The cheapest and best way to get rid of mold in the bathtub area is to mix bleach and water in a spray bottle. Spray and walk away! Turn on the fan in the bathroom or open a window for an hour or so. You can rinse if you like, but you don't have to.
I've used a solution of bleach and water in the bathroom for years. It works! But be sure to dilute the bleach as much as possible. The bleach actually dries out the grout in tiles. You don't want to regrout those tiles!
I just saw a program on the History channel about removing mold which indicated that bleach (sodium hypochlorite) stays on the surface and does not kill the mold. According to: http://www.bleach-mold-myth.com/
Bleach is too diluted and thus too weak to permanently kill mold unless the mold is simply sitting on top of a hard surface like a counter top or sink.
What little killing power chlorine bleach does have is diminished significantly as the bleach sits in warehouses and on grocery store shelves or inside your home or business [50% loss in killing power in just the first 90 days inside a never opened jug or container] Chlorine ions constantly escapes through the plastic walls of its containers.
Chlorine bleach's ion structure also prevents chlorine from penetrating into porous materials such as dry wall and wood--- It just stays on the outside surface, whereas mold has protected enzyme roots growing inside the porous construction materials. When you spray porous surface molds with bleach, ONLY the water in the water solution soaks into the wood while the bleach chemical sits atop the surface, gasses off, and thus only partially kills the surface layer of mold while the water penetration of the building materials fosters further mold growth.