AMC 10 · 2015 · #4
Grade 4 arithmeticPick an answer.
AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Comparing 1/5, 1/3, 1/4 in their raw fraction form is awkward. Tool #4 (Introduce a Variable) fixes that by letting the pizza be a concrete number of equal slices — pick 60, a common multiple of 5, 3, and 4, so every share becomes a whole number of slices and the sizes are easy to read off. Tool #16 (Count the Complement) handles Dan: instead of computing his share directly, take the whole pizza and subtract the three known shares. Tool #3 (Eliminate Possibilities) finishes the job — once the four slice counts are known, only one of the five orderings matches, so the rest are crossed off.
Cut the pizza into 60 slices
Denominators 5, 3, 4 all divide 60, so picture the pizza as 60 equal slices.
A number that all three denominators divide evenly makes each share land on a whole slice count instead of a messy fraction.
4.OA.B.4Use Matrix LogicCount each known share
Alex eats 1/5 of 60 = 12, Beth 1/3 = 20, Cyril 1/4 = 15 slices.
Finding a fraction of a whole pizza is just multiplying that fraction by the slice count.
4.NF.B.4Use Matrix LogicDan gets the leftovers
Dan eats the rest: 60 - (12 + 20 + 15) = 60 - 47 = 13 slices.
The leftovers are easiest to find by taking the whole and removing the parts you already know.
4.NBT.B.4Count The ComplementRank from most to least
Ranking 20 ＞ 15 ＞ 13 ＞ 12 gives Beth, Cyril, Dan, Alex — choice (C).
Whole-slice counts line up in plain size order, so the ranking reads straight off the numbers.
4.NBT.A.2Eliminate PossibilitiesTurn the fractions into a pizza of 60 slices and the whole AMC 10 problem becomes Grade 4 counting and comparing.
- Cut the pizza into 60 slices
- Count each known share
- Dan gets the leftovers
- Rank from most to least