100 students were interviewed. 28 took Physical Education, 31 took Biology, 42 to English, 9 took Physical Education and Biology, 10 took Physical Education and English and 6 took Biology and English.
i. How many students to none of the three subjects
ii. How took physical education, but not biology or English
iii. How many students took biology and physical education but not English
Answers
Answer:
If x is the number of students who took all three classes, then
i. 24-x students took none of the three subjects.
ii. 9+x students took physical education, but not biology or english.
iii. 9-x students took biology and physical education but not english.
Step-by-step explanation:
There is a lack of information: We don't know how many students took all the three classes, then we will call it x. Our answers will depend on x.
Once we have 6 students who took both biology and english, we have 6-x students who took exclusively biology and english, that is, they did not take physical education. Similarly, we have 9-x students who took exclusively biology and physical education and 10-x students who took exclusively english and physical education.
Now, in order to know how many students took only biology classes, we have to consider the total number of students who took biology classes and subtract it from the number of students who took all three classes, both biology and physical education classes and both biology and english classes, that it, 31-x-(9-x)-(6-x) = 16+x. Similarly, we get 26+x students took only english classes and 9+x took only physical education classes.
Now we can answer the questions.
i. If we sum all the numbers at Venn diagram, we get (16+x)+(26+x)+(9+x)+(6-x)+(9-x)+(10-x)+x = 76+x. Then 100-(76+x) = 24-x students took none of the three subjects.
ii. As we can see in the diagram, 9+x students took physical education, but not biology or english.
iii. Again, as we can see in the diagram, 9-x students took biology and physical education but not english.
