Which of the following is a type of grant that is appropriated by Congress to states for a specific purpose and allocated by a precise formula with detailed conditions imposed by the national government?

The correct answer is categorical grants, which are specifically appropriated by Congress to states for particular purposes. These grants come with strict conditions and requirements that the states must meet in order to receive and use the funds effectively. Categorical grants are often aimed at addressing specific issues such as education, health care, or transportation, and the national government outlines detailed criteria that states must adhere to, which can include how the funds are spent, reporting requirements, and compliance with federal regulations.

In contrast, other options do not fit this definition. Free grants do not represent a formal designated category in federal funding. Dual federalism refers to a system where state and national governments operate independently within their own spheres, without intertwining responsibilities and funding mechanisms like categorical grants. Block grants, while also a type of federal funding, are allocated to states with fewer restrictions, allowing more flexibility regarding how the funds can be used, which differentiates them from the highly regulated nature of categorical grants.