2002 AMC 12A Problem 9

Question

Jamal wants to store

30

computer files on floppy disks, each of which has a capacity of

1.44

megabytes (mb). Three of his files require

0.8

mb of memory each,

12

more require

0.7

mb each, and the remaining

15

require

0.4

mb each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?

12

13

14

15

16

The total memory needed is

3(0.8)+12(0.7)+15(0.4)

mb

A disk holding a

0.8

file has room for only one

0.4

file, wasting

0.24

mb; add up this waste

The files need

3(0.8)+12(0.7)

+15(0.4) = 16.8

mb, so at least

\dfrac{16.8}{1.44} = 11\tfrac{2}{3}

disks by volume alone.

A disk containing a

0.8

-mb file has room for only one more

0.4

-mb file, leaving at least

0.24

mb unused. Across the three

0.8

-mb files this wastes at least

3(0.24) = 0.72

mb, over half a disk, forcing at least

13

disks.

Thirteen suffice: six disks each hold two

0.7

-mb files, three disks each hold one

0.8

-mb file plus one

0.4

-mb file, and four disks each hold three

0.4

-mb files.

Thus, the correct answer is B.

Problem 9 in other years