AMC 10 · 2002 · #9
Grade 5 arithmeticPick an answer.
We want the smallest disk count, so tool #14 (Extreme Principle) says push each disk to hold as much as it can and check the boundary of 1.44 MB: which files can share a disk and which cannot. The files come in three sizes, so tool #7 (Identify Subproblems) splits the job into three groups. Start with the biggest files, because they are the hardest to pair up, and see what each big file is forced to do. Then fit the medium files, then the small ones, filling any leftover room. Counting the disks each group forces gives both a working plan and a proof that fewer disks is impossible.
Count the files of each size
Subtract the named groups from 30: 30-3-12 = 15 files of 0.4 MB.
"The rest" just means everything not already counted, so subtract the named groups from the total.
4.NBT.B.4Identify SubproblemsEach big file needs its own disk
Two big files or a big plus a medium both exceed 1.44, so each big file takes its own disk plus one small: 3 disks.
The biggest files are the pickiest, so pin them down first and give each the only partner that fits.
Each of the largest files must have a disk to itself, because no second file of any size will fit beside it.
▸ Why?
If two big files shared a disk they would together exceed its capacity, so each one needs a place of its own.
▸ Why?
Every other way of pairing a big file is ruled out by the capacity, so the packing has no choice left to make here.
Pair up the medium files
Mediums fit only two per disk (0.7+0.7 = 1.4), so twelve of them need 6 disks.
Two mediums fill a disk almost exactly, and a third never fits, so pairing is the tightest they can pack.
5.NBT.A.3Identify SubproblemsGroup the leftover small files
Twelve small files remain and pack three to a disk (1.2 MB), needing 4 disks.
Fit as many small files as stay under the limit, which is three, then count how many groups of three you need.
5.NBT.B.7Identify SubproblemsAdd up the disks
Every group was forced, so the total is 3+6+4 = 13, choice (B).
Once each group is packed as tightly as it can be, adding the group totals gives the true minimum.
4.NBT.B.4Extreme PrinciplePack the biggest, pickiest files first and fill each disk as tightly as the limit allows — then adding up the forced groups gives the fewest disks.
- Count the files of each size
- Each big file needs its own disk
- Pair up the medium files
- Group the leftover small files
- Add up the disks
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