In a certain town, 25% of the families own a phone and 15% own a car, 65% families own neither a phone nor a car and 2,000 families own both a car and a phone. Consider the following three statements :
(A) 5% families own both a car and a phone
(B) 35% families own either a car or a phone
(C ) 40,000 families live in the town
Then,

Options

A. Only (A) and (C ) are correct
B. Only (B) and (C ) are correct
C. All (A), (B) and (C ) are correct
D. Only (A) and (B) are correct

Answer

To solve the problem step by step, we will analyze the information given and use set theory to find the required values.

### Step 1: Define the Variables
Let \( x \) be the total number of families in the town.

### Step 2: Identify the Percentages
- Families owning a phone: 25% of \( x \)
- Families owning a car: 15% of \( x \)
- Families owning neither a phone nor a car: 65% of \( x \)
- Families owning both a car and a phone: 2000 families

### Step 3: Calculate Families Owning Either a Phone or a Car
Since 65% of families own neither a phone nor a car, the percentage of families that own either a phone or a car is:
\[ 100\% - 65\% = 35\% \]
Thus, the number of families owning either a phone or a car is:
\[ 0.35x \]

### Step 4: Use the Principle of Inclusion-Exclusion
According to the principle of inclusion-exclusion for two sets:
\[ n(P \cup C) = n(P) + n(C) - n(P \cap C) \]
Where:
- \( n(P) = 0.25x \) (families with a phone)
- \( n(C) = 0.15x \) (families with a car)
- \( n(P \cap C) = 2000 \) (families with both)

Substituting the values we have:
\[ 0.35x = 0.25x + 0.15x - 2000 \]

### Step 5: Simplify the Equation
Combine like terms:
\[ 0.35x = 0.40x - 2000 \]
Rearranging gives:
\[ 0.35x - 0.40x = -2000 \]
\[ -0.05x = -2000 \]

### Step 6: Solve for \( x \)
Dividing both sides by -0.05:
\[ x = \frac{2000}{0.05} = 40000 \]

### Step 7: Verify Each Statement
1. **Statement A**: 5% of families own both a car and a phone.
- Calculation:
\[ \frac{2000}{40000} \times 100 = 5\% \]
- **True**.

2. **Statement B**: 35% of families own either a car or a phone.
- Calculation:
\[ 0.35 \times 40000 = 14000 \text{ families} \]
- **True**.

3. **Statement C**: 40,000 families live in the town.
- **True**.

### Conclusion
All three statements (A, B, and C) are correct.

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