The well-known calculus of fractions of Gabriel and Zisman provides a convenient way to formally invert morphisms in a category. This was extended to bicategories by Pronk. On the other hand, the second author has developed a calculus of lax fractions for order-enriched categories that formally turns a given class of morphisms into left adjoint right inverses. We extend these constructions by presenting a calculus of lax fractions for 2-categories that formally turns a class of morphisms and pseudo-commutative squares into left adjoint right inverses and Beck-Chevalley squares.