
Question: A survey shows that 61%, 46% and 29% of the people watched 3 idiots, Raajneeti and Avatar respectively. 25% people watched exactly two of the three movies and 3% watches none. What percentage of people watched all the three movies?
A) \[39\% \]
B) \[11\% \]
C) \[14\% \]
D) \[7\% \]
Answer
162.9k+ views
Hint: In this question, we have to find what percentage of people watched all three movie. In order to find this concept of Venn diagram is used. Draw intersection of Venn diagram and then by using diagram find solution.
Formula used: In this question we are going to use the Venn diagram. This diagram give the relation between various set and their subset.
Complete step by step solution:
Draw a Venn-diagram taking three intersecting sets 3 idiots, Rajneeti and Avatar. After intersection even regions will be developed.

Now we know that
\[\{ (a + d + e + g) + (b + d + f + g) + (c + e + f + g)\} - (d + e + f) - 2g = a + b + c + d + e + f + g\]
\[61x + 46x + 29x - 25x - 2g = 97x\]
\[2g = 14x\]
\[g = 7x\]
Required value is \[7\% \]
Thus, Option (D) is correct.
Note: Here we must remember the algebra used in Venn diagram.
Some important properties of Sets are given below:
A. Idempotent Law is given as
(i) Union of two same sets \[A{\rm{ }} \cup {\rm{ }}A{\rm{ }} = {\rm{ }}A\]
(ii) Intersection of two same sets \[A{\rm{ }} \cap {\rm{ }}A{\rm{ }} = {\rm{ }}A\]
B. Associative Law is given as
(i) \[\left( {A{\rm{ }} \cup {\rm{ }}B} \right){\rm{ }} \cup {\rm{ }}C{\rm{ }} = {\rm{ }}A{\rm{ }} \cup {\rm{ }}\left( {B{\rm{ }} \cup {\rm{ }}C} \right)\]
(ii) \[\left( {A{\rm{ }} \cap {\rm{ }}B} \right){\rm{ }} \cap {\rm{ }}C{\rm{ }} = {\rm{ }}A{\rm{ }} \cap {\rm{ }}\left( {B{\rm{ }} \cap {\rm{ }}C} \right)\]
C. Commutative Law is given as
(i) \[A{\rm{ }} \cup {\rm{ }}B{\rm{ }} = {\rm{ }}B{\rm{ }} \cup {\rm{ }}A\]
(ii) \[A{\rm{ }} \cap {\rm{ }}B{\rm{ }} = {\rm{ }}B{\rm{ }} \cap {\rm{ }}A\]
D. Distributive law is given as
(i) \[A{\rm{ }} \cup {\rm{ }}\left( {B{\rm{ }} \cap {\rm{ }}C} \right){\rm{ }} = {\rm{ }}\left( {A{\rm{ }} \cup {\rm{ }}B} \right){\rm{ }} \cap {\rm{ }}\left( {A{\rm{ }} \cup {\rm{ }}C} \right)\]
(ii) \[A{\rm{ }} \cap {\rm{ }}\left( {B{\rm{ }} \cup {\rm{ }}C} \right){\rm{ }} = \left( {A{\rm{ }} \cap {\rm{ }}B} \right){\rm{ }} \cup {\rm{ }}\left( {A{\rm{ }} \cap {\rm{ }}C} \right)\]
Where A, B, C are set or subset of any universal set
E. De Morgan’s law is given as
(i) \[{\left( {A{\rm{ }} \cup B} \right)^c} = {A^c} \cap {\rm{ }}{B^c}\]
(ii) \[{\left( {A{\rm{ }} \cap B} \right)^c} = {A^c} \cup {\rm{ }}{B^c}\]
Where, \[{A^c},{B^c}\] is complement of set A and B respectively
Formula used: In this question we are going to use the Venn diagram. This diagram give the relation between various set and their subset.
Complete step by step solution:
Draw a Venn-diagram taking three intersecting sets 3 idiots, Rajneeti and Avatar. After intersection even regions will be developed.

Now we know that
\[\{ (a + d + e + g) + (b + d + f + g) + (c + e + f + g)\} - (d + e + f) - 2g = a + b + c + d + e + f + g\]
\[61x + 46x + 29x - 25x - 2g = 97x\]
\[2g = 14x\]
\[g = 7x\]
Required value is \[7\% \]
Thus, Option (D) is correct.
Note: Here we must remember the algebra used in Venn diagram.
Some important properties of Sets are given below:
A. Idempotent Law is given as
(i) Union of two same sets \[A{\rm{ }} \cup {\rm{ }}A{\rm{ }} = {\rm{ }}A\]
(ii) Intersection of two same sets \[A{\rm{ }} \cap {\rm{ }}A{\rm{ }} = {\rm{ }}A\]
B. Associative Law is given as
(i) \[\left( {A{\rm{ }} \cup {\rm{ }}B} \right){\rm{ }} \cup {\rm{ }}C{\rm{ }} = {\rm{ }}A{\rm{ }} \cup {\rm{ }}\left( {B{\rm{ }} \cup {\rm{ }}C} \right)\]
(ii) \[\left( {A{\rm{ }} \cap {\rm{ }}B} \right){\rm{ }} \cap {\rm{ }}C{\rm{ }} = {\rm{ }}A{\rm{ }} \cap {\rm{ }}\left( {B{\rm{ }} \cap {\rm{ }}C} \right)\]
C. Commutative Law is given as
(i) \[A{\rm{ }} \cup {\rm{ }}B{\rm{ }} = {\rm{ }}B{\rm{ }} \cup {\rm{ }}A\]
(ii) \[A{\rm{ }} \cap {\rm{ }}B{\rm{ }} = {\rm{ }}B{\rm{ }} \cap {\rm{ }}A\]
D. Distributive law is given as
(i) \[A{\rm{ }} \cup {\rm{ }}\left( {B{\rm{ }} \cap {\rm{ }}C} \right){\rm{ }} = {\rm{ }}\left( {A{\rm{ }} \cup {\rm{ }}B} \right){\rm{ }} \cap {\rm{ }}\left( {A{\rm{ }} \cup {\rm{ }}C} \right)\]
(ii) \[A{\rm{ }} \cap {\rm{ }}\left( {B{\rm{ }} \cup {\rm{ }}C} \right){\rm{ }} = \left( {A{\rm{ }} \cap {\rm{ }}B} \right){\rm{ }} \cup {\rm{ }}\left( {A{\rm{ }} \cap {\rm{ }}C} \right)\]
Where A, B, C are set or subset of any universal set
E. De Morgan’s law is given as
(i) \[{\left( {A{\rm{ }} \cup B} \right)^c} = {A^c} \cap {\rm{ }}{B^c}\]
(ii) \[{\left( {A{\rm{ }} \cap B} \right)^c} = {A^c} \cup {\rm{ }}{B^c}\]
Where, \[{A^c},{B^c}\] is complement of set A and B respectively
Recently Updated Pages
JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Chemical Properties of Hydrogen - Important Concepts for JEE Exam Preparation

JEE Energetics Important Concepts and Tips for Exam Preparation

JEE Isolation, Preparation and Properties of Non-metals Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths Chapter 6 Permutations and Combinations

NCERT Solutions for Class 11 Maths In Hindi Chapter 1 Sets

NEET 2025 – Every New Update You Need to Know
