Problem 2 · 1986 AJHSME
Easy
Fractions, Decimals & Percents
reciprocal-monotonicity
Which of the following numbers has the largest reciprocal?
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Answer: A — 1⁄3.
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Hint 1 of 2
Picture cutting a pizza: dividing 1 into *more* pieces (a bigger number on the bottom) makes each piece *smaller*. So which number gives the biggest reciprocal?
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Hint 2 of 2
Reciprocal flips a number over 1, and flipping reverses the order: the smallest positive number has the largest reciprocal.
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Approach: flipping reverses size order
- The reciprocal of a positive number is 1 over it. Taking 1-over reverses the size order — the *smallest* number turns into the *largest* reciprocal. So you never have to compute a single reciprocal; just find the smallest number.
- The smallest of the choices is 1⁄3; its reciprocal is 3, larger than the reciprocals of 1, 5, and 1986 (which are all 1 or less).
- Why this transfers: whenever you flip a list of positive numbers, biggest and smallest swap places — handy for spotting the answer without arithmetic.
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