Problem 20 · 2007 AMC 8
Medium
Algebra & Patterns
percent-equation
Before district play, the Unicorns had won 45% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?
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Answer: A — 48 games.
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Hint 1 of 2
The unknown isn't the answer they want — let x be the games before district play, write wins both before and after, and set them equal to 'half the season.'
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Hint 2 of 2
Name the natural variable, then translate each sentence into an equation: 'won 45%' → 0.45x wins; 'won half' → wins = total/2.
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Approach: let x = pre-district games, equate the two win counts
- Before district: x games, 0.45x wins. District adds 6 wins and 2 losses, so the season is x+8 games with 0.45x+6 wins.
- 'Won half their games': 0.45x + 6 = (x+8)/2.
- Clear decimals/fractions (×10): 4.5x + 60 = 5x + 40 ⇒ 0.5x = 20 ⇒ x = 40.
- Total = 40 + 8 = 48.
- Why 45% is the clue to x: 0.45x must be a whole number of wins, so x is a multiple of 20. That alone narrows pre-district games to 20, 40, 60… — a fast reality check on the algebra.
Another way — test the divisibility-friendly value:
- For 45% = 9/20 of x to be a whole number, x must be a multiple of 20. Try x = 40: wins = 18.
- After district: 18 + 6 = 24 wins out of 40 + 8 = 48 games — exactly half. So the total is 48.
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