Question

In a certain class, some students donated cans of food to a local food bank. What was the average (arithmetic mean) number of cans donated per student in the class?
(1) The students donated a total of 56 cans of food.
(2) The total number of cans donated was 40 greater than the total number of students in the class.

• 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:

To find an average, you need two pieces of information: the total sum and the number of items. In a Data Sufficiency context, look for one statement that gives you the total and another that gives you the count, or two equations that allow you to solve for both.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency question. Let \(C\) be the total number of cans donated and \(S\) be the total number of students. The question asks for the average number of cans per student, which is the value of \(C/S\).
Step 2: Detailed Explanation:
Analyze Statement (1):
This statement tells us that \(C = 56\).
The average is \(C/S = 56/S\).
Since the number of students \(S\) is unknown, we cannot find a unique value for the average. Therefore, statement (1) alone is not sufficient.
Analyze Statement (2):
This statement tells us that \(C = S + 40\).
The average is \(C/S = (S + 40)/S = 1 + 40/S\).
Since the value of \(S\) is unknown, we cannot determine a unique average. Therefore, statement (2) alone is not sufficient.
Analyze Statements (1) and (2) Together:
From statement (1), we have \(C = 56\).
From statement (2), we have \(C = S + 40\).
We can set the two expressions for \(C\) equal to each other:
\[ 56 = S + 40 \]
Solving for \(S\), we get:
\[ S = 56 - 40 = 16 \]
Now we have unique values for both the total cans (\(C = 56\)) and the total students (\(S = 16\)). We can find the average:
\[ \text{Average} = \frac{C}{S} = \frac{56}{16} = \frac{7}{2} = 3.5 \]
Since we found a single, unique value for the average, both statements together are sufficient.
Step 3: Final Answer:
Neither statement is sufficient on its own, but combined they provide enough information. This corresponds to option (C).