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?

$12$

$13$

$14$

$15$

$16$

Answer: B

Concepts:optimization bounding to limit cases

Difficulty rating: 1570

Solution:

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.

2002 AMC 12A Problem 9

Answer: B

Difficulty rating: 1570

Solution:

Problem 9 in Other Years