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In a certain town 25% families own a cel...

In a certain town 25% families own a cellphone, 15% families own a scooter and 65% families own neither a cellphone nor a scooter. If 500 families own both a cellphone and scooter, then total number of families in the town is

A

10000

B

20000

C

30000

D

40000

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we can use the principle of inclusion-exclusion and a Venn diagram approach. ### Step 1: Define Variables Let \( X \) be the total number of families in the town. ### Step 2: Determine the Percentages According to the problem: - 25% of families own a cellphone: \( N(A) = \frac{25}{100}X = 0.25X \) - 15% of families own a scooter: \( N(B) = \frac{15}{100}X = 0.15X \) - 65% of families own neither a cellphone nor a scooter: This means that 35% of families own either a cellphone or a scooter (or both): \( N(A \cup B) = \frac{35}{100}X = 0.35X \) ### Step 3: Use the Inclusion-Exclusion Principle According to the inclusion-exclusion principle: \[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \] Where \( N(A \cap B) \) is the number of families that own both a cellphone and a scooter, which is given as 500. Substituting the values we have: \[ 0.35X = 0.25X + 0.15X - 500 \] ### Step 4: Simplify the Equation Combine the terms on the right side: \[ 0.35X = 0.40X - 500 \] ### Step 5: Rearrange the Equation Rearranging gives: \[ 0.35X - 0.40X = -500 \] \[ -0.05X = -500 \] ### Step 6: Solve for \( X \) Dividing both sides by -0.05: \[ X = \frac{-500}{-0.05} = 10000 \] ### Conclusion The total number of families in the town is \( \boxed{10000} \). ---
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