🇺🇸 AMC 8 ⇄ switch contest
2020 AMC 8

Problem 13

Problem 13 · 2020 AMC 8 Medium
Fractions, Decimals & Percents percent-multipliersubstitution

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?

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Answer: B — 9 purple socks added.
Show hints
Hint 1 of 2
Adding purple socks changes the purple count and the total — two moving numbers. Look for what doesn't move: the green and orange socks. Anchor everything to them.
Still stuck? Show hint 2 →
Hint 2 of 2
The 6 + 12 = 18 non-purple socks never change. If purple is now 60%, then those 18 are exactly the other 40%. That one fact gives the new total directly.
Show solution
Approach: anchor to the part that never changes (non-purple = 40%)
  1. Only purple socks are added, so the 6 green + 12 orange = 18 non-purple socks stay fixed. After the change, purple is 60%, so the unchanged 18 must be the remaining 40%.
  2. New total = 18 ÷ 0.4 = 45 socks. He started with 36, so he added 45 − 36 = 9 purple socks.
  3. Why this transfers: when one group grows and you're given the other group's percentage, pin your work to the unchanging group. “18 socks = 40%” pins the whole down in one step — far lighter than introducing a variable and cross-multiplying.
Another way — set up the new probability and solve (MAA):
  1. If s purple are added, (18 + s) / (36 + s) = 0.6.
  2. Cross-multiply: 18 + s = 21.6 + 0.6s, so 0.4s = 3.6 and s = 9.
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