In an examination involving 30 candidates,31 passed mathematics,29 passed physics,while 3 failed both subjects. How many candidates passed (a)both subject,(b)physics only.
We can use a Venn diagram to solve this problem. Let the circle on the left represent the mathematics students, the circle on the right represent the physics students, and the overlapping region represent those who passed both subjects.
From the information given, we know that:
- 31 students passed mathematics, so the left circle has a total of 31 students.
- 29 students passed physics, so the right circle has a total of 29 students.
- 3 students failed both subjects, so the overlapping region has 3 students.
To find the number of students who passed both subjects, we subtract the number who failed both subjects from the total number of students:
Total students = 30
Failed both = 3
Passed at least one subject = 30 - 3 = 27
Passed both = 27 - 29 (passed physics only) = 2
Therefore, 2 students passed both mathematics and physics.
To find the number of students who passed physics only, we subtract the number who passed both subjects and the number who failed both subjects from the total number of students who passed physics:
Total passed physics = 29
Failed both = 3
Passed both = 2
Passed physics only = 29 - 2 - 3 = 24
Therefore, 24 students passed physics only.
To solve this problem, we will use a formula known as the principle of inclusion-exclusion. Let's break down the information given:
Total number of candidates = 30
Number of candidates who passed mathematics = 31
Number of candidates who passed physics = 29
Number of candidates who failed both subjects = 3
(a) To find the number of candidates who passed both subjects, we need to subtract the number of candidates who failed both subjects from the total number of candidates. Therefore:
Number of candidates who passed both subjects = Total number of candidates - Number of candidates who failed both subjects
Number of candidates who passed both subjects = 30 - 3
Number of candidates who passed both subjects = 27
So, 27 candidates passed both subjects.
(b) To find the number of candidates who passed physics only, we need to subtract the number of candidates who passed both subjects from the number of candidates who passed physics. Therefore:
Number of candidates who passed physics only = Number of candidates who passed physics - Number of candidates who passed both subjects
Number of candidates who passed physics only = 29 - 27
Number of candidates who passed physics only = 2
So, 2 candidates passed physics only.
To find the number of candidates who passed both subjects, we need to subtract the number of candidates who failed both subjects from the total number of candidates who passed mathematics and physics individually.
Step 1: Calculate the number of candidates who failed both subjects.
Since 3 candidates failed both subjects, the number of candidates who failed both subjects is 3.
Step 2: Calculate the number of candidates who passed both subjects.
The total number of candidates who passed mathematics is 31, and the total number of candidates who passed physics is 29. Since 3 candidates failed both subjects, we subtract 3 from the total number of candidates who passed mathematics and physics individually.
Candidates who passed both subjects = Total candidates who passed mathematics + Total candidates who passed physics - Candidates who failed both subjects
Candidates who passed both subjects = 31 + 29 - 3
Candidates who passed both subjects = 57 - 3
Candidates who passed both subjects = 54
Step 3: Calculate the number of candidates who passed physics only.
Candidates who passed physics only = Total candidates who passed physics - Candidates who passed both subjects
Candidates who passed physics only = 29 - 54 (Note: We already calculated Candidates who passed both subjects in Step 2)
Candidates who passed physics only = 0 (Since the result is negative, we can conclude that there are no candidates who passed physics only.)
So, the answers are:
(a) The number of candidates who passed both subjects is 54.
(b) There are no candidates who passed physics only.