Use logical reasoning (Venn Diagrams) to solve each problem.
1. The 52 children in the village learned some dances. 31 learned the Star Dance and 27 learned the Handkerchief Dance. How many children learned both dances?
2. A total of 43 people bought pies at the dance festival. 27 bought plum pies and 25 bought mincemeat pies. How many people bought both types of pies?