Question

In a recent town election, what was the ratio of the number of votes in favor of a certain proposal to the number of votes against the proposal?
(1) There were 60 more votes in favor of the proposal than against the proposal.
(2) There were 240 votes in favor of the proposal.

• 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:

In Data Sufficiency problems asking for a ratio (\(x/y\)), a single statement is only sufficient if it allows you to determine the ratio itself (e.g., \(x=3y\)), not just a relationship like \(x=y+k\). To get a specific numerical ratio, you often need the numerical values of both variables, which may require two separate equations.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency question. We need to determine if the given statements, either alone or together, provide enough information to find a unique answer to the question. Let \(F\) be the number of votes in favor and \(A\) be the number of votes against. The question asks for the ratio \(F/A\).
Step 2: Detailed Explanation:
Analyze Statement (1):
This statement tells us that \(F = A + 60\).
The ratio is \(F/A = (A + 60)/A = 1 + 60/A\).
Since we do not know the value of \(A\), we cannot determine a unique numerical value for the ratio. For example, if \(A=60\), the ratio is \(120/60 = 2\). If \(A=120\), the ratio is \(180/120 = 1.5\). Therefore, statement (1) alone is not sufficient.
Analyze Statement (2):
This statement tells us that \(F = 240\).
The ratio is \(F/A = 240/A\).
Since we do not know the value of \(A\), we cannot determine a unique ratio. Therefore, statement (2) alone is not sufficient.
Analyze Statements (1) and (2) Together:
From statement (1), we have \(F = A + 60\).
From statement (2), we have \(F = 240\).
We can substitute the value of \(F\) from the second equation into the first:
\[ 240 = A + 60 \]
Solving for \(A\), we get:
\[ A = 240 - 60 = 180 \]
Now we have unique values for both \(F\) (\(240\)) and \(A\) (\(180\)). We can find the specific ratio:
\[ \frac{F}{A} = \frac{240}{180} = \frac{24}{18} = \frac{4}{3} \]
Since we can find a single, unique ratio, both statements together are sufficient.
Step 3: Final Answer:
Neither statement alone is sufficient, but both statements together are sufficient to answer the question. This corresponds to option (C).