## CD To Tape

#### Problem

Matilda wanted to transfer her favourite CD to tape.

The CD has six songs and the length of the tracks are 7:55, 9:40, 9:15, 12:45, 8:20 and 11:30; a total playing time of 59:25. After changing the order of the songs, Matilda was able to fit all the songs, without any breaks, on a sixty minute tape. How did she arrange the songs?

#### Solution

For the songs to fit on the tape, each side must be almost completely filled, as the total playing time will leave 35 seconds unused on a sixty minute tape. If the two longest songs were placed on the same side, 12:45 and 11:30 take up 24:15 together, leaving only 5:45, which is not enough for any other single song. So they must go on opposite sides.

By trial we get:`Side 1` 12:45, 7:55 and 9:15 (29:55 total)`Side 2` 11:30, 9:40 and 8:20 (29:30 total)

If the total playing time is less than the length of the tape, is a solution always possible?