2021 AMC 12A Spring Problem 6

Question

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

Let

r

be the number of red cards; the original deck has

3r

cards

Adding

4

black cards keeps

r

red out of

3r + 4,

and this equals

\dfrac14

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.

Problem 6 in other years