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A cinema house has 600 people. \(26\frac{2}{3}{\rm{\% }}\) People buy premium tickets and \(33\frac{1}{3}{\rm{\% }}\) People buy gold tickets and 40% of people buy platinum tickets. Only 5% of people buy both platinum and gold tickets and 10% of people buy both premium and platinum tickets and \(6\frac{2}{3}{\rm{\% }}\) of people buy both premium and gold tickets and 30 people buy all three types of tickets. Find the number of people who buy only one type of ticket.

A cinema house has 600 people. \(26\frac{2}{3}{\rm{\% }}\) People buy premium tickets and \(33\frac{1}{3}{\rm{\% }}\) People buy gold tickets and 40% of people buy platinum tickets. Only 5% of people buy both platinum and gold tickets and 10% of people buy both premium and platinum tickets and \(6\frac{2}{3}{\rm{\% }}\) of people buy both premium and gold tickets and 30 people buy all three types of tickets. Find the number of people who buy only one type of ticket. Correct Answer 370

Given:

Total number of people = 600

People buy premium tickets.

People buy gold tickets.

40% of people buy platinum tickets.

Only 5% of people buy both platinum and gold tickets.

10% of people buy both premium and platinum tickets.

of people buy both premium and gold.

30 people buy all three types of tickets.

Formula used:

Total people who buy only one type tickets

⇒ n(A) + n(B) + n(C) - 2 × n(A ∩ B) - 2 × n(B ∩ C) - 2 × n(C ∩ A) - 3 × n(A ∩ B ∩ C)

Where n(A) = number of elements in set A.

Calculation:

Total number of peoples who buy premium tickets

⇒ (80 / 300) × 600 = 160

Total number of peoples who buy gold tickets

⇒ (100 / 300) × 600 = 200

Total number of peoples who buy platinum tickets = 40% of 600

⇒ (40 / 100) × 600 = 240

Total number of peoples who buy only both platinum and gold tickets = 5% of 600

⇒ (5 / 100) × 600 = 30

Total number of peoples who buy both premium and platinum tickets = 10% of 600

⇒ (10 / 100) × 600 = 60

Total number of peoples who buy both premium and gold tickets

⇒ (20 / 300) × 600 = 40

 Total number of peoples who buy all three types of tickets = 30

Total number of peoples who buy only both premium and platinum tickets = 60 - 30 = 30

A

Premium

B

Gold

C

Platinum

 

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Total number of peoples who buy only both premium and gold tickets = 40 - 30 = 10

Now by formula,

⇒ 160 + 200 + 240 - 2 × 10 - 2 × 30 - 2 × 30 - 3 × 30

⇒ 600 - 230 = 370

∴ The number of people who buy only one type of ticket is 370.

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