Question

In college M the average (arithmetic mean) number of students per course is 30 and the ratio of the number of students to the number of faculty is 20 to 1.
Column A: The total number of students in College M
Column B: 600

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

In problems like this, check if you have enough information. You generally need as many independent equations as you have variables to find a unique solution. With three variables (\(S, C, F\)) and only two equations relating them, a unique value for \(S\) cannot be found.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We are given two pieces of information about a college: an average and a ratio. We need to determine if this information is sufficient to find a unique value for the total number of students.
Step 2: Detailed Explanation:
Let's define variables for the unknown quantities:
\begin{itemize} \item \(S\) = Total number of students \item \(C\) = Total number of courses \item \(F\) = Total number of faculty \end{itemize} Now, let's translate the given information into equations:
1. Average students per course is 30:
\[ \frac{S}{C} = 30 \implies S = 30C \] This tells us that the total number of students must be a multiple of 30.
2. Ratio of students to faculty is 20 to 1:
\[ \frac{S}{F} = \frac{20}{1} \implies S = 20F \] This tells us that the total number of students must be a multiple of 20.
We have two equations but three unknown variables (\(S, C, F\)). We cannot solve for a unique value of \(S\). Let's demonstrate this with examples:
\begin{itemize} \item Scenario 1: Assume there are \(F=30\) faculty members. Then the number of students would be \(S = 20 \times 30 = 600\). In this case, Column A equals Column B. \item Scenario 2: Assume there are \(F=60\) faculty members. Then the number of students would be \(S = 20 \times 60 = 1200\). In this case, Column A is greater than Column B. \item Scenario 3: Assume there are \(F=15\) faculty members. Then the number of students would be \(S = 20 \times 15 = 300\). In this case, Column A is less than Column B. \end{itemize} Step 3: Final Answer:
Since the total number of students can be less than, equal to, or greater than 600 depending on the number of faculty or courses (which is unknown), the relationship cannot be determined from the information given.