2003 AMC 12A Problems/Problem 10
Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of , respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?
Because the ratios are , Al, Bert, and Carl believe that they need to take , , and of the pile when they each arrive, respectively. After each person comes, , , and of the pile's size (just before each came) remains. The pile starts at , and at the end of the original pile goes unclaimed. (Note that because of the properties of multiplication, it does not matter what order the three come in.) Hence the answer is .