Question

What percent of the female residents of a particular county own a driver's license?
(1) Of the owners of a driver's license resident in the county, 25 percent are female.
(2) Of the non-owners of a driver's license resident in the county, 25 percent are male.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

For percentage or ratio problems with multiple population subgroups, drawing a 2x2 table (e.g., Male/Female vs. Owner/Non-Owner) can be very helpful to organize the information. If you cannot establish a link between the rows or columns of the table, the information is likely insufficient.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Let's define our variables to represent the different groups of residents.
- \(F\): Total number of female residents
- \(M\): Total number of male residents
- \(D_F\): Females who own a driver's license
- \(D_M\): Males who own a driver's license
- \(N_F\): Females who do not own a driver's license
- \(N_M\): Males who do not own a driver's license
The total number of license owners is \(D = D_F + D_M\).
The total number of non-owners is \(N = N_F + N_M\).
The question asks for the value of \(\frac{D_F}{F} \times 100%\), which is \(\frac{D_F}{D_F + N_F} \times 100%\).
Step 2: Detailed Explanation:
Analyzing Statement (1):
"Of the owners of a driver's license... 25 percent are female."
This translates to the equation: \(\frac{D_F}{D_F + D_M} = 0.25\).
This gives a relationship between \(D_F\) and \(D_M\) (\(D_F = 0.25(D_F + D_M)\) \(\implies\) \(0.75 D_F = 0.25 D_M\) \(\implies\) \(D_M = 3D_F\)).
However, we have no information about the females who do not own a license (\(N_F\)), so we cannot find the total number of females \(F\). Thus, statement (1) is not sufficient.
Analyzing Statement (2):
"Of the non-owners of a driver's license... 25 percent are male."
This translates to the equation: \(\frac{N_M}{N_F + N_M} = 0.25\).
This implies that 75 percent of non-owners are female: \(\frac{N_F}{N_F + N_M} = 0.75\).
This gives a relationship between \(N_F\) and \(N_M\) (\(N_F = 0.75(N_F + N_M)\) \(\implies\) \(0.25 N_F = 0.75 N_M\) \(\implies\) \(N_F = 3N_M\)).
This statement gives no information about females who own a license (\(D_F\)). Thus, statement (2) is not sufficient.
Combining Statements (1) and (2):
From (1), we have \(D_M = 3D_F\).
From (2), we have \(N_F = 3N_M\).
We want to find the value of \(\frac{D_F}{D_F + N_F}\).
We have relationships within the 'owner' group and within the 'non-owner' group, but there is no information linking these two groups. The total number of residents or the ratio of owners to non-owners is unknown.
Let's test with numbers. Assume \(D_F = 100\). From (1), \(D_M = 300\).
Assume \(N_M = 100\). From (2), \(N_F = 300\).
In this case, the percentage is \(\frac{D_F}{D_F + N_F} = \frac{100}{100 + 300} = \frac{100}{400} = 25%\).
Now, let's change one assumption. Let \(D_F = 100\) (\(D_M = 300\)), but let \(N_M = 200\). From (2), \(N_F = 600\).
In this case, the percentage is \(\frac{D_F}{D_F + N_F} = \frac{100}{100 + 600} = \frac{100}{700} \approx 14.3%\).
Since we get different answers depending on the relative sizes of the owner and non-owner populations, the statements together are not sufficient.
Step 3: Final Answer:
Even with both statements, we cannot determine a unique percentage. Therefore, the correct option is (E).