Case based question of Sets

Harit Thakuri

Instructor

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A politician is trying to win election to the city council, and as his campaign manager, you need to decide how to promote the candidate.
There are three ways you can do so:
a. You can send glossy.full-color pamphlets to registered voters of the city,
b. You can run a commercial during the television news on a local cable network, and /or
c. You can buy a full-page ad in the newspaper.
Two hundred fifty thousand voters live in the city, and 36% of them read the newspaper.
Fifty thousand voters watch the local cable network news, and 30% of them also read the newspaper.
You also know that the television commercial would cost $40,000, the newspaper ad $27,000, and the pamphlets mailed to voters 90 cents each, including printing and bulk-rate postage.
Suppose that the success of the candidate depends on your campaign reaching at least 125,000 voters and that because your budget is limited, you must achieve this goal at a minimum cost. Based on above information, answer the following questions. (Use Venn diagram, if required)
i. How many voters in the city read the newspaper but do not watch the local cable television news?
ii. How many voters read the newspaper or watch the local cable television news, or both?
iii. Complete the following chart by indicating the number of voters reached by each promotional option, the total the cost per voter reached.


Promotional mediumNo. of voters reachedTotal CostCost per voter








iv. What would be your plan and the cost of plan?

  • 25 Dec, 2023
Replies (1)
Harit Thakuri

Instructor

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i. To find the number of voters who read the newspaper but do not watch the local cable television news, we can use the information given in the problem.

36% of the 250,000 registered voters read the newspaper, which is: 0.36 x 250,000 = 90,000 

30% of the 50,000 voters who watch the local cable news also read the newspaper, which is: 0.30 x 50,000 = 15,000 

So, the number of voters who read the newspaper but do not watch the local cable television news is: 90,000 - 15,000 = 75,000 


ii. To find the number of voters who read the newspaper or watch the local cable television news, or both, we can use a Venn diagram.

Let N represent the set of voters who read the newspaper, and C represent the set of voters who watch the local cable news. The overlap between N and C represents the number of voters who both read the newspaper and watch the local cable news. We can fill in the diagram with the following information:

  • 36% of voters read the newspaper, which is 90,000
  • 50,000 voters watch the local cable news
  • 30% of local cable news viewers also read the newspaper, which is 15,000

Using these values, we can calculate the number of voters in each section of the Venn diagram:

  • N only = 90,000 - 15,000 = 75,000
  • C only = 50,000 - 15,000 = 35,000
  • N and C = 15,000

The total number of voters who read the newspaper or watch the local cable news, or both, is: N only + C only + N and C = 75,000 + 35,000 + 15,000 = 125,000


 iii. We can complete the following chart by using the information given in the problem: 


Promotional Option
Number of Voters Reached
Total Cost
Cost per Voter Reached
 Pamphlets
250,000
$225,000
 $0.90
Television
 50,000
$40,000
$0.80
  Newspaper
 90,000
 $27,000
$0.30


 iv. Based on the information in the chart, the most cost-effective plan would be to use the newspaper advertisement to reach 90,000 voters at a cost of $0.30 per voter, and then use the television commercial to reach an additional 35,000 voters (the number who watch the local cable news but do not read the newspaper) at a cost of $0.80 per voter. This would bring the total number of voters reached to 125,000, which is the minimum required for success. The cost of this plan would be: $27,000 + $40,000 = $67,000.

  • 25 Dec, 2023

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