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:

Translating word problems into algebraic expressions is the key. Notice that "twice as many A as B" means \(A = 2B\), and "A is 1/3 of the total of A and B" means \(A = \frac{1}{3}(A+B)\). Practice these translations to improve speed and accuracy.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Let \(W\) represent the number of women and \(M\) represent the number of men. The question asks for the value of the ratio \(\frac{W}{M}\).
Step 2: Detailed Explanation:
Analyzing Statement (1):
This statement says that the number of women is two times the number of men.
We can write this as an equation:
\[ W = 2M \]
To find the required ratio, we can divide both sides of the equation by \(M\):
\[ \frac{W}{M} = 2 \]
This statement provides a specific value for the ratio. Therefore, statement (1) alone is sufficient.
Analyzing Statement (2):
This statement says that the number of men is one-third of the total number of people (men and women).
We can write this as an equation:
\[ M = \frac{1}{3} (M + W) \]
To find the ratio \(\frac{W}{M}\), we need to rearrange this equation.
First, multiply both sides by 3:
\[ 3M = M + W \]
Next, subtract \(M\) from both sides:
\[ 2M = W \]
Finally, divide by \(M\):
\[ 2 = \frac{W}{M} \]
This statement also provides a specific value for the ratio. Therefore, statement (2) alone is sufficient.
Step 3: Final Answer:
Each statement, independently, is sufficient to find the ratio of women to men. Thus, the correct choice is (D).