Question

In a lecture hall, what is the ratio of the number of women to the number of men?
(1) There are twice as many women as men in the lecture hall.
(2) The number of men is \(\frac{1}{3}\) of the number of men and women in the lecture hall.

• 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 ratio problems, you don't need to find the actual number of items, just the relationship between them. Both statements, although worded differently, describe the same relationship: for every one man, there are two women.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Let \(W\) be the number of women and \(M\) be the number of men. The question asks for the value of the ratio \(\frac{W}{M}\).
Step 2: Detailed Explanation:
Analyzing Statement (1):
"There are twice as many women as men in the lecture hall."
This statement can be translated directly into an algebraic equation:
\[ W = 2M \]
To find the ratio of women to men, we can divide both sides by \(M\):
\[ \frac{W}{M} = 2 \]
This gives a unique value for the ratio. Therefore, statement (1) alone is sufficient.
Analyzing Statement (2):
"The number of men is \(\frac{1}{3}\) of the number of men and women in the lecture hall."
This can also be translated into an equation:
\[ M = \frac{1}{3} (M + W) \]
We can solve this equation to find the relationship between \(W\) and \(M\).
Multiply both sides by 3:
\[ 3M = M + W \]
Subtract \(M\) from both sides:
\[ 2M = W \]
This is the same relationship we found in statement (1). Dividing by \(M\), we get:
\[ \frac{W}{M} = 2 \]
This also gives a unique value for the ratio. Therefore, statement (2) alone is sufficient.
Step 3: Final Answer:
Since each statement alone is sufficient to determine the ratio of women to men, the correct answer is (D).