Out of \(100\) textile companies, \(10\) companies are involved in spinning, weaving and chemical processing, \(25\) companies are involved in spinning and chemical processing, and \(30\) companies are involved in weaving and chemical processing. If \(65\) companies are involved in chemical processing, the number of companies involved {ONLY} in chemical processing is ________________________.
Textile Companies: Understanding the Problem
This problem involves calculating the number of companies involved exclusively in chemical processing using principles of set theory.
Defining Sets and Given Information
- Let S be the set of companies involved in spinning.
- Let W be the set of companies involved in weaving.
- Let C be the set of companies involved in chemical processing.
- Total companies = 100.
We are given the following:
- Number of companies in spinning, weaving, AND chemical processing: \( |S \cap W \cap C| = 10 \)
- Number of companies in spinning AND chemical processing: \( |S \cap C| = 25 \)
- Number of companies in weaving AND chemical processing: \( |W \cap C| = 30 \)
- Total number of companies in chemical processing: \( |C| = 65 \)
Calculating Companies in Chemical Processing with Other Activities
We need to find the number of companies involved in chemical processing along with spinning OR weaving (or both). This is represented by \( |C \cap (S \cup W)| \). We use the inclusion-exclusion principle:
\( |C \cap (S \cup W)| = |(C \cap S) \cup (C \cap W)| \)
\( |C \cap (S \cup W)| = |C \cap S| + |C \cap W| - |(C \cap S) \cap (C \cap W)| \)
Since \( (C \cap S) \cap (C \cap W) \) is the same as \( S \cap W \cap C \), the formula becomes:
\( |C \cap (S \cup W)| = |S \cap C| + |W \cap C| - |S \cap W \cap C| \)
Substituting the given values:
\( |C \cap (S \cup W)| = 25 + 30 - 10 \)
\( |C \cap (S \cup W)| = 55 - 10 = 45 \)
So, 45 companies are involved in chemical processing AND at least one other activity (spinning or weaving).
Calculating Companies ONLY in Chemical Processing
To find the number of companies involved ONLY in chemical processing, we subtract the companies involved in chemical processing along with other activities from the total number of companies involved in chemical processing:
Number of companies ONLY in C = \( |C| - |C \cap (S \cup W)| \)
Number of companies ONLY in C = \( 65 - 45 \)
Number of companies ONLY in C = \( 20 \)
Therefore, 20 companies are involved solely in chemical processing.
Important Questions from Venn Diagrams
Each row of Column-I has three items and each item is represented by a circle in Column-II. The arrangement of circles in Column-II represents the relationship among the items in Column-I.
Identify the option that has the most appropriate match between Column-I and Column-II.
Note: The figure shown are representative.
Column-I Column-II (1) Animal, Zebra, Giraffe (P) 
(2) Director, Producer, Actor (Q) 
(3) Word, Sentence, Novel (R) 
(4) Pianist, Guitarist, Instrumentalist (S) 
- To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly. A total of 6,30,000 candidates appeared for the test. Question A was correctly answered by 3,30,000 candidates. Question B was answered correctly by 2,50,000 candidates. Question C was answered correctly by 2,60,000 candidates. Both questions A and B were answered correctly by 1,00,000 candidates. Both questions B and C were answered correctly by 90,000 candidates. Both questions A and C were answered correctly by 80,000 candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test?
- Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films?
- 500 students are taking one or more courses out of Chemistry, Physics, and Mathematics. Registration records indicate course enrolment as follows: Chemistry (329), Physics (186), Mathematics (295), Chemistry and Physics (83), Chemistry and Mathematics (217), and Physics and Mathematics (63). How many students are taking all 3 subjects?
In the given diagram, teachers are represented in the triangle, researchers in the circle and administrators in the rectangle. Out of the total number of the people, the percentage of administrators shall be in the range of ___________.
