Final answer:
Piano rags made their way into respectable dancing establishments primarily through sheet music, which was the main distribution method before radio and phonograph records became prevalent.
Step-by-step explanation:
The medium that piano rags used to get into respectable dancing establishments was sheet music. During the time when piano rags were popular, before the widespread use of radio broadcasts and phonograph records became common, sheet music was the primary method by which musical compositions were distributed. People purchased sheet music to play piano rags and other styles at home and in public venues, such as dance halls, where bands or pianists would perform them live for dancing. It was also during this era that certain radio programs, such as Amos 'n' Andy, which began in the late 1920s, began to homogenize American culture by reaching a national audience, but this was after the peak of the ragtime period.