Problem 18 · 1985 AJHSME
Hard
Algebra & Patterns
interval-from-inequalities
Nine copies of a certain pamphlet cost less than $10.00 while ten copies of the same pamphlet (at the same price) cost more than $11.00. How much does one copy of this pamphlet cost?
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Answer: E — $1.11.
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Hint 1 of 2
Each clue gives a boundary on the single-copy price: '9 copies under $10' caps it from above, '10 copies over $11' lifts it from below. Together they squeeze the price into a narrow window — then just see which choice fits.
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Hint 2 of 2
Translate words to bounds: 9p < 10 means p is below 10⁄9; 10p > 11 means p is above 11⁄10. The answer is the only listed price caught between the two.
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Approach: bracket p from both inequalities
- Upper bound: 9 copies cost less than $10, so p < 10⁄9 ≈ $1.111.
- Lower bound: 10 copies cost more than $11, so p > 11⁄10 = $1.10.
- Price is trapped in 1.10 < p < 1.111. The only choice in that sliver is $1.11.
- Sanity check: at $1.11, nine copies cost $9.99 (under $10 ✓) and ten copies cost $11.10 (over $11 ✓). The window is barely a penny wide, which is exactly why only one option survives.
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