Problem:
Jamal has a drawer containing green socks, purple socks, and orange socks. After adding more purple socks, Jamal noticed that there is now a chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
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Solution:
A chance of choosing a purple sock implies a chance of choosing a green or orange sock. It follows that the non-purple socks represent of all the socks. This means that socks represent of the total, and socks represent of the total. Jamal therefore added purple socks to the drawer.
Let equal the number of additional purple socks, increasing the number of purple socks to and the total number of socks to Because the proportion of purple socks must equal , the following equation must be satisfied:
Solving for yields
Jamal added purple socks to the drawer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions