In a group of 200 people, 120 like tea, 90 like coffee, 70 like juice. If 50 people like both tea and coffee…

Question

In a group of 200 people, 120 like tea, 90 like coffee, 70 like juice. If 50 people like both tea and coffee, 40 like both tea and juice and 30 like both coffee and juice and 20 like all the three all the beverages, how many people do not like any of tge beverages? With Venn diagram

ScanSolve AI Tutor · Accepted Answer

Let T be the set of people who like tea, C for coffee, and J for juice.
We are given:
Total people = 200 |T| = 120 |C| = 90 |J| = 70 |T ∩ C| = 50 |T ∩ J| = 40 |C ∩ J| = 30 |T ∩ C ∩ J| = 20 (people who like all three)

We need to find the number of people who do not like any of the beverages. First, we'll find the number of people who like at least one beverage using the Principle of Inclusion-Exclusion.

Calculate the number of people who like at least one beverage.
The formula for the union of three sets is: |T ∪ C ∪ J| = |T| + |C| + |J| - |T ∩ C| - |T ∩ J| - |C ∩ J| + |T ∩ C ∩ J|

Substitute the given values: |T ∪ C ∪ J| = 120 + 90 + 70 - 50 - 40 - 30 + 20 |T ∪ C ∪ J| = 280 - 120 + 20 |T ∪ C ∪ J| = 160 + 20 |T ∪ C ∪ J| = 180 people

So, 180 people like at least one of the beverages.
Calculate the number of people who do not like any beverage.
Number of people who do not like any beverage = Total people - |T ∪ C ∪ J|
Number of people who do not like any beverage = 200 - 180
Number of people who do not like any beverage = 20 people
Venn Diagram Representation
To visualize this with a Venn diagram, we fill in the regions starting from the innermost intersection:
- People who like all three (T ∩ C ∩ J): 20
- People who like only Tea and Coffee (T ∩ C only): |T ∩ C| - |T ∩ C ∩ J| = 50 - 20 = 30
- People who like only Tea and Juice (T ∩ J only): |T ∩ J| - |T ∩ C ∩ J| = 40 - 20 = 20
- People who like only Coffee and Juice (C ∩ J only): |C ∩ J| - |T ∩ C ∩ J| = 30 - 20 = 10
- People who like only Tea: |T| - (T ∩ C only) - (T ∩ J only) - (T ∩ C ∩ J) = 120 - 30 - 20 - 20 = 50
- People who like only Coffee: |C| - (T ∩ C only) - (C ∩ J only) - (T ∩ C ∩ J) = 90 - 30 - 10 - 20 = 30
- People who like only Juice: |J| - (T ∩ J only) - (C ∩ J only) - (T ∩ C ∩ J) = 70 - 20 - 10 - 20 = 20
Summing these individual regions gives the total number of people who like at least one beverage:
20 + 30 + 20 + 10 + 50 + 30 + 20 = 180 people.

The number of people who do not like any beverage is the total number of people minus those who like at least one:
200 - 180 = 20 people.

In a group of 200 people, 120 like tea, 90 like coffee, 70 like juice. If 50 people like both tea and coffee, 40 like both tea and juice and 30 like both coffee and juice and 20 like all the three all the beverages, how many people do not like any of tge beverages? With Venn diagram

Step-by-step solution

Calculate the number of people who like at least one beverage.

Calculate the number of people who do not like any beverage.

Venn Diagram Representation

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