Problem 11 · 1986 AJHSME
Hard
Algebra & Patterns
custom-operation
If A ✶ B means (A + B) ⁄ 2, then (3 ✶ 5) ✶ 8 is
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Answer: A — 6.
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Hint 1 of 2
The ✶ symbol is a made-up rule, not real multiplication — translate it into plain English first. "(A + B) ⁄ 2" is just the *average* of A and B.
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Hint 2 of 2
Like nested parentheses, do the inside first: find 3 ✶ 5, then combine that result with 8.
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Approach: decode the symbol, work inside-out
- Translate ✶: A ✶ B is simply the average (midpoint) of A and B. That turns a strange symbol into something familiar.
- Inside first: 3 ✶ 5 is the average of 3 and 5, which is 4. Then 4 ✶ 8 is the average of 4 and 8, which is 6.
- Watch the trap: ✶ is *not* ×, so 3 ✶ 5 is 4, not 15. With custom operators, always rewrite the rule in words before computing.
- Sanity check: each ✶ produces a number between its two inputs, so the final answer must sit between 4 and 8 — 6 fits, while choices like 12, 16, 30 can't.
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