In an examination of certain number of candidates 18 passed mathematics, 17 passed physics, 11 passed both mathematics and physics, 1 failed both subjects. Find the number of candidates that passed. i. Mathematics only. ii. Total number of candidates that sat for the examination.
math only: 18-11
total: 18+17-11+1
To solve this problem, we need to use set theory and the principle of inclusion-exclusion. Let's break down the given information step by step:
Let A be the set of candidates who passed mathematics,
B be the set of candidates who passed physics.
We are given that:
The number of candidates who passed mathematics (A) is 18.
The number of candidates who passed physics (B) is 17.
The number of candidates who passed both mathematics and physics (A ∩ B) is 11.
The number of candidates who failed both subjects is 1.
Now, let's find the number of candidates who passed:
1. Find the number of candidates who passed either mathematics or physics (A ∪ B):
Using the principle of inclusion-exclusion, we can calculate this as:
|A ∪ B| = |A| + |B| - |A ∩ B|
= 18 + 17 - 11
= 24
2. Find the number of candidates who passed mathematics only:
To calculate this, we need to subtract the number of candidates who passed both mathematics and physics (A ∩ B) from the number of candidates who passed mathematics (A):
|Mathematics only| = |A| - |A ∩ B|
= 18 - 11
= 7
Therefore, the number of candidates who passed mathematics only is 7.
3. Find the total number of candidates that sat for the examination:
To calculate this, we need to consider all possible candidates, i.e., the number of candidates who passed mathematics, who passed physics, and who failed both subjects:
Total candidates = |A ∪ B ∪ Neither(A or B)|
= |A ∪ B ∪ (A' ∩ B')|
= |A ∪ B ∪ (A ∩ B)'|
= |A ∪ B| + |Neither(A or B)|
= |A ∪ B| + |Neither(A and B)|
= |A ∪ B| + 1 (since 1 candidate failed both subjects)
= 24 + 1
= 25
Therefore, the total number of candidates that sat for the examination is 25.
To find the number of candidates that passed mathematics only, we need to subtract the number of candidates who passed both mathematics and physics from the total number of candidates who passed mathematics.
Given that 18 candidates passed mathematics and 11 passed both mathematics and physics, we can subtract 11 from 18 to get the number of candidates who passed mathematics only:
Number of candidates that passed mathematics only = 18 - 11 = 7.
Therefore, 7 candidates passed mathematics only.
To find the total number of candidates that sat for the examination, we need to consider those who passed mathematics, those who passed physics, and those who failed both subjects.
Given that 18 candidates passed mathematics, 17 passed physics, and 1 failed both subjects, we can sum up these numbers to get the total number of candidates:
Total number of candidates = Number that passed mathematics + Number that passed physics + Number that failed both subjects
= 18 + 17 + 1
= 36.
Therefore, the total number of candidates that sat for the examination is 36.