Problem 11 · 1996 AJHSME
Medium
Algebra & Patterns
compare-magnitudes
Let x be the number 0.00…01, where there are 1996 zeros after the decimal point before the 1. Which of the following expressions represents the largest number?
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Answer: D — 3/x.
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Hint 1 of 2
You never need the actual value of x — it's just 'a microscopically tiny positive number.' The question is which OPERATION makes something huge. Adding, subtracting, or multiplying by a tiny number barely changes things. What about dividing?
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Hint 2 of 2
Dividing BY a tiny number is the one move that explodes: 3 ÷ (almost zero) is enormous. Watch the trap — x/3 has the x on top, so it stays tiny. You want x on the BOTTOM.
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Approach: ask which operation explodes
- Treat x as basically zero. Then 3 + x ≈ 3, 3 − x ≈ 3, 3·x ≈ 0, and x/3 ≈ 0 — none of those gets big. The only expression that behaves differently is the one dividing BY x.
- 3/x divides 3 by something microscopic, which makes it gigantic, so 3/x is the largest. (Don't be fooled by x/3 — that has x on top, so it's tiny, not large.)
- Why this transfers: dividing by a near-zero number sends a result toward infinity; multiplying by it sends it toward zero. When a problem says 'very small/large,' track which slot the extreme value sits in — top or bottom — instead of computing.
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