Question

in a survey of 250 people, it was found that 125 had read Time magazine, 175 had read Newsweek, 100 had read U. S. News, 75 had read Time and Newsweek, 60 had read Newsweek and U.S. News, 55 had read Time and U.S. News, and 25 had read all three. a) How many had read Time but not the other two? b) How many had read Time or Newsweek but not the U.S. News And World Report? c) How many had read none of these three magazines?

Answer 1

Let time magazine be represented by T

Let Newsweek be represented by N

Let US News be represented by U

We need to apply the Venn Diagram to arrive at the answers

n(T) = 125

n(N) = 175

n(U) = 100

n(T intersection N) = 75 (people reading Times and Newsweek)

n(N intersection U) = 60 (people reading Newsweek and US News)

n(T intersection U) = 55 (people reading Times and US News)

n(T intersection N intersection U) = 25 (people reading all three)

Based on the above data, we draw the venn diagram as below

Times Newsweek 250 15 20 50 65 25 35 US News

Since number of people reading all three is 25 and n(T intersection N) = 75, then 75-25 = 50. Number of people reading times and newsweek only is 50.

Since number of people reading all three is 25 and n(N intersection U) = 60, then 60-25 = 35. Number of people reading newsweek and US News only is 35.

Since number of people reading all three is 25 and n(T intersection U) = 55, then 55-25 = 35. Number of people reading Time and US News only is 30.

The total number of people surveyed is 250. The people who have read either of the three magazines is a sum of all the above i.e 20+50+65+30+25+35+10 = 235. Hence the number of people who have read neither of the three news is 250-235 = 15.

Based on these numbers, the above venn diagram is drawn. With the above diagram, the answer become very easy.

1. How many read times but not the other two

Ans : i.e n(T) exclusively. From the above diagram, it is clear that 20 people read Time magazine only.

2. How many had read Time or Newsweek but not the US News and World report.

Ans : From the above diagram, it is clear that 20+50+65 i.e 135 people had read either of Time or Newsweek and not US News.

3. How many had read none of these magazines

Ans : As explained above 250 people were surveyed, and 235 (Sum of all numbers) have had either of the three choices. Hence 250-235 i.e 15 people had not read any of these magazines.