AMC 8 2020 Problem 13

AMC8 2020 年第 13 题

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Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that there is now a 60% chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?

Jamal 有个抽屉，里面有 6 只绿色袜子，18 只紫色袜子，12 只橙色袜子。当加了更多紫色袜子后，Jamal 发现，从抽屉里随机抽取一只袜子，有 60％的概率是紫色的。那么 Jamal 加了多少只紫袜子？

A.6
B.9
C.12
D.18
E.24

正确答案 · Correct Answer

解析 · Solution

Solution 1

After Jamal adds

x

purple socks, he has (18+x) purple socks and 6+18+12+x=(36+x) total socks. This means the probability of drawing a purple sock is

\displaystyle \frac{18+x}{36+x}

, so we obtain

\displaystyle \frac{18+x}{36+x}=\frac{3}{5}

Since

\displaystyle \frac{18+9}{36+9}

\displaystyle \frac{27}{45}

\displaystyle \frac{3}{5}

, the answer is 9. As in, we have the equation

\displaystyle \frac{18+x}{36+x}

\displaystyle \frac{3}{5}

. Cross-multiplying yields

90+5x=108+3x \Rightarrow 2x=18 \Rightarrow x=9

. Thus, Jamal added 9 purple socks. very fast The answer is 9 The probability that a randomly selected sock is purple is: The # of purple socks ÷ The total # of socks. We can test the answer choices until we find the correct answer. [A] If Jamal adds 6 purple socks, he will have 18 + 6 = 24 purple socks. The probability of a randomly selected sock being purple after adding 6 socks is

\displaystyle \frac{24}{6+24+12}

\displaystyle \frac{24}{42}

\displaystyle \frac{4}{7}

, which isn't equal to

\displaystyle \frac{3}{5}

. [B] If Jamal adds 9 purple socks, he will have 18 + 9 = 27 purple socks. The probability of a randomly selected sock being purple after adding 9 socks is

\displaystyle \frac{27}{6+27+12}

\displaystyle \frac{27}{45}

\displaystyle \frac{3}{5}

, which is indeed equal to

\displaystyle \frac{3}{5}

. So our final answer is 9 purple socks.

~ TealBear16 🐰 ~ FunnyLizard15 🐶 (Assistance)

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