AMC 8 2007 Problem 20
AMC8 2007 年第 20 题
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Before the 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?
在地区赛之前，独角兽队已经赢得了他们所打过的篮球比赛的 45％。在地区赛期间，他们又赢了六场比赛，输了两场，最终整个赛季赢得了一半的比赛。则独角兽队总共打了多少场比赛？
- A.48
- B.50
- C.52
- D.54
- E.60
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
At the beginning of the problem, the Unicorns had played games and they had won of these games. From the information given, we can say that Next, the Unicorns win 6 more games and lose 2 more, for a total of 6+2=8 games played during district play. We are told that they end the season having won half of their games, or 0.5 of their games. We can write another equation: This gives us a system of equations: and We first multiply both sides of the first equation by to get x=0.45y. Then, we multiply both sides of the second equation by (y+8) to get x+6=0.5(y+8). Applying the Distributive Property yields x+6=0.5y+4. Now we substitute $0.45y$ for to get 0.45y+6=0.5y+4. Solving gives us y=40. Since the problem asks for the total number of games, we add on the last 8 games to get the solution 48. As a check, before district play the Unicorns would have won 18 games out of 40, and . Thus, the answer is 48.
At the start of the problem we figure out the Unicorns have won 45% of their games, which can be represented by . Since this is a simplified fraction and not necessarily the amount of games they have played, we can add a variable (in this case I am using ) to the numerator and denominator to get . We then know that during the district play they have won 6 more games and lost 2 of them for a total of 6+2=8 games played. We then can add it to our fraction we have built to make . We also know that the fraction is equal to . Putting these equations together we get . Now we can cross multiply to come out with 18x+12=20x+8. We now solve for ! . After we get which comes out to be 2 we then plug it back into the denominator to get 20(2)+8 which is equal to 48, so the answer must be 48.
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