A group of students were asked whether they liked History, Science,or Geography. Their responses are as shown in the table below .
All three subject=7
History and Geography=11
Geography and Science=9
History and science=10
History=34
Geography only=18
Science only =16
None of the 3 subjects=3
To solve this problem, we can use a Venn diagram to visualize the information given.
Let's assume that the three circles in the Venn diagram represent History, Science, and Geography.
From the given information:
- All three subjects: 7 students
We can place this number in the overlapping region of the three circles.
- History and Geography: 11 students
We can place this number in the overlapping region between the History and Geography circles.
- Geography and Science: 9 students
We can place this number in the overlapping region between the Geography and Science circles.
- History and Science: 10 students
We can place this number in the overlapping region between the History and Science circles.
- History: 34 students
We can subtract the numbers we already have from this total to find the number of students who only study History:
34 - 7 (all three) - 11 (History and Geography) - 10 (History and Science) = 6
- Geography only: 18 students
We can subtract the numbers we already have from this total to find the number of students who only study Geography:
18 - 7 (all three) - 11 (History and Geography) - 9 (Geography and Science) = -9
It seems like there's an inconsistency here, as we can't have a negative number of students studying only Geography. There might be an error in the given information.
- Science only: 16 students
We can subtract the numbers we already have from this total to find the number of students who only study Science:
16 - 7 (all three) - 10 (History and Science) - 9 (Geography and Science) = -10
Again, there's an inconsistency here with a negative number of students studying only Science. We should double-check the given information to make sure there are no errors.
- None of the three subjects: 3 students
We can place this number outside of all the circles, representing students who do not study any of the three subjects.
From the given information, we can conclude:
- The total number of students who like History is the sum of those who study History only and those who have History as one of their choices: 6 (History only) + 11 (History and Geography) + 10 (History and Science) + 7 (all three) = 34 (given total for History)
- The total number of students who like Geography is the sum of those who study Geography only and those who have Geography as one of their choices: 9 (Geography only) + 11 (History and Geography) + 18 (Geography only) + 7 (all three) = 45
- The total number of students who like Science is the sum of those who study Science only and those who have Science as one of their choices: 10 (History and Science) + 9 (Geography and Science) + 16 (Science only) + 7 (all three) = 42
However, due to the inconsistencies in the information provided, it is difficult to draw any further conclusions or determine the number of students who like each subject individually.
To find the number of students who liked each subject, we can use the principle of inclusion-exclusion.
1. Start with the total number of students who liked at least one subject:
Total = All three subjects + History and Geography + Geography and Science + History and Science + History only + Geography only + Science only + None of the 3 subjects
Total = 7 + 11 + 9 + 10 + 34 + 18 + 16 + 3
2. Calculate the number of students who liked each subject:
Number of students who liked History = All three subjects + History and Geography + History and Science + History only
Number of students who liked History = 7 + 11 + 10 + 34
Number of students who liked Science = All three subjects + Geography and Science + History and Science + Science only
Number of students who liked Science = 7 + 9 + 10 + 16
Number of students who liked Geography = All three subjects + History and Geography + Geography and Science + Geography only
Number of students who liked Geography = 7 + 11 + 9 + 18
Therefore,
Number of students who liked History = 62
Number of students who liked Science = 42
Number of students who liked Geography = 45