Wild Pawn Races

One of my students had a game with what Silman call "Wild Pawn Races". My student and his opponent both are rated about 1400 on ICC (Internet Chess Club) but "Silman's Complete Endgame Course" covers the winning idea in the Class A 1800-1990 section. Below are two positions. The first one from the book and the second one from my student's game. See if you can find the winning idea.