Question

In a certain two 65% families own cell phones, 15000 families own scooter and 15% families own both. Taking into consideration that the families own at least one of the two, the total number of families in the town is

• A. 20000
• B. 30000
• C. 40000
• D. 50000

Answer 1

The Correct Option is B

Let \( T \) be the total number of families in the town.

Let \( C \) be the set of families who own cellphones.

Let \( S \) be the set of families who own scooters.

From the problem statement, we have:

• The number of families owning cellphones is \( n(C) = 65\% \) of \( T \), which is \( n(C) = 0.65 T \).
• The number of families owning scooters is \( n(S) = 15000 \).
• The number of families owning both cellphones and scooters is \( n(C \cap S) = 15\% \) of \( T \), which is \( n(C \cap S) = 0.15 T \).

We are told that every family owns at least one of the two items. This means the union of the sets \( C \) and \( S \) is equal to the total number of families \( T \):

\[ n(C \cup S) = T \]

We apply the Principle of Inclusion-Exclusion for two sets:

\[ n(C \cup S) = n(C) + n(S) - n(C \cap S) \]

Now, substitute the known values into the formula:

\[ T = (0.65 T) + 15000 - (0.15 T) \]

Combine the terms involving \( T \) on the right side:

\[ T = (0.65 - 0.15) T + 15000 \] \[ T = 0.50 T + 15000 \]

Subtract \( 0.50 T \) from both sides of the equation:

\[ T - 0.50 T = 15000 \] \[ 0.50 T = 15000 \]

Solve for \( T \) by dividing both sides by \( 0.50 \):

\[ T = \frac{15000}{0.50} \] \[ T = 30000 \]

Therefore, the total number of families in the town is 30,000.

The correct option is (B) 30000.

Answer 2

Let T be the total number of families in the town.

Let C be the set of families who own cell phones.

Let S be the set of families who own a scooter. 

We are given the following information:

• Number of families owning cell phones, \( n(C) = 65\% \) of T \( = 0.65 T \)
• Number of families owning a scooter, \( n(S) = 15000 \)
• Number of families owning both, \( n(C \cap S) = 15\% \) of T \( = 0.15 T \)

We are also told that every family owns at least one of the two items. This means the union of the two sets is equal to the total number of families:

\[ n(C \cup S) = T \]

We use the Principle of Inclusion-Exclusion for two sets:

\[ n(C \cup S) = n(C) + n(S) - n(C \cap S) \]

Now, substitute the given values and the condition \( n(C \cup S) = T \) into the formula:

\[ T = (0.65 T) + 15000 - (0.15 T) \]

Combine the terms involving T on the right side:

\[ T = (0.65 - 0.15) T + 15000 \]

\[ T = 0.50 T + 15000 \]

Subtract \( 0.50 T \) from both sides of the equation to isolate T:

\[ T - 0.50 T = 15000 \]

\[ (1 - 0.50) T = 15000 \]

\[ 0.50 T = 15000 \]

Finally, solve for T by dividing by 0.50:

\[ T = \frac{15000}{0.50} \]

\[ T = \frac{15000}{1/2} \]

\[ T = 15000 \times 2 \]

\[ T = 30000 \]

Therefore, the total number of families in the town is 30000.

Comparing this result with the given options:

• 20000
• 30000
• 40000
• 50000

The correct option is 30000.