Problem:
A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is . When 4 black cards are added to the deck, the probability of choosing red becomes . How many cards were in the deck originally.
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let and be the numbers of red cards and black cards, respectively. Before adding the additional cards, the probability of choosing a red card is . This implies , so . After 4 black cards are added, the probability of choosing a red card is
Therefore , so . Then , and the deck has cards in all.
OR
Let be the number of cards and be the number of red cards in the deck originally. Then the ratio of the original probability of choosing a red card to the probability of choosing a red card after the 4 black cards have been added is
This simplifies to
Therefore and .
The problems on this page are the property of the MAA's American Mathematics Competitions