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Zorluk: ZorOverlapping Sets and Venn Diagrams

An international airline surveyed 360360 frequent flyers regarding three premium services they utilized during the past year: First-Class Lounge Access, Priority Baggage Handling, and In-Flight Wi-Fi. The survey revealed the following data:

- Every surveyed passenger utilized at least one of the three services.
- 220220 passengers utilized First-Class Lounge Access.
- 180180 passengers utilized Priority Baggage Handling.
- 140140 passengers utilized In-Flight Wi-Fi.
- 6060 passengers utilized all three services.

If the ratio of the number of passengers who utilized exactly one service to the number of passengers who utilized exactly two services is 4:14 : 1, how many passengers utilized exactly two services?

  1. A
    4848
  2. 6060Cevap
  3. C
    7575
  4. D
    120120
  5. E
    240240

Cevap

The number of passengers who utilized exactly two services is 60.
By breaking the total set of passengers into disjoint groups—those using exactly one service (E1E_1), exactly two services (E2E_2), all three services (E3=60E_3 = 60), and none (None=0\text{None} = 0)—we set up the equation 360=E1+E2+60+0360 = E_1 + E_2 + 60 + 0. Subtracting 6060 from both sides gives E1+E2=300E_1 + E_2 = 300. Given the ratio E1:E2=4:1E_1 : E_2 = 4 : 1, we substitute E1=4E2E_1 = 4E_2, yielding 5E2=3005E_2 = 300, so E2=60E_2 = 60.

Adım Adım Çözüm

1
Set up the fundamental 3-set total elements equation in terms of disjoint regions.
Total=E1+E2+E3+None\text{Total} = E_1 + E_2 + E_3 + \text{None}, where E1E_1 is exactly 1 service, E2E_2 is exactly 2 services, and E3E_3 is all 3 services.
Categorizing members into mutually exclusive regions simplifies group accounting without double-counting.
2
Substitute the known values into the total elements formula.
360=E1+E2+60+0  ⟹  E1+E2=300360 = E_1 + E_2 + 60 + 0 \implies E_1 + E_2 = 300.
Since every passenger used at least one service, None=0\text{None} = 0, leaving 300300 passengers who used either exactly 1 or exactly 2 services.
3
Apply the given ratio of E1:E2=4:1E_1 : E_2 = 4 : 1 to solve for E2E_2.
E1=4E2  ⟹  4E2+E2=300  ⟹  5E2=300  ⟹  E2=60E_1 = 4E_2 \implies 4E_2 + E_2 = 300 \implies 5E_2 = 300 \implies E_2 = 60.
The ratio implies that out of 55 total parts for E1+E2E_1 + E_2, 11 part represents E2E_2.

Anahtar Kavram

3-Set Overlapping Sets Disjoint Regions Formula
Tahmini Süre:2m 0s
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Konular

Overlapping Sets and Venn Diagrams

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