Question

A certain voting bloc has how many voters?
(1) If no additional voters are added to the bloc, and 4 of the current voters leave the bloc, there will be fewer than 20 voters.
(2) If 4 more voters join the bloc and all of the present voters remain, there will be at least 27 voters.

• 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:

When dealing with inequalities in Data Sufficiency, combining statements means finding the intersection of the possible values. If this intersection contains only one integer (for questions about integer quantities), the statements are sufficient together.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency problem where we need to find a specific integer value (the number of voters). The statements provide inequalities related to this value.
Step 2: Key Formula or Approach:
Let \(V\) be the number of voters in the bloc. The question asks for the value of \(V\). We must translate the statements into mathematical inequalities.
Step 3: Detailed Explanation:
Analyzing Statement (1):
"If... 4 of the current voters leave the bloc, there will be fewer than 20 voters."
This translates to the inequality:
\[ V - 4<20 \] Adding 4 to both sides gives:
\[ V<24 \] This tells us the number of voters is less than 24. It could be 23, 22, 21, etc. Since there is no unique value for \(V\), statement (1) is not sufficient.
Analyzing Statement (2):
"If 4 more voters join the bloc... there will be at least 27 voters."
"At least 27" means 27 or more. This translates to the inequality:
\[ V + 4 \geq 27 \] Subtracting 4 from both sides gives:
\[ V \geq 23 \] This tells us the number of voters is 23 or greater. It could be 23, 24, 25, etc. Since there is no unique value for \(V\), statement (2) is not sufficient.
Analyzing Both Statements Together:
From statement (1), we have \(V<24\).
From statement (2), we have \(V \geq 23\).
We need to find a value for \(V\) that satisfies both inequalities. Since the number of voters must be an integer, the only integer that is greater than or equal to 23 AND less than 24 is 23.
\[ 23 \leq V<24 \implies V = 23 \] Together, the statements lead to a unique value for \(V\). Therefore, the statements together are sufficient.
Step 4: Final Answer:
Neither statement alone is sufficient, but both statements together are sufficient to determine the number of voters.