Question

A survey of 525 adults aged 18-24 year olds was conducted in which they were asked what they did last Friday night. It found:

166 hung out with friends

152 ate pizza

26 watched TV and ate pizza, but did not hang out with friends

50 watched TV and hung out with friends, but did not eat pizza

32 hung out with friends and ate pizza, but did not watch TV

25 watched TV, hung out with friends, and ate pizza

96 did not do any of these three activities

How may 18-24 year olds (of these three activities) only watched TV last Friday night?

Answer 1

168 of these adults only watched TV last Friday night

Explanation:

To solve this problem, we must build the Venn's Diagram of this set.

I am going to say that:

-The set A represents the adults that watched TV

-The set B represents the adults that hung out with friends.

-The set C represents the adults that ate pizza

-The set D represents the adults that did not do any of these three activities.

We have that:

`A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)`

In which a is the number of adults that only watched TV,
`A \cap B` is the number of adults that both watched TV and hung out with friends,
`A \cap C` is the number of adults that both watched TV and ate pizza, and
`A \cap B \cap C` is the number of adults that did all these three activies.

By the same logic, we have:

`B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)`

`C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)`

This diagram has the following subsets:

`a,b,c,D,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)`

There were 525 adults suveyed. This means that:

`a + b + c + D + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 525`.

Solution

We build the sets from what the problem states:

96 did not do any of these three activities:

`D = 96`

25 watched TV, hung out with friends, and ate pizza:

[/tex]A \cap B \cap C = 25[/tex]

32 hung out with friends and ate pizza, but did not watch TV:

`B \cap C = 32`

50 watched TV and hung out with friends, but did not eat pizza:

`A \cap B = 50`

26 watched TV and ate pizza, but did not hang out with friends:

`A \cap C = 26`

152 ate pizza:

`C = 152`

`C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)`

`152 = c + 26 + 32 + 25`

`c = 69`

166 hung out with friends

`B = 166`

`B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)`

`166 = b + 32 + 50 + 25`

`b = 59`

How may 18-24 year olds (of these three activities) only watched TV last Friday night?

We can find the value of a from the following equation:

`a + 59 + 69 + 96 + 50 + 26 + 32 + 25 = 525`

`a = 525 - 357`

`a = 168`

168 of these adults only watched TV last Friday night