Problem 13 · 1987 AJHSME
Hard
Fractions, Decimals & Percents
compare-to-half
Which of the following fractions has the largest value?
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Answer: E — 151⁄301.
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Hint 1 of 2
Don't hunt for a common denominator across all five. Pick one easy yardstick they're all near and measure each against it.
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Hint 2 of 2
Each fraction is close to 1⁄2. To test a⁄b against 1⁄2, just compare 2a to b — double the top and see if it beats the bottom.
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Approach: compare every fraction to the landmark 1⁄2
- Use the test a⁄b > 1⁄2 exactly when 2a > b (double the numerator, compare to the denominator). Double each top: 6 vs 7, 8 vs 9, 34 vs 35, 200 vs 201 — every one falls just short, so each is a hair below 1⁄2.
- But 151⁄301 doubles to 302 vs 301: 302 > 301, so it's just over 1⁄2.
- Four fractions sit below 1⁄2 and one sits above it, so 151⁄301 is the largest.
- Why this transfers: comparing many quantities to a single landmark (here 1⁄2) replaces ten messy pairwise comparisons with five quick checks.
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