3 Circle Venn Diagram

[Courtneyb022]

A beverage distributor conducts a survey to determine the most popular carbonated beverage. It asks 700 randomly chosen people to list their three favorite carbonated beverages and obtains the following data:

175 like 7-Up
232 like Pepsi
205 like Coke
61 like both 7-Up and Coke
78 like both Pepsi and Coke
32 like all three beverages
312 prefer other beverages

A.) Draw a Venn diagram to show this information.
B.) How many like both Pepsi and 7-Up, but not Coke?
C.) How many like Pepsi, but do not like 7-Up or Coke?

So far what I have is this:

32 people like all three
29 people like Coke and 7-Up
98 people like Coke
46 people like Pepsi and Coke
I need to find out how many people like Pepsi and 7-Up, but not sure what formula it is.

 

[Courtneyb022]

I think the number of people who like 7-Up and Pepsi is 25.....is that right?

 

[mmm4444bot]

A beverage distributor … obtains the following data:

205 like Coke

61 like both 7-Up and Coke

78 like both Pepsi and Coke

The numbers 205, 61 and 78 are fixed because they are given.

So far what I have is this:

98 people like Coke

29 people like Coke and 7-Up

46 people like Pepsi and Coke

You've changed the given numbers above. You may not do that.

Did you draw and label a Venn diagram to model the given information? (It's three overlapping circles, creating seven regions, with the value 32 in the center.)

A Venn diagram makes it easy to find the relationships, in this exercise. If you don't yet understand what a Venn diagram is, or how to draw it, that's what you should start asking about.

 

[Courtneyb022]

No I didn't....
you have 32 people that like all three, so you need to take 78-32 to get 46 that like Pepsi and Coke, 61-32 to get 29 people that like 7-Up and Coke.

 

[soroban]

Hello, Courtneyb022!

Your preliminary work is correct . . . Good for you!

A beverage distributor conducts a survey to determine the most popular carbonated beverage.
It asks 700 randomly chosen people to list their three favorite carbonated beverages and obtains the following data:

. . 175 like 7-Up.
. . 232 like Pepsi.
. . 205 like Coke.
. . 61 like both 7-Up and Coke.
. . 78 like both Pepsi and Coke.
. . 32 like all three beverages.
. . 312 prefer other beverages.

(A) Draw a Venn diagram to show this information.

(B) How many like both Pepsi and 7-Up, but not Coke?

(C) How many like Pepsi, but do not like 7-Up or Coke?

In symbols, we are given:

. . \(\displaystyle \begin{array}{ccc} n(S) \;=\;175 \\ n(P) \;=\;232 \\ n(C) \;=\;205 \\ n(S\cap C) \;=\;61 \\ n(P \cap C) \;=\;78 \\ n(S\cap P \cap C) \;=\;32 \\ n(S' \cap P' \cap C') \;=\;312 \end{array}\)


\(\displaystyle \text{There are 700 people in the survey.}\)
\(\displaystyle \text{312 of them like }none\text{ of the three beverages.}\)
\(\displaystyle \text{Hence: }\:700 - 312 \,=\,388\text{ like at least one of the three beverages.}\)
. . \(\displaystyle n(S \cup P \cup C) \:=\:388\)


\(\displaystyle \text{The formula for a three-set Venn diagram is:}\)

. . \(\displaystyle n(S\,\cup P\,\cup\,C) \;=\;n(S)\,+\,n(P)\,+\,n(C)\,-\,n(S \,\cap\,P)\,-\,n(P\,\cap\,C)\,-\,n(S\,\cap\,C)\,+\,n(S\,\cap\,P\cap\,C)\)

\(\displaystyle \text{Substitute: }\;388 \;=\;175 + 232 + 205 - n(S\cap P) - 78 - 61 + 32\)
. a . . . . . . \(\displaystyle 388 \;=\;505 - n(S \cap P)\)
. . . . .\(\displaystyle n(S \cap P) \;=\;117\)

\(\displaystyle \text{Got it?}\)