Problem 3 · 2019 AMC 8
Medium
Fractions, Decimals & Percents
fraction-comparison
Which of the following is the correct order of the fractions 1511, 1915, and 1713, from least to greatest?
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Answer: E — 19/15 < 17/13 < 15/11.
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Hint 1 of 2
Don't cross-multiply three times — first notice each fraction sits just barely above 1. The whole question is really "which one sticks up the most past 1?"
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Hint 2 of 2
In every fraction the top is exactly 4 bigger than the bottom, so each equals 1 + 4denominator. Same numerator, so the comparison collapses to comparing denominators.
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Approach: split off the whole 1, compare the leftover
- Every fraction has top − bottom = 4, so write each as 1 + 4denom. The "1" is shared, so only the leftover 4denom decides the order.
- Same numerator 4 with a bigger denominator gives a smaller piece: 415 < 413 < 411.
- Adding back the shared 1 keeps that order, so least to greatest is 1915 < 1713 < 1511 (choice E).
- Why this transfers: when fractions cluster near a round number, subtract that number off and compare the tiny remainders — far easier than cross-multiplying, and the rule "same top, bigger bottom = smaller" does the rest.
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