Some of the most cost-effective ideas for cheap soundproofing include using a nonhardening caulk, filling gaps around windows and installing a floor sweeper at the base of a door with a gap. Some inexpensive soundproofing methods can present a fire risk, such as using paper egg cartons or newspaper insulation.
When soundproofing, the first priority is to eliminate gaps in the air in which sound can pass through. Once those gaps have been filled or blocked, the next step should be to create an insulating layer of nonflammable material. Additionally, flat surfaces carry sound more easily than surfaces with texture.