A survey shows that in a city 60% familes own a car, 80% families have a scooter, and 40% have a bicycle. Also 30% own both a Car and scooter, 35% Car and bicycle and 25% scooter and bicycle, and some families owns all the three. Now the families who have neither of the three can be

Options

A. 0.07
B. 0.11
C. 0.12
D. 0.15

Explanation

To solve the problem, we will use the principle of inclusion-exclusion to find the number of families that own at least one of the three vehicles (car, scooter, bicycle) and then determine how many families own none of the vehicles.

### Step-by-Step Solution:

1. **Define Sets**:
- Let \( A \) be the set of families that own a car.
- Let \( B \) be the set of families that own a scooter.
- Let \( C \) be the set of families that own a bicycle.

2. **Given Data**:
- \( N(A) = 60\% \) (families that own a car)
- \( N(B) = 80\% \) (families that own a scooter)
- \( N(C) = 40\% \) (families that own a bicycle)
- \( N(A \cap B) = 30\% \) (families that own both a car and a scooter)
- \( N(A \cap C) = 35\% \) (families that own both a car and a bicycle)
- \( N(B \cap C) = 25\% \) (families that own both a scooter and a bicycle)
- Let \( N(A \cap B \cap C) = x\% \) (families that own all three)

3. **Using the Inclusion-Exclusion Principle**:
We need to find \( N(A \cup B \cup C) \):
\[ N(A \cup B \cup C) = N(A) + N(B) + N(C) - N(A \cap B) - N(A \cap C) - N(B \cap C) + N(A \cap B \cap C) \]
Substituting the values:
\[ N(A \cup B \cup C) = 60 + 80 + 40 - 30 - 35 - 25 + x \]
Simplifying this gives:
\[ N(A \cup B \cup C) = 60 + 80 + 40 - 30 - 35 - 25 + x = 90 + x \]

4. **Finding Families with None of the Vehicles**:
The total percentage of families is 100%. Therefore, the percentage of families that own neither a car, scooter, nor bicycle is:
\[ N(\text{neither}) = 100 - N(A \cup B \cup C) \]
Substituting the expression we found:
\[ N(\text{neither}) = 100 - (90 + x) = 10 - x \]

5. **Constraints on \( x \)**:
Since \( x \) represents the percentage of families that own all three vehicles, it must be greater than 0 (as stated in the problem that some families own all three). Thus:
\[ 0 < x < 10 \]
This means \( 10 - x \) will be between 0 and 10.

6. **Conclusion**:
The families who have neither of the three vehicles can be expressed as:
\[ N(\text{neither}) = 10 - x \]
Since \( x \) can take any value from 0 to 10 (but not including 0), the number of families who own neither vehicle will range from 0% to 10%.

### Final Answer:
The families who have neither of the three can be between 0% to 10%.