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 umber of families in the town is

Options

A. 10000
B. 20000
C. 30000
D. 40000

Answer

To solve the problem step by step, we will use the information provided about the families in the town.

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

### Step 2: Calculate the Percentages
From the problem, we know:
- 25% of families own a cellphone:
\[ n(A) = \frac{25}{100} \times x = \frac{25x}{100} \]
- 15% of families own a scooter:
\[ n(B) = \frac{15}{100} \times x = \frac{15x}{100} \]
- 65% of families own neither a cellphone nor a scooter:
\[ n(\text{neither}) = \frac{65}{100} \times x \]

### Step 3: Calculate Families Owning Either or Both
Since 65% own neither, the percentage of families that own either a cellphone or a scooter (or both) is:
\[ n(A \cup B) = x - n(\text{neither}) = x - \frac{65}{100}x = \frac{35}{100}x \]

### Step 4: Use the Information About Families Owning Both
We are given that 500 families own both a cellphone and a scooter:
\[ n(A \cap B) = 500 \]

### Step 5: Apply the Formula for Union of Two Sets
Using the formula for the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Substituting the values we have:
\[ \frac{35}{100}x = \frac{25}{100}x + \frac{15}{100}x - 500 \]

### Step 6: Simplify the Equation
Combine the terms on the right:
\[ \frac{35}{100}x = \frac{40}{100}x - 500 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ \frac{35}{100}x - \frac{40}{100}x = -500 \]
\[ -\frac{5}{100}x = -500 \]

### Step 8: Solve for \( x \)
Multiplying both sides by -1:
\[ \frac{5}{100}x = 500 \]
Now, multiply both sides by 100:
\[ 5x = 50000 \]
Finally, divide by 5:
\[ x = 10000 \]

### Final Answer
The total number of families in the town is \( \boxed{10000} \).