Problem 20 · 2012 AMC 8
Medium
Fractions, Decimals & Percents
fraction-comparisonrewrite-as-1-minus
What is the correct ordering of the three numbers 519, 721, and 923, in increasing order?
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Answer: B — 5/19 < 7/21 < 9/23.
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Hint 1 of 2
Look at each fraction's gap between top and bottom: 19 − 5, 21 − 7, 23 − 9 — all 14! Every fraction is exactly 14 short of being 1, so compare how far each falls below 1.
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Hint 2 of 2
Rewrite each as 1 − (the missing piece): the bigger the piece you subtract, the smaller the fraction. This compare-against-1 trick beats finding a common denominator of 19, 21, 23.
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Approach: every fraction is 1 minus a piece — compare the pieces
- Spot the shared structure: each numerator is 14 below its denominator, so each fraction equals 1 − 14/(denominator). Concretely 5/19 = 1 − 14/19, 7/21 = 1 − 14/21, 9/23 = 1 − 14/23.
- Now compare the subtracted pieces: 14/19 > 14/21 > 14/23 (same top 14, so the smallest bottom gives the biggest piece).
- Subtracting a bigger piece leaves a smaller fraction, so 5/19 is smallest and 9/23 largest: 5/19 < 7/21 < 9/23.
- Intuition / sanity check: all three sit just under 1; the one closest to 1 (smallest leftover gap) is biggest. Reusable when a batch of fractions share a constant top−bottom difference — rewrite as 1 − (piece) and rank the pieces.
Another way — cross-multiply pairwise:
- Compare 5/19 vs 7/21 by cross-multiplying: 5×21 = 105 vs 7×19 = 133. Since 105 < 133, 5/19 < 7/21.
- Compare 7/21 vs 9/23: 7×23 = 161 vs 9×21 = 189, so 7/21 < 9/23.
- Chaining gives 5/19 < 7/21 < 9/23 — more arithmetic, but no clever rewrite needed.
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