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Mix ¾ cup baking soda, ¼ cup hydrogen peroxide, and 1 tablespoon dish soap. This highly effective paste cleans the grout in three ways:  The baking soda works as a natural abrasive to scrub the grout. The hydrogen peroxide chemically reacts with the baking soda releasing bleaching oxygen ions. The dish soap helps loosen dirt and remove grease. NOTE: The bleaching action of the chemical reaction may affect colored grout. Test a hidden corner before cleaning the entire floor. A toothbrush or a nylon scrub brush work well. Be sure to apply the paste to the grout between all tiles and around the edges for a uniform clean look. You may see a bubbling reaction as the baking soda and hydrogen peroxide interact. Give the paste time to work its way into the grout to fully remove any stains. Pour a small amount of water directly onto the tile floor to rinse the solution off the grout. Use caution as wet tiled floors can be very slippery. Clean the remaining paste out of the grout by gently scrubbing the floor with a towel. This can be done by standing on a towel and shuffling the towel across the floor with both feet or by crawling and scrubbing as you go. Ensure that no soap or residue is left behind by thoroughly mopping the floor with a cotton or sponge mop. Rinse the mop and change the water often to get achieve a clean finish.

Summary:
Make a paste. Apply the paste with a nylon bristled brush. Let the mixture sit on the grout for 15 minutes. Rinse the grout with warm or hot water to remove all of the solution. Use a rag to wipe the grout to remove any leftover residue or dirt. Mop the floor with clean water.