Five tablespoons and 1 teaspoon are the equivalent of 1/3 cup. There is no way to make a round number of tablespoons fit into 1/3 cup. A tablespoon is equal to 3 teaspoons, however, so 5 1/3 tablespoons are also equivalent to 1/3 cup.
It takes exactly 16 tablespoons to fill a cup in standard measurement. Splitting the measurement into 1/3 or 1/6 cup results in the need to use teaspoon measurements or fractions of tablespoons in conjunction with tablespoon measurements. These measurements are meant to be used when measuring liquids or powders for recipes. Tablespoon measurement is common in the United States but is not a commonly used form of measurement in other countries.