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In a group of 80 people, 45 people like car driving and 20 people like both car driving and cycling. Everyone in the group like at least one of the two. How many people like only cycling?

In a group of 80 people, 45 people like car driving and 20 people like both car driving and cycling. Everyone in the group like at least one of the two. How many people like only cycling? Correct Answer 35

Given:

Total people in group = 80

Like car driving = 45

Like both = 20

Formula used:

n(C∪T) = n(C) + n(T) − n(C∩T)

Only T for

n(T−C) = n(T) − n(T∩C)

Calculation:

Let C denote the set the people like car driving, and T denote the set of people who like cycling

Here, n(C∪T) = 80, n(C) = 45, n(C∩T) = 20

n(C∪T) = n(C) + n(T) − n(C∩T)

⇒ 80 = 45 + n(T) − 20

⇒ 80 = 25 + n(T)

⇒ n(T) = 80 – 25

⇒ n(T) = 55

Then 65 people like cycling.

According to question,

n(T−C) = n(T) − n(T∩C)

⇒ n(T−C) = 55 – 20

⇒ n(T−C) = 35

∴ 35 people like only cycling.

Shortcut Trick

Using venn diagram

[ alt="F3 Savita State Govt. 15-4-22 D1" src="//storage.googleapis.com/tb-img/production/22/04/F3_Savita_State%20Govt._15-4-22_D1.png" style="width: 289px; height: 163px;">

⇒ 80 = 25 + 20 + only cycle

⇒ only cycle = 80 - 20 - 25 

⇒ only cycling = 35%

∴ 35 people like only cycling.

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