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In a survey of 120 people, the number of users for WhatsApp (W), Instagram (I), and TikTok (T) are given. Calculate the number of people using only one app and exactly two apps.

Mathematics

ScanSolve AI Answer

Answer

60\text{60}

40\text{40}

5\text{5}

Step-by-step solution

8 steps

Okay selorm, let's solve .

Let WW be the set of people who use WhatsApp, II for Instagram, and TT for TikTok.
Given:
Total people = 120
∣W∣=70|W| = 70
∣I∣=65|I| = 65
∣T∣=50|T| = 50
∣W∩I∣=35|W \cap I| = 35
∣W∩T∣=28|W \cap T| = 28
∣I∩T∣=22|I \cap T| = 22
∣W∩I∩T∣=15|W \cap I \cap T| = 15

1. How many use only one app?
This is the sum of people who use only WhatsApp, only Instagram, and only TikTok.

  1. Calculate the number of people using only each app.

    Only WhatsApp: ∣W only∣=∣W∣−(∣W∩I∣+∣W∩T∣)+∣W∩I∩T∣|W \text{ only}| = |W| - (|W \cap I| + |W \cap T|) + |W \cap I \cap T|

    ∣W only∣=70−(35+28)+15=70−63+15=22|W \text{ only}| = 70 - (35 + 28) + 15 = 70 - 63 + 15 = 22

    Only Instagram: ∣I only∣=∣I∣−(∣W∩I∣+∣I∩T∣)+∣W∩I∩T∣|I \text{ only}| = |I| - (|W \cap I| + |I \cap T|) + |W \cap I \cap T|

    ∣I only∣=65−(35+22)+15=65−57+15=23|I \text{ only}| = 65 - (35 + 22) + 15 = 65 - 57 + 15 = 23

    Only TikTok: ∣T only∣=∣T∣−(∣W∩T∣+∣I∩T∣)+∣W∩I∩T∣|T \text{ only}| = |T| - (|W \cap T| + |I \cap T|) + |W \cap I \cap T|

    ∣T only∣=50−(28+22)+15=50−50+15=15|T \text{ only}| = 50 - (28 + 22) + 15 = 50 - 50 + 15 = 15
  2. Sum these values to find the total number using only one app.

    Only one app=22+23+15\text{Only one app} = 22 + 23 + 15 Only one app=60\text{Only one app} = 60

    The number of people who use only one app is 60\boxed{\text{60}}.

    2. How many use exactly two apps?
    This is the sum of people who use (WhatsApp and Instagram only), (WhatsApp and TikTok only), and (Instagram and TikTok only).

  3. Calculate the number of people using exactly two apps for each pair.

    WhatsApp and Instagram only: ∣W∩I∣−∣W∩I∩T∣=35−15=20|W \cap I| - |W \cap I \cap T| = 35 - 15 = 20
    WhatsApp and TikTok only: ∣W∩T∣−∣W∩I∩T∣=28−15=13|W \cap T| - |W \cap I \cap T| = 28 - 15 = 13
    Instagram and TikTok only: ∣I∩T∣−∣W∩I∩T∣=22−15=7|I \cap T| - |W \cap I \cap T| = 22 - 15 = 7

  4. Sum these values to find the total number using exactly two apps.

    Exactly two apps=20+13+7\text{Exactly two apps} = 20 + 13 + 7 Exactly two apps=40\text{Exactly two apps} = 40

    The number of people who use exactly two apps is 40\boxed{\text{40}}.

    3. How many use none?
    First, we find the number of people who use at least one app using the Principle of Inclusion-Exclusion.

  5. Apply the Principle of Inclusion-Exclusion formula to find ∣W∪I∪T∣|W \cup I \cup T|.

    ∣W∪I∪T∣=∣W∣+∣I∣+∣T∣−(∣W∩I∣+∣W∩T∣+∣I∩T∣)+∣W∩I∩T∣|W \cup I \cup T| = |W| + |I| + |T| - (|W \cap I| + |W \cap T| + |I \cap T|) + |W \cap I \cap T|
  6. Substitute the given values.

    ∣W∪I∪T∣=70+65+50−(35+28+22)+15|W \cup I \cup T| = 70 + 65 + 50 - (35 + 28 + 22) + 15
  7. Perform the calculations.

    ∣W∪I∪T∣=185−85+15|W \cup I \cup T| = 185 - 85 + 15 ∣W∪I∪T∣=100+15|W \cup I \cup T| = 100 + 15 ∣W∪I∪T∣=115|W \cup I \cup T| = 115
  8. Subtract the number of people using at least one app from the total number of people.

    Number who use none=Total people−∣W∪I∪T∣\text{Number who use none} = \text{Total people} - |W \cup I \cup T| Number who use none=120−115\text{Number who use none} = 120 - 115 Number who use none=5\text{Number who use none} = 5

    The number of people who use none is 5\boxed{\text{5}}.

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