Question

一副牌只有红牌和黑牌。随机抽到一张红牌的概率是 $\dfrac13$ 。当向这副牌中加入 $4$ 张黑牌后，抽到红牌的概率变为 $\dfrac14$ 。这副牌原来有多少张牌？

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?

$6$

$9$

$12$

$15$

$18$

答案：C

知识点：基本概率方程组

难度评级：1270

解答：

设红牌数为 $r$ ，总牌数为 $t$ 。由 $\dfrac{r}{t} = \dfrac13$ 得 $t = 3r$ 。加入 $4$ 张黑牌后， $\dfrac{r}{t+4} = \dfrac14$ ，所以 $t + 4 = 4r$ 。

代入 $t = 3r$ 得 $3r + 4 = 4r$ ，因此 $r = 4$ ， $t = 12$ 。

因此，正确答案是 C。

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.

2021 AMC 12A Spring 第 6 题

答案：C

难度评级：1270

解答：

其他年份的第 6 题