In a certain town \(30 \%\) families own a scooter and \(40 \%\) on a car \(50 \%\) own neither a scooter nor a car 2000 families own both a scooter and car consider the following statements in this regard (1) \(20 \%\) families own both scooter and car (2) \(35 \%\) families own either a car or a scooter (3) 10000 families live in town. Which of the above statements are correct? (a) 2 and 3 (b) 1,2 and 3 (c) 1 and 2 (d) 1 and 3

We have the following information: - 30% families own a scooter - 40% families own a car - 50% families own neither a scooter nor a car - 2000 families own both a scooter and car Let's convert these percentages to fractions: - \(f_s = 0.30\) (fraction of families who own a scooter) - \(f_c = 0.40\) (fraction of families who own a car) - \(f_n = 0.50\) (fraction of families who own neither a scooter nor a car)

We know that 2000 families own both a scooter and a car. Let \(f_b\) be the fraction of families who own both a scooter and a car. We are not given the total number of families in the town, so let's use the variable \(N\) to represent it: $$f_b = \frac{2000}{N} $$

According to the principle of inclusion and exclusion, the fraction of families who own either a scooter or a car is obtained by summing the fraction of families who own a scooter and the fraction of families who own a car and subtracting the fraction of families who own both a scooter and a car: $$f_{sc} = f_s + f_c - f_b $$ Since we know the values of \(f_s\) and \(f_c\), we can express \(f_b\) in terms of these fractions: $$f_b = f_s + f_c - f_{sc} $$

Since the sum of the fractions of families who own either a scooter, a car or neither must be equal to 1, we can write the following equation: $$f_{sc} + f_n = 1 $$ From this equation, we can obtain the value of \(f_{sc}\): $$f_{sc} = 1 - f_n = 1 - 0.50 = 0.50 $$ Substitute the values of \(f_s\), \(f_c\), and \(f_{sc}\) in the equation for \(f_b\): $$f_b = 0.30 + 0.40 - 0.50 = 0.20 $$

Now we can verify if the given statements are correct: (1) 20% families own both a scooter and a car As obtained in step 4, \(f_b = 0.20\), which means that 20% families own both a scooter and a car. So, this statement is correct. (2) 35% families own either a car or a scooter As obtained in step 4, \(f_{sc} = 0.50\), which means that 50% families own either a car or a scooter. So, this statement is incorrect. (3) There are 10000 families in the town We are given that 2000 families own both a scooter and car and \(f_b = 0.20\). We can use these values to find N: $$ N = \frac{2000}{0.20} = 10000 $$ So, there are 10000 families in the town, and this statement is correct. Thus, the correct answer is (d) 1 and 3.