A survey shows that 41%, 35% and 60% of the people watch “A”, “B” and “C” respectively. 27% people watch exactly two of the three movies and 3% watch none. What percentage of people watch all the three movies?

Options

A. `40%`
B. `6%`
C. `9%`
D. `12%`

Answer

To find the percentage of people who watch all three movies A, B, and C, we can follow these steps:

### Step 1: Define the total population
Let’s assume the total number of people surveyed is 100%.

### Step 2: Identify the percentages of people watching each movie
- Percentage of people watching movie A = 41%
- Percentage of people watching movie B = 35%
- Percentage of people watching movie C = 60%
- Percentage of people watching none of the movies = 3%

### Step 3: Calculate the percentage of people watching at least one movie
Since 3% of people watch none of the movies, the percentage of people watching at least one movie is:
\[ 100\% - 3\% = 97\% \]

### Step 4: Set up the equation for those watching exactly two movies
Let \( x \) be the percentage of people watching all three movies (A, B, and C). According to the problem, 27% of people watch exactly two movies.

### Step 5: Use the principle of inclusion-exclusion
The total percentage of people watching at least one movie can be expressed using the principle of inclusion-exclusion:
\[ P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C) \]
Where:
- \( P(A \cap B) \), \( P(A \cap C) \), and \( P(B \cap C) \) are the percentages of people watching exactly two movies.

Since we know that the total percentage of people watching exactly two movies is 27%, we can express this as:
\[ P(A \cap B) + P(A \cap C) + P(B \cap C) = 27\% + 3x \]
This accounts for the overlap of those who watch all three movies.

### Step 6: Substitute values into the inclusion-exclusion formula
Now substituting the values we have:
\[ 97\% = 41\% + 35\% + 60\% - (27\% + 3x) + x \]
Simplifying this:
\[ 97\% = 136\% - 27\% - 2x \]
\[ 97\% = 109\% - 2x \]
\[ 2x = 109\% - 97\% \]
\[ 2x = 12\% \]
\[ x = 6\% \]

### Conclusion
Thus, the percentage of people who watch all three movies A, B, and C is **6%**.

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