A hundred students of Midtown college were asked to pick their preference for one of three genres of movies: romantic, drama, and tragedy. Each of them picked one of the three genres. More than half picked romantic, more than a quarter picked drama and at least one picked tragedy. When asked about the genre of the movie Titanic, however, more than three quarter viewed it as a romantic movie.
If all of the statements above are true, which one of the following must also be true?
Select an option, then click Submit answer.
Reference / correct answer:
Some students whose preference was romantic genres of movies viewed Titanic as a romantic movie.
Argument construction
The survey covers 100 students in Midtown college. They were asked to pick their preference for one of the three genres of movies: romantic, drama, and tragedy. More than 50% picked romantic, more than 25% picked drama and at least 1 student picked tragedy. When asked about the movie Titanic, however, more than 75% viewed it as a romantic movie.
Predict an Inference
Presenting the information mathematically will help.
Say R represents Romantic genre; D represents Drama genre; and T represents Tragedy genre.
Let's first find out the minimum values of R, D and T.
It is given that there are 100 students, thus, the percent figures represent the actual numbers.
We know that
More than 50% picked romantic, thus R ≥ 51;
More than 25% picked drama, thus D ≥ 26;
At least 1 student picked tragedy, thus T ≥1
Let's now find out the maximum values of R, D and T. Remember that a student could choose only one of the three genres, and every student did choose one of these three genres. So, the sum of R, D and T would be 100.
We know that
Maximum value of R would happen when the values of D and T are minimum. So, it would be 100 − 26 − 1 = 73
Likewise, Maximum value of D would be 100 −51 − 1 = 48
Maximum value of T would be 100 − 51 − 26 = 23
51 ≤ R ≤ 73
26 ≤ D ≤ 48 1 ≤ T ≤ 23
We have to see in how many ways the number of votes for Titanic viewed as a romantic movie can be more than 75%.
Since there can be many inferences, we need to see the option statements to get the correct answer.
Answer choices explanation:
[Only the students whose preferences were romantic and drama genres of movies viewed Titanic as a romantic movie.] This is incorrect. This is not necessarily true. Even some students from each genre can make the figure more than 75%. For example, 60 students who voted R, 30 students who voted D and 5 students who voted T could have viewed Titanic as a romantic movie. Further, since 'more than 75%' can stretch up to 100%, even all the 100 students covered in the survey can view Titanic as a romantic movie.
[Only the students whose preferences were romantic and tragedy genres of movies viewed Titanic as a romantic movie.] This is incorrect. For the same reasoning, as with option [Only the students whose preferences were romantic and drama genres …] this option is incorrect.
[At least one student from each genre of movies viewed Titanic as a romantic movie.] This is incorrect. This is not necessarily true. If the maximum number of students (73) whose preference is romantic movies and the minimum number of students (26) whose preference is drama movies view Titanic as a romantic movie, the total number of students equals 73 + 26 = 96 number of students > 75%. So, even if none of the students whose preference is tragedy movies viewed Titanic as a romantic movie, the given survey fact - more than 75% of students view Titanic as a romantic movie – can still hold.
[Some students whose preference was romantic genres of movies viewed Titanic as a romantic movie.] This is correct. The minimum number of students whose preference is Romantic movies is 51. This means that the maximum number of students whose preference is drama or tragedy movies is 100 −51= 49. This is less than 75% of all surveyed students. Since the number of students who view Titanic as a romantic movie is more than 75%, some students whose preference is romantic movies must view Titanic as a romantic movie.
[Some students whose preference was tragedy genres of movies viewed Titanic as a romantic movie] This is incorrect. This is not necessarily true. With the same reasoning as used in option C, this is incorrect.