in a class of 100 students, 35 offers history, 43 offers Geography and 50 offers economic, 14 students offers history and geography, 13 offers geography and economic and 11 offers history and economic. the number of students that offers none of the subjects is four times the number of those that offers three subjects.
(i) how many students offers three subjects
(ii) how many students offers none
(iii) how many student offer only one of the subjects
(iv) how many students offers history and Geography
To approach this problem, we can use a Venn diagram to visualize the information given:
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From the diagram, we can see that:
(i) The number of students who offer three subjects is the intersection of all three sets, which is 2.
(ii) The number of students who offer none of the subjects is four times the number of those who offer three subjects. So, if x is the number of students who offer three subjects, then the number of students who offer none is 4x. Hence, the number of students who offer none of the subjects is 4*2 = 8.
(iii) To find the number of students who offer only one subject, we can add up the individual sets (excluding the intersections): 35 + 43 + 50 = 128. Then we can subtract the students who offer two or three subjects: (14 + 13 + 11) + (2*3) = 44. Therefore, the number of students who offer only one subject is 128 - 44 = 84.
(iv) The number of students who offer history and geography is 14, which is the intersection of the history and geography sets.
Therefore, the answers are:
(i) 2 students offer three subjects
(ii) 8 students offer none of the subjects
(iii) 84 students offer only one of the subjects
(iv) 14 students offer history and geography.
in a class of 100 students, 35 offers history, 43 offers Geography and 50 offers economic, 14 students offers history and geography, 13 offers geography and economic and 11 offers history and economic. the number of students that offers none of the subjects is four times the number of those that offers three subjects.
(i) how many students offers three subjects
(ii) how many students offers none
(iii) how many student offer only one of the subjects
(iv) how many students offers history and Geography
in a class of 100 students, 35 offers history, 43 offers Geography and 50 offers economic, 14 students offers history and geography, 13 offers geography and economic and 11 offers history and economic. the number of students that offers none of the subjects is four times the number of those that offers three subjects.
(i) how many students offers three subjects
(ii) how many students offers none
(iii) how many student offer only one of the subjects
(iv) how many students offers history and Geography
with the aid of a venn diagram
To solve this problem, we can use the principle of inclusion and exclusion.
(i) To find the number of students who offer three subjects, we need to find the intersection of all three subjects. We can do this using the formula:
Number of students offering three subjects = Total number of students - (Number offering History + Number offering Geography + Number offering Economic) + (Number offering History and Geography and Economic)
Given:
Total number of students = 100
Number offering History (H) = 35
Number offering Geography (G) = 43
Number offering Economic (E) = 50
Number offering History and Geography (H ∩ G) = 14
Number offering Geography and Economic (G ∩ E) = 13
Number offering History and Economic (H ∩ E) = 11
Number of students offering three subjects = 100 - (35 + 43 + 50) + 14 + 13 + 11
Number of students offering three subjects = 100 - 128 + 38
Number of students offering three subjects = 10
(ii) To find the number of students who offer none of the subjects, we are given that this number is four times the number offering three subjects:
Number offering none = 4 * (Number offering three subjects) = 4 * 10
Number offering none = 40
(iii) To find the number of students offering only one subject, we need to add up the number of students offering each subject individually and subtract the number of students offering two or more subjects:
Number of students offering only one subject = (Number offering History + Number offering Geography + Number offering Economic) - (Number offering History and Geography + Number offering Geography and Economic + Number offering History and Economic) + (Number offering none)
Number of students offering only one subject = (35 + 43 + 50) - (14 + 13 + 11) + 40
Number of students offering only one subject = 128 - 38 + 40
Number of students offering only one subject = 130
(iv) To find the number of students offering History and Geography, we can directly use the given value:
Number offering History and Geography (H ∩ G) = 14
Therefore, the number of students offering History and Geography is 14.