Question

A certain cafeteria sells donuts and pizzas. Is the number of people who bought donuts are more than the number of people who bought pizzas?
I. Of the people who bought donuts, 30 percent of them also bought pizzas.
II. Of the people who bought pizzas, 40 percent of them also bought donuts.

• A. Statement I alone is sufficient but statement II alone is not sufficient to answer the question asked.
• B. Statement II alone is sufficient but statement I alone is not sufficient to answer the question asked.
• C. Both statements I and II together are sufficient to answer the question but neither statement is sufficient alone.
• D. Each statement alone is sufficient to answer the question.
• E. Statements I and II are not sufficient to answer the question asked and additional data is needed to answer the statements.

Hint:

In Data Sufficiency problems involving overlapping sets, translating the percentage statements into algebraic equations is the key. Often, the number of people in the intersection ("both") can be used to link the two sets and establish a ratio between them.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency question comparing the sizes of two groups (donut buyers and pizza buyers). We need to determine if we can definitively answer "yes" or "no" to the question: Is \(D>P\)? where \(D\) is the number of people who bought donuts and \(P\) is the number of people who bought pizzas.
Step 2: Detailed Explanation:
Let \(B\) be the number of people who bought both donuts and pizzas.
Analyze Statement I: "Of the people who bought donuts, 30 percent of them also bought pizzas."
This translates to the equation: \[ B = 0.30 \times D \] This statement gives us a relationship between the number of people who bought both items and the number who bought donuts. However, it provides no information about \(P\), the number of pizza buyers. Therefore, Statement I alone is not sufficient.
Analyze Statement II: "Of the people who bought pizzas, 40 percent of them also bought donuts."
This translates to the equation: \[ B = 0.40 \times P \] This statement relates the number of people who bought both to the number who bought pizzas. It provides no information about \(D\). Therefore, Statement II alone is not sufficient.
Analyze Statements I and II Together:
From both statements, we have two different expressions for \(B\), the number of people who bought both. We can set them equal to each other: \[ 0.30 \times D = 0.40 \times P \] We can now find the ratio of \(D\) to \(P\): \[ \frac{D}{P} = \frac{0.40}{0.30} = \frac{4}{3} \] Since \(\frac{D}{P} = \frac{4}{3}\), which is greater than 1, it must be true that \(D>P\). This provides a definite "yes" to the question. Therefore, both statements together are sufficient.
Step 3: Final Answer:
Neither statement is sufficient alone, but together they provide enough information to answer the question. This corresponds to option (C).