Problem:
A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only of the marbles in the bag are blue. Then yellow marbles are added to the bag until only of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue?
Answer Choices:
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Solution:
If there are initially blue marbles in the bag, after red marbles are added, then the total number of marbles in the bag must be . Then after the yellow marbles are added, the number of marbles in the bag must be . Finally, adding blue marbles to the bag gives blue marbles out of total marbles. Thus of the marbles are blue.
Just before the number of blue marbles is doubled, the ratio of blue marbles to non-blue marbles is to . Doubling the number of blue marbles makes the ratio to , so of the marbles are blue.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions