In a class of 30 students, 18 offer mathematics (M), 15 offer science (S) and 13 offer english (€). The number of students who offer all the three subjects is equal to the number of students who do not offer any of these subject. 10 students offer both mathematics and english, 8 offer both mathematics and science and 3 offers only english and science. Determine: (i) the number of students who offer all the three subjects (ii) the number of students who offer only one subject (iii) the number of learners who offer at least two subject
18 (M)
15 (S)
13 (E)
10 (M,E)
8 (M,S)
3 (E,S)
I made a venn diagram with 3 circles for this lol.
Math
10 (M,E) + 8 (M,S) = 18
18 (M) - 18 = 0
Science
8 (M,S) + 3 (E,S) = 11
15 (S) - 11 = 4
Eng
10 (M, E) + 3 (E,S) = 13
13 (E) - 13 = 0
10 (M,E) + 8 (M,S) + 3 (E,S) + 0 (M) + 4 (S) + 0 (E) = 25
30 - 25 = 5
I. 5
II. 4 (only 4 were left for Sci; 0 for Math and Eng)
III. 21 (add all the students who offer 2 subjects; given)
I don't understand on the first answer
A class at a college has 30 students and of these, 18 study business mathematics and 20 students
study Micro-economics. Find the percentage of students who study business mathematics only
To solve this problem, we can use the principle of inclusion-exclusion.
(i) To find the number of students who offer all three subjects, we need to find the intersection of all three sets (M, S, and €). Let's denote this number as x.
(ii) To find the number of students who offer only one subject, we need to add up the number of students in each individual subject and subtract any overlaps. Let's denote the number of students who offer only mathematics as a, only science as b, and only english as c.
(a) The number of students who offer only mathematics can be found by subtracting the overlaps:
a = (students offering M) - (students offering both M and S) - (students offering both M and €) + x
(b) The number of students who offer only science can be found similarly:
b = (students offering S) - (students offering both M and S) - (students offering both S and €) + x
(c) The number of students who offer only english can be found similarly:
c = (students offering €) - (students offering both M and €) - (students offering both S and €) + x
(iii) To find the number of students who offer at least two subjects, we need to count the students in each individual subject and add any overlaps:
(students offering at least two subjects) = (students offering M) + (students offering S) + (students offering €)
- (students offering both M and S) - (students offering both M and €) - (students offering both S and €)
+ x
Now, let's solve the equations based on the given information:
(i) We are given that the number of students who offer all three subjects is equal to the number of students who do not offer any of these subjects. Mathematically, this can be expressed as:
x = (students not offering any subject)
(ii) We are given the following information:
18 students offer mathematics, 15 students offer science, and 13 students offer english.
10 students offer both mathematics and english, 8 students offer both mathematics and science, and 3 students offer only english and science.
Using this information, we can calculate the values of a, b, and c as follows:
a = 18 - 8 - 10 + x
b = 15 - 8 - 3 + x
c = 13 - 10 - 3 + x
(iii) To find the number of students who offer at least two subjects, we can use the given information and calculate as follows:
(students offering at least two subjects) = 18 + 15 + 13
- 8 - 10 - 3
+ x
Solving these equations will give us the answers to the questions asked.