From a group of 65 students who failed Business Mathematics in 2010 we found the following reasons: 35 for not attending class (C); 30 for not doing homework (H) and 25 for not visiting tutors (T). Fourteen did not attend class and did not visit tutors; 15 did not visit tutors and did
not do homework; 20 no homework and class attendance while 8 failed for all three reasons.
(b)How many failed……..
(i) for other reasons?
(ii) not because of homework?
(iii) attending class but not doing homework?
(iv) attending class and doing homework but not visiting tutors? (v) only because of no class attendance?
(vi) because of no class attendance or not visiting tutors?
(vii) visiting tutors but not doing homework?
(viii) not doing homework and no class attendance but visiting tutors
Use a Venn diagram with 3 overlapping circles named after the reason for failing, use C, H, T as suggested as their labels
Place 8 in the intersection of all three.
Now work on the intersection of two at a time.
e.g. 14 would go in the intersection of C and T, but we already counted 8 of those. So 6 would go in the intersection of C and T, not part of the intersection of all three.
Continue like that for the other 2 pairs of intersection.
Now look at each of the whole circles. Keep in mind that some of the counts have already been entered, so fill in all the missing parts.
Now each of the individual questions can be answered.
To answer the questions, we can use a Venn diagram, which is a graphical representation of the relationships between different sets. Let's break down the information given and solve each part step by step.
Step 1: Draw the Venn diagram
Start by drawing three overlapping circles representing the three reasons: class attendance (C), homework (H), and tutor visits (T). Label each circle accordingly.
Step 2: Fill in the given information
Now let's fill in the information we have:
- 35 students did not attend class (C).
- 30 students did not do homework (H).
- 25 students did not visit tutors (T).
- 14 students did not attend class and did not visit tutors (C ∩ T).
- 15 students did not visit tutors and did not do homework (H ∩ T).
- 20 students did not do homework and did not attend class (H ∩ C).
- 8 students failed for all three reasons (C ∩ H ∩ T).
Fill in these numbers in the appropriate sections of the Venn diagram.
Step 3: Solve the questions
(i) How many failed for other reasons?
To find the number of students who failed for other reasons, we need to subtract the sum of students in the overlapping sections from the total number of students who failed (65 in this case).
So, to find the answer, add the numbers in the intersecting sections (C ∩ H, H ∩ T, C ∩ T) and subtract this from the total (65).
Other reasons = Total - (C ∩ H + H ∩ T + C ∩ T)
(ii) How many failed not because of homework?
To find the number of students who failed for reasons other than homework, we need to find the sum of all sections except H (the homework circle).
So, to find the answer, add the numbers in the sections excluding H (C, T, C ∩ T, C ∩ H ∩ T).
Not because of homework = C + T + C ∩ T + C ∩ H ∩ T
(iii) How many failed attending class but not doing homework?
To find the number of students who failed because they attended class but didn't do homework, we need to find the number of students in the section C ∩ H.
So, the answer is the value in section C ∩ H.
(iv) How many failed attending class and doing homework but not visiting tutors?
To find the number of students who failed because they attended class, did homework, but didn't visit tutors, we need to find the number of students in the section C ∩ H ∩ ¬T (where ¬T represents the complement of T, i.e., students not visiting tutors).
So, the answer is the value in section C ∩ H ∩ ¬T.
(v) How many failed only because of no class attendance?
To find the number of students who failed only because of no class attendance, we need to subtract the number of students who failed for other reasons (not attending class and doing homework, or not visiting tutors) from the total number of students who failed. This is calculated by subtracting (C ∩ H ∩ T + C ∩ H ∩ ¬T + C ∩ T) from the total.
Only because of no class attendance = Total - (C ∩ H ∩ T + C ∩ H ∩ ¬T + C ∩ T)
(vi) How many failed because of no class attendance or not visiting tutors?
To find the number of students who failed because of no class attendance or not visiting tutors, we need to find the sum of the sections C ∩ T and C ∩ H ∩ ¬T.
So, the answer is C ∩ T + C ∩ H ∩ ¬T.
(vii) How many failed visiting tutors but not doing homework?
To find the number of students who failed because they visited tutors but didn't do homework, we need to find the number of students in the section T ∩ H ∩ ¬C (where ¬C represents the complement of C, i.e., students not attending class).
So, the answer is the value in section T ∩ H ∩ ¬C.
(viii) How many failed not doing homework and no class attendance but visiting tutors?
To find the number of students who failed because they did not do homework, did not attend class, but visited tutors, we need to find the number of students in the section H ∩ ¬C ∩ T.
So, the answer is the value in section H ∩ ¬C ∩ T.
By following these steps and using the Venn diagram, you can determine the answers to each question based on the given information.