Problem 3 · 2000 AMC 8
Easy
Arithmetic & Operations
careful-countingestimate-and-pick
How many whole numbers lie in the interval between 53 and 2π?
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Answer: D — 5.
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Hint 1 of 2
You don't need exact values — you only need to know which two whole numbers the endpoints sit *between*. Turn each messy endpoint into 'the first integer past it.'
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Hint 2 of 2
Both endpoints are exclusive, so the count is (largest whole number below the top) − (smallest whole number above the bottom) + 1.
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Approach: round each endpoint inward, then count
- 5/3 ≈ 1.67, so the first whole number strictly above it is 2. And 2π ≈ 6.28, so the last whole number strictly below it is 6.
- Count from 2 to 6: that's 2, 3, 4, 5, 6 — 5 whole numbers.
- Watch the trap: 6 − 2 = 4 is wrong here. Counting endpoints inclusive needs the '+1' (6 − 2 + 1 = 5) — the classic fencepost catch.
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