Problem 12 · 2001 AMC 8
Medium
Algebra & Patterns
custom-operation
If a ♦ b = a + ba − b, then (6 ♦ 4) ♦ 3 =
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Answer: A — 4.
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Hint 1 of 2
A brand-new symbol is just a recipe — don't be thrown by ♦. Substitute the two numbers into the given rule.
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Hint 2 of 2
Same rules as arithmetic: parentheses first (inside-out), and keep the order — the denominator is the first minus the second, a − b.
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Approach: evaluate the custom operation inside-out
- Resolve the inner operation first. 6 ♦ 4 plugs into (a + b)/(a − b) with a = 6, b = 4: (6 + 4)/(6 − 4) = 10/2 = 5.
- That 5 becomes the new left input: 5 ♦ 3 = (5 + 3)/(5 − 3) = 8/2 = 4.
- Made-up operators only test whether you substitute carefully and honor order — keeping a − b (not b − a) is the whole game here.
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