In a survey, 70% like tea and 60% like coffee. What percentage... | Mathematics MCQ | Examoogle

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In a survey, 70% like tea and 60% like coffee. What percentage like both?

10%

20%

30%

40%

Correct Answer

📐 MA • Math Direct FormulaMathematicsSet Theory

Option C

30%

Quick Summary:

Subtract 100% from the sum of the two groups. Here, $(70 + 60) - 100 = 30$ . This result represents the overlap.

📐MAMath SolutionDirect Formula

📋 Given

Percentage of people who like tea is 70% and percentage of people who like coffee is 60%.

🔢 Formula Used

n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)n(A∪B)=n(A)+n(B)−n(A∩B)

⚡ Exam Hall Shortcut / Speed Trick

Subtract 100% from the sum of the two groups. Here, $(70 + 60) - 100 = 30$ . This result represents the overlap.

⚠️ Common Student Trap / Pitfall

Students often treat the total percentage as 100 but forget that the sum of individual sets $(70 + 60 = 130)$ exceeds the total capacity of 100, which is the definition of the overlap.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Define variables

Let $A$ be the set of people who like tea ( $70\%$ ) and $B$ be the set of people who like coffee ( $60\%$ ). The total survey represents $100\%$ .

n(A)=70%,n(B)=60%,n(A∪B)=100%n(A) = 70\%, \quad n(B) = 60\%, \quad n(A \cup B) = 100\%n(A)=70%,n(B)=60%,n(A∪B)=100%

Apply Inclusion-Exclusion principle

We use the standard set theory formula to find the intersection, which denotes those who like both.

n(A∩B)=n(A)+n(B)−n(A∪B)n(A \cap B) = n(A) + n(B) - n(A \cup B)n(A∩B)=n(A)+n(B)−n(A∪B)

Calculate the intersection

Substitute the given values into the formula to compute the final percentage.

n(A∩B)=70%+60%−100%=30%n(A \cap B) = 70\% + 60\% - 100\% = 30\%n(A∩B)=70%+60%−100%=30%

✅

C is correct because the overlap between the two sets, calculated by subtracting the total percentage from the sum of individual preferences, is 30%.

Core Concepts Used

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Set Theory Inclusion-Exclusion Principle Venn Diagrams

💡 EXAM TIP

This principle is the foundation for solving problems in Probability and Data Interpretation (DI) involving overlapping data sets.

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