I don't understand how to solve the problem in the description

Question

A survey of 125 people was conducted to determine who watched three different networks. 55 watched ABC; 30 watched nbc; 40 watched cbs; 10 watched abc and cbs; 12 watched abc and nbc; 8 watched nbc and cvs; 5 watched nbc and cbs. How many of the 125 people did not watch any of the three networks?

Answer 1

Amanda,

 

It doesn't seem like elementary math from the problem statement.  There are a lot of numbers there and one error. 5 watched all 3 channels, not just nbc and cbs.  Otherwise, it would contradict the preceding clause of 8.  With this correction we can proceed.

 

The answer is 25 of the 125 people didn't watch any of the 3 networks.

 

You can solve this problem in two related ways.  

 

Visually, using a Venn diagram where the rectangle represents the universe of all 125 people surveyed, and within it 3 intersecting circles representing each of the three networks. Then just calculate and fill in the jigsaw pieces from the above data, add them all up, and subtract the total from 125.

 

Or using sets algebraically which I will do.

 

Let A = ABC, C=CBS, N=NBC for simplicity, and ∩ represent set intersection. Then from above data:

 

{A ∩ C ∩ N} = 5  , represents the number of people who watched all 3 networks.

 

{A ∩ C} - {A ∩ C ∩ N} = 10-5 = 5 , represents the number of people who watched only ABC & CBS.

{A ∩ N} - {A ∩ C ∩ N} = 12-5 = 7 , represents the number of people who watched only ABC & NBC.

{C ∩ N} - {A ∩ C ∩ N} =   8-5 = 3 , represents the number of people who watched only CBS & NBC.

 

{A'} = {A} - [{A ∩ C} - {A ∩ C ∩ N}] - [{A ∩ N} - {A ∩ C ∩ N}] - [{A ∩ C ∩ N}]

       = 55 - 5 - 7 - 5  = 38. represents the number of people who watched only ABC.  Similarly

{C'} = 40 - 5 - 3 - 5 = 27,  represents the number of people who watched only CBS.

{N'} = 30 - 7 - 3 - 5 = 15, represents the number of people who watched only NBC.

 

Now you have all the non-overlapping, independent jigsaw pieces within the 3 intersecting circles in the Venn diagram. Total them up to get 100 which represents all the people who watched one or more networks.  Then its complement relative to the universe of 125 people or (125 - 100 = 25) is all the people who didn't watch any network.

 

Answer 2

This math question, as many do, has information in it to mislead you. Try making a chart with the 3 network names and see what total you get.