GMAT Question of the Week #6: Data Sufficiency

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether:

• Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
• Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

A cafeteria served hamburgers and hot dogs on a certain day. If the cost of each hamburger was \$5 and the cost of each hot dog was \$3, what was the total cost of the hamburgers and hot dogs served by the cafeteria that day?

(1) The total number of hamburgers and hot dogs served was 130.
(2) The ratio of the number of hamburgers to the number of hot dogs served was 3 to 2.

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.

Explanation

Let $x$ and $y$ represent the number of hamburgers and hot dogs served in the cafeteria, respectively. We need to know if we can determine the total cost of the hamburgers and hot dogs given the hamburgers cost \$5 each and the hot dogs cost \$3 each.

(1) It is given that the total number of hamburgers and hot dogs served was 130. So, $x+y=130$. However, we don’t know the value of either $x$ or $y$, so the total cost cannot be determined.

Statement (1) alone is NOT sufficient. Eliminate Choices (A) and (D).

(2) It is given that the ratio of the number of hamburgers to the number of hot dogs served was 3 to 2. So, $\tfrac{x}{y}=\tfrac{3}{2}$. If $x=30$ and $y=20$, then $\tfrac{x}{y}=\tfrac{3}{2}$ is true. But if $x=60$ and $y=40$, then $\tfrac{x}{y}=\tfrac{3}{2}$ is also true. Since we don’t know the value of either $x$ or $y$, the total cost cannot be determined.

Statement (2) alone is NOT sufficient. Eliminate Choice (B), leaving Choices (C) and (E).

Combining the statements, we know $x+y=130$ and $\tfrac{x}{y}=\tfrac{3}{2}$, or equivalently, $x=\tfrac{3}{2}y$. We can substitute $\tfrac{3}{2}y$ for $x$ into $x+y=130$ and get the value of $y$. We can then use that to get the value of $x$ and the total cost can be determined.  Both statements together are sufficient.