2021 AMC 12A Spring Problem 6

Below is the professionally curated solution for Problem 6 of the 2021 AMC 12A Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Spring solutions, or check the answer key.

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Concepts:basic probability system of equations

Difficulty rating: 1270

A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\dfrac13.$ When $4$ black cards are added to the deck, the probability of choosing red becomes $\dfrac14.$ How many cards were in the deck originally?

$6$

$9$

$12$

$15$

$18$

Solution:

Let $r$ be the number of red cards and $t$ the total. From $\dfrac{r}{t} = \dfrac13$ we get $t = 3r.$ After adding $4$ black cards, $\dfrac{r}{t+4} = \dfrac14,$ so $t + 4 = 4r.$

Substituting $t = 3r$ gives $3r + 4 = 4r,$ so $r = 4$ and $t = 12.$

Thus, the correct answer is C.

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