2018 AMC 10A Problems/Problem 6
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of , and the score increases by for each like vote and decreases by for each dislike vote. At one point Sangho saw that his video had a score of , and that of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
If of the votes were likes, then of the votes were dislikes. , so votes is of the total number of votes. Doing quick arithmetic shows that the answer is .
Let's consider that Sangho has received 100 votes. This means he has received 65 upvotes and 35 downvotes. Part of these upvotes and downvotes cancel out, so Sangho is now left with a total of 30 upvotes, or a score increase of 30. In order for his score to be 90, Sangho must receive three sets of 100 votes. Therefore, the answer is .
Let be the amount of votes cast, of would be dislikes and of would be likes. Since a like earns the video 1 point and a dislike takes 1 point away from the video, the amount of points Sangho's video will have in terms of is of , or of . Because Songho's video had a score of 90 points, . After we solve for , we get .
Let x be the amount of like votes and y be the amount of dislike votes. Then . Since of the total votes are like votes, . then or y = . Plugging that in to the original equation, = and Then and the total number of votes is . The answer is .
Obviously, . The answer is .
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