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

In a certain town `25%` families own a phone and `15%` own a car, `65%` families own neither a phone nor a car, `2000` families own both a car and a phone . How many families live in the town ?

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To solve the problem step by step, we will use the information provided about the families owning phones, cars, both, or neither. ### Step 1: Define the total number of families Let the total number of families in the town be denoted by \( x \). ### Step 2: Calculate the number of families owning a phone According to the problem, 25% of families own a phone. Therefore, the number of families that own a phone (denoted as \( P \)) can be calculated as: \[ P = \frac{25}{100} \times x = 0.25x \] ### Step 3: Calculate the number of families owning a car Similarly, 15% of families own a car. Thus, the number of families that own a car (denoted as \( C \)) is: \[ C = \frac{15}{100} \times x = 0.15x \] ### Step 4: Calculate the number of families owning neither a phone nor a car The problem states that 65% of families own neither a phone nor a car. Therefore, the number of families that own neither (denoted as \( N \)) is: \[ N = \frac{65}{100} \times x = 0.65x \] ### Step 5: Calculate the number of families owning both a phone and a car We are given that 2000 families own both a phone and a car. We denote this number as \( B \): \[ B = 2000 \] ### Step 6: Use the principle of inclusion-exclusion According to the principle of inclusion-exclusion, the total number of families can be expressed as: \[ x = P + C - B + N \] Substituting the values we have: \[ x = 0.25x + 0.15x - 2000 + 0.65x \] ### Step 7: Simplify the equation Combining the terms on the right side: \[ x = (0.25 + 0.15 + 0.65)x - 2000 \] \[ x = 1.05x - 2000 \] ### Step 8: Rearranging the equation Now, we can rearrange the equation to isolate \( x \): \[ x - 1.05x = -2000 \] \[ -0.05x = -2000 \] \[ 0.05x = 2000 \] ### Step 9: Solve for \( x \) Now, divide both sides by 0.05: \[ x = \frac{2000}{0.05} = 40000 \] ### Conclusion Thus, the total number of families living in the town is: \[ \boxed{40000} \]
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