Competition · AMC preparation · step 4 of 4
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: n/(n+14). 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: 7/21 simplifies to the familiar landmark 1/3, which we can use as a sanity check against the other two.
Spot the shared structure
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 PatternRewrite each fraction as 1 minus a piece
Use 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 PatternOrder the leftover pieces
As 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.
Writing each fraction as 1 - 14/(n+14), the leftover pieces run 14/19 ＞ 14/21 ＞ 14/23, and taking a bigger leftover away from the same 1 leaves a smaller result, so 1 - 14/19 ＜ 1 - 14/21 ＜ 1 - 14/23.
▸ Why?
A fraction and its leftover piece are the two parts that make up the one whole 1 — n/(n+14) + 14/(n+14) = 1, because the bottom n+14 is just the top n plus an extra 14 with nothing missing and nothing counted twice — so each fraction is exactly 1 minus its leftover, and out of the same whole a bigger piece taken away must leave a smaller remainder.
▸ Why?
The leftover pieces all share the same top 14 but sit over the growing bottoms 19, 21, 23, and a fixed amount split into more equal parts gives smaller parts, so 14/19 ＞ 14/21 ＞ 14/23.
▸ Why?
14/d is 14 copies of one part 1/d, and those d equal parts are exactly what the single whole 1 splits into, so a larger d shares that one whole among more parts and makes every part smaller.
Translate the order back
A 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 PatternCheck against one third
Sanity-check with Tool #9: = , and cross-multiplying gives ＜ and ＞ — matching the pattern.
Reducing the middle fraction to the easier 1/3 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.
- Spot the shared structure
- Rewrite each fraction as 1 minus a piece
- Order the leftover pieces
- Translate the order back
- Check against one third
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