Question

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,

• A. Only (a) and (b) are correct
• B. Only (a) and (c) are correct
• C. Only (b) and (c) are correct
• D. All (a), (b) and (c) are correct

Answer 1

The Correct Option is D

To solve the given problem, let's analyze the information provided and the statements (a), (b), and (c) to determine their correctness.

The given information is as follows:

• 25\% of families own a phone.
• 15\% of families own a car.
• 65\% of families own neither a phone nor a car.
• 2,000 families own both a car and a phone.

Using this information, we can calculate:

1. The percentage of families that own both a phone and a car (P(A \cap B)). Given that 2,000 families own both, it is \left(\frac{2000}{\text{Total number of families}}\right) \times 100\%.
2. The percentage of families that own either a phone or a car using the formula for union: P(A \cup B) = P(A) + P(B) - P(A \cap B).
3. The total number of families using the fact that 65\% own neither, meaning 100\% - 65\% = 35\% own either a phone, a car, or both.

Let's perform these calculations:

1. Since 65\% of families own neither, 35\% of families own either a phone or a car or both. Given this is equal to the union, statement (b) is correct if this percentage matches. We'll verify this calculation:
2. Using the union formula:
   P(A \cup B) = P(A) + P(B) - P(A \cap B)
   35\% = 25\% + 15\% - P(A \cap B)
   \Hence, P(A \cap B) = 25\% + 15\% - 35\% = 5\%. This matches 5\% which verifies (a).
3. To find the total number of families, equate:
   P(A \cap B) = \frac{2000}{\text{Total number of families}} \times 100\% = 5\%
   Therefore, \text{Total number of families} = \frac{2000}{0.05} = 40,000. This verifies (c).

Thus, all these calculations confirm statements (a), (b), and (c) are correct:

• (a) 5\% of families own both a car and a phone.
• (b) 35\% of families own either a car or a phone.
• (c) 40,000 families live in the town.

Correct Option: All (a), (b), and (c) are correct.