How many of the surveyed adults watched the following?

Question

In a survey, 676 adults were asked what television programs they had recently watched. The following information was obtained: 222 watched neither the Big Game nor the New Movie, and 285 watched the New Movie. If 183 of those who watched the New Movie did not watch the Big Game, how many of the surveyed adults watched the following?

Answer 1

I assume that this is an exercise in Venn diagrams, so I will use that terminology.

It would be helpful for you to draw a Venn diagram of the situation.

Let

B be the set of people who watched the Big Game,

N be the set of people who watched the New Movie.

The set of people who watched at least one movie is a union of three disjoint sets:

B∩N, which denotes the set of people who watched both movies,

B\N, which denotes the set of people who watched the Big Game, but not the New Movie,

N\B, which denotes the set of people who watched the New Movie, but not the Big Game.

We will calculate the sizes of those three sets |B\N|, |B∩N|, |N\B|,

which will allow you to answer any questions about the survey.

The information

"676 adults were asked what television programs they had recently watched"

together with

"222 watched neither the Big Game nor the New Movie"

means

|B\N| + |B∩N| + |N\B| = 676 - 222 = 454 (1)

The statement

"285 watched the New Movie"

means

|B∩N| + |N\B| = 285 (2)

Combined with (1) gives

|B\N| = 454 - 285 = 169

The statement

"183 of those who watched the New Movie did not watch the Big Game"

means

|N\B| = 183

Combined with (2)

|B∩N| = 285 - 183 = 102