Consider a connected graph G and let e = uv € E(G) such that G' = G-e is disconnected. Let G₁ and G₂ be two induced subgraphs of G such that V(G₁) = {x EG: G' contains an cu-path} and V(G₂) = {x EG: G' contains an xv-path}. (1) Show that G₁ and G₂ are both connected. (2) Show that V(G) = V(G₁) UV(G₂). (3) Deduce that G' contains only two connected components.

The sum of money of Bill and Zan is 4,230. If 2/3 of Bill's money is equal to 5/6 of Zan's money, how much money does Bill have?

Jenny, Scott, and Alan have a total of $78 dollars in their wallets. Alan has $6 less than Jenny. Scott has 2 times what Alan has. How much do they have in their wallets?