2007 AMC 8 Problems/Problem 20
Contents
Problem
Before the district play, the Unicorns had won % 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?
Solution 1
At the beginning of the problem, the Unicorns had played games and they had won of these games. From the information given in the problem, we can say that Next, the Unicorns win 6 more games and lose 2 more, for a total of games played during district play. We are told that they end the season having won half of their games, or 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 Then, we multiply both sides of the second equation by to get Applying the Distributive Property gives yields Now we substitute for to get Solving gives us Since the problem asks for the total number of games, we add on the last 8 games to get the solution .
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Solution 2
Simplifying 45% to , we see that the numbers of games before district play are a multiple of 20. After that the Aces played 8 more games to the total number of games is in the form of 20x+8 where x is any positive integer. The only answer choice is , which is 20(2)+8.
Solution 3
First we simplify % to . Ratio of won to total is , but ratio of total number won to total number played is for some . After they won 6 more games and lost 2 more games the number of games they won is , and the total number of games is . Turning it into a fraction we get , so solving for we get Plugging in 2 for we get .
Solution 4
Because 45% can be simplified to , and we know that we cannot play a fraction amount of games, we know that the amount of games before district play is divisible by 20. After district play, there was games, so in total there must be . The only answer in this format is .
Solution 5
Let be the number of pre-district games. Therefore, we can write the percentage of total games won as a weighted average, namely . Solving this equation for gives , but since the problem asked for all games, the answer is
Solution 6
Let the number of games they won be and the number of games they lost be . We are told that , which can be manipulated to which simplifies down to . Then, after district games, we are told , which can be changed into which simplifies down to . Then we can solve for using substitution: Now that we know , we can figure out that . . Now we need to add on the district games: .
Solution 7
Let be the total number of games before the district play. The Unicorns have wins, therefore the rest are losses. But after the district play, they won 6 and lost 2 more games. We can solve for x by forming the equation . Subtracting and from both sides gives us , and from here we multiply both sides by 10 to get . We are not finished yet as the problem is asking for the total games (which includes the games after the district play), so we add 8 to our value of x to get our answer which is .
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Solution 8 (Answer Choices)
We can check each answer choice from left to right to see which one is correct. Suppose the Unicorns played games in total. Then, after district play, they would have won games. Now, consider the situation before district play. The Unicorns would have won games out of . Converting to a percentage, %. Thus, the answer is .
Note: If A didn't work, we would have similarly tested the other choices until we found one that did.
Video Solution by OmegaLearn
https://youtu.be/rQUwNC0gqdg?t=1993
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See Also
2007 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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