AMC 8 · 2012 · #20
Grade 6 arithmeticPick an answer.
AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Before doing any arithmetic, scan the three fractions for structure. The numerators are 5, 7, 9 and the denominators are 19, 21, 23 — each denominator is exactly 14 more than its numerator. That turns the question into one easy-to-reason-about family: . Tool #5 (Look for a Pattern) lets us answer the whole ordering with one observation about that family, without grinding through three pairwise cross-multiplications. Tool #9 (Easier Related Problem) supports it: simplifies to the familiar landmark , which we can use as a sanity check against the other two.
Rewrite each denominator as numerator + 14, so every fraction has the shared form for n = 5, 7, 9.
Naming the common form is the Tool #5 move — once the pattern is named, one rule will sort all three.
4.NF.A.1Look For A PatternUse the identity = 1 - : the leftover piece has a fixed numerator of 14, so only its denominator n+14 changes.
Splitting a fraction into "1 minus a leftover piece" turns an ordering problem into a much simpler ordering of the leftover pieces.
5.NF.B.3Look For A PatternAs n grows, the denominator n+14 grows, so shrinks; subtracting a smaller piece from 1 leaves a larger result.
Same numerator, bigger denominator means smaller fraction — a Grade 4 fraction-sense fact, just used at scale.
4.NF.A.2Look For A PatternA smaller leftover means a larger value, so larger n gives a larger : the fractions rise as , , .
Ordering a list of numbers by ordering a single varying quantity is Grade 6 rational-number reasoning.
6.NS.C.7Look For A PatternSanity-check with Tool #9: = , and cross-multiplying gives ＜ and ＞ — matching the pattern.
Reducing the middle fraction to the easier is Tool #9 — solve the easier related comparison first, then confirm the harder claim.
4.NF.A.2Solve An Easier Related ProblemLook for hidden structure first: once you see all three fractions are , ordering them is a one-line Grade 6 observation — no big arithmetic needed.