Set `S={2,3,5,2,11,1}`
`{:("Quantity A","Quantity B"),("The average of Set S","The mode of Set S if every"),(,"number in the set were"),(,"doubled"):}`

Options

A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the infromation given.

Answer

To solve the problem, we need to find the average of the set \( S = \{2, 3, 5, 2, 11, 1\} \) and the mode of the set when every number is doubled. Let's go through the steps one by one.

### Step 1: Calculate the Average of Set S

1. **Find the sum of all elements in Set S.**
\[ \text{Sum} = 2 + 3 + 5 + 2 + 11 + 1 \]
\[ \text{Sum} = 24 \]

2. **Count the total number of elements in Set S.**
\[ \text{Total number of elements} = 6 \]

3. **Calculate the average using the formula:**
\[ \text{Average} = \frac{\text{Sum of all elements}}{\text{Total number of elements}} = \frac{24}{6} = 4 \]

Thus, **Quantity A** (the average of Set S) is **4**.

### Step 2: Calculate the Mode of Set S when Every Number is Doubled

1. **Double each element in Set S:**
\[ \text{New Set} = \{2 \times 2, 2 \times 3, 2 \times 5, 2 \times 2, 2 \times 11, 2 \times 1\} = \{4, 6, 10, 4, 22, 2\} \]

2. **Identify the frequency of each element in the new set:**
- 4 occurs **2 times**
- 6 occurs **1 time**
- 10 occurs **1 time**
- 22 occurs **1 time**
- 2 occurs **1 time**

3. **Determine the mode (the most frequently occurring element):**
The mode is the element that appears most frequently. Here, **4** occurs the most (2 times).

Thus, **Quantity B** (the mode of the new set) is also **4**.

### Step 3: Compare Quantity A and Quantity B

- **Quantity A = 4**
- **Quantity B = 4**

Since both quantities are equal, we conclude that:

**The two quantities are equal.**

### Final Answer:
The correct option is **C: The two quantities are equal.**

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