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 step by step, we will use the principle of inclusion-exclusion.

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

From the problem, we have:
- \( n(A) = 60\% = 0.6 \)
- \( n(B) = 80\% = 0.8 \)
- \( n(C) = 40\% = 0.4 \)

### Step 2: Define the Intersections
We also have the following intersections:
- \( n(A \cap B) = 30\% = 0.3 \) (families owning both a car and a scooter)
- \( n(A \cap C) = 35\% = 0.35 \) (families owning both a car and a bicycle)
- \( n(B \cap C) = 25\% = 0.25 \) (families owning both a scooter and a bicycle)

Let \( x \) be the percentage of families that own all three vehicles (car, scooter, and bicycle).

### Step 3: Use the Inclusion-Exclusion Principle
According to the principle of inclusion-exclusion, the total percentage of families owning at least one of the vehicles is given by:

\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \]

Substituting the known values:

\[ n(A \cup B \cup C) = 0.6 + 0.8 + 0.4 - 0.3 - 0.35 - 0.25 + x \]

### Step 4: Simplify the Equation
Now, simplify the equation:

\[ n(A \cup B \cup C) = 1.8 - 0.3 - 0.35 - 0.25 + x \]
\[ = 1.8 - 0.9 + x \]
\[ = 0.9 + x \]

### Step 5: Calculate Families with Neither Vehicle
The percentage of families that own neither a car, scooter, nor bicycle is given by:

\[ n(\text{neither}) = 1 - n(A \cup B \cup C) \]
\[ = 1 - (0.9 + x) \]
\[ = 0.1 - x \]

### Step 6: Determine the Constraints
Since \( x \) (the percentage of families owning all three vehicles) must be a non-negative number, we have:

\[ 0.1 - x \geq 0 \]
\[ x \leq 0.1 \]

### Step 7: Find the Maximum Value of \( x \)
To find the maximum possible value of \( x \), we consider that the total percentage of families owning at least one vehicle must not exceed 100%. Thus, we set:

\[ 0.9 + x \leq 1 \]
\[ x \leq 0.1 \]

### Step 8: Conclusion
The families who own neither vehicle can be expressed as:

\[ n(\text{neither}) = 0.1 - x \]

Since \( x \) can be any value from 0 to 0.1, the maximum families who own neither would be when \( x \) is at its minimum (0), giving us:

\[ n(\text{neither}) = 0.1 - 0 = 0.1 \]

However, since we know some families own all three, the actual value must be less than 0.1. Testing values, we find that the only valid option provided in the question that satisfies all conditions is \( 0.07 \).

### Final Answer
The families who have neither of the three vehicles can be **7%**.

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