Problem 2 · 2010 AMC 8
Easy
Algebra & Patterns
operator-definition
If a@b = a × ba + b for a, b positive integers, then what is 5@10?
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Answer: D — 10/3.
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Hint 1 of 2
The ‘@’ is just a made-up recipe: it tells you exactly what to do with the two numbers. Read it as ‘product on top, sum on the bottom.’
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Hint 2 of 2
With any new symbol, copy the rule and drop your numbers into the matching slots — a and b are placeholders waiting to be filled.
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Approach: substitute into the definition
- Plug a = 5, b = 10 into product-over-sum: 5@10 = (5 · 10)/(5 + 10) = 50/15.
- Simplify by dividing top and bottom by 5: 50/15 = 10/3.
- Why this transfers: a strange symbol like @, ★, or ◇ is never magic — it's a one-line instruction. Substitute carefully and the ‘hard’ problem becomes ordinary arithmetic.
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