b, b+c, c+d. You want a+d — which two should you add, and which should you subtract, so the b's and c's vanish?
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Approach: combine the pair-sums so the middle terms cancel
- Insight: three clues can't fix four numbers, but the target a+d is a special combination that is determined. Turn each average into a sum: a+b = 42, b+c = 52, c+d = 60.
- Add the outer two and subtract the inner one — the b and c cancel cleanly: (a+b) + (c+d) − (b+c) = a+d = 42 + 60 − 52 = 50.
- Average of first and last = 50 ÷ 2 = 25.
- You'll see this again: when a problem gives you sums and asks for a particular combination, don't solve for the individual unknowns — add and subtract the given sums so the unwanted variables cancel.
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