Maths Olympiad Prep

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Algebra Difficulty 3.4 AMC 10/12 Find the answer

Jamal has a drawer containing 66 green socks, 1818 purple socks, and 1212 orange socks. After adding more purple socks, Jamal noticed that there is now a 60%60\% chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?

Pick one

Solutions — 2

Solution 1

After Jamal adds xx purple socks, he has (18+x)(18+x) purple socks and 6+18+12+x=(36+x)6+18+12+x=(36+x) total socks. This means the probability of drawing a purple sock is 18+x36+x\frac{18+x}{36+x}, so we obtain 18+x36+x=35\frac{18+x}{36+x}=\frac{3}{5} Since 18+936+9=2745=35\frac{18+9}{36+9}=\frac{27}{45}=\frac{3}{5}, the answer is (B) 9\boxed{\textbf{(B) }9}.

Solution 2

1. Jamal initially has 6 green socks, 18 purple socks, and 12 orange socks. Therefore, the total number of socks initially is:
6+18+12=36 6 + 18 + 12 = 36

2. Let x x be the number of purple socks Jamal added. After adding x x purple socks, the total number of socks becomes:
36+x 36 + x

3. The number of purple socks after adding x x purple socks is:
18+x 18 + x

4. According to the problem, the probability that a randomly selected sock is purple is now 60%. Therefore, we can set up the following equation:
18+x36+x=0.60 \frac{18 + x}{36 + x} = 0.60

5. Converting 60% to a fraction, we get:
18+x36+x=35 \frac{18 + x}{36 + x} = \frac{3}{5}

6. Cross-multiplying to solve for x x :
5(18+x)=3(36+x) 5(18 + x) = 3(36 + x)

7. Expanding both sides:
90+5x=108+3x 90 + 5x = 108 + 3x

8. Subtracting 3x 3x from both sides:
90+2x=108 90 + 2x = 108

9. Subtracting 90 from both sides:
2x=18 2x = 18

10. Dividing both sides by 2:
x=9 x = 9

Therefore, Jamal added 9 \boxed{9} purple socks.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.