Problem 11 · 1987 AJHSME
Hard
Fractions, Decimals & Percents
bound-the-fractional-part
The sum 2 1⁄7 + 3 1⁄2 + 5 1⁄19 is between
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Answer: B — 10 1⁄2 and 11.
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Hint 1 of 2
You don't need the exact sum — just which half-unit it lands in. Add the whole numbers, then ask only how big the leftover fractions can get.
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Hint 2 of 2
The whole parts give 10 exactly. For the three fractions, find a floor and a ceiling: one of them is already 1⁄2, and the other two together are tiny.
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Approach: split off the whole parts, then bound the fractions
- Whole parts: 2 + 3 + 5 = 10 — that's locked in. Now the fractions 1⁄7 + 1⁄2 + 1⁄19 only need bounding, not exact addition.
- Floor: it's more than 1⁄2, because one term is already 1⁄2 and the others are positive. Ceiling: it's less than 1, because 1⁄7 + 1⁄19 is well under 1⁄2 (each is smaller than 1⁄4).
- So the fractions land between 1⁄2 and 1, putting the total between 10 1⁄2 and 11 — answer B.
- Why this transfers: when a question asks 'between which values,' estimate with bounds instead of finding a common denominator. Far less work, no arithmetic slips.
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