Solve the following quantitative comparison problem by plugging in the number 0,1,2, -2, and `1//2` in that order - when possible.
`{:("Column A",y!=0,"Column B"),(x//y,,xy):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the quantitative comparison problem, we will evaluate the expressions in Column A and Column B by plugging in the specified values for \( x \) and \( y \). The expressions are:

- Column A: \( \frac{x}{y} \)
- Column B: \( xy \)

We will plug in the values \( 0, 1, 2, -2, \) and \( \frac{1}{2} \) in that order, ensuring that \( y \neq 0 \).

### Step 1: Plug in \( x = 0 \), \( y = 1 \)
- Column A: \( \frac{0}{1} = 0 \)
- Column B: \( 0 \times 1 = 0 \)

**Comparison:** Both columns are equal (0 = 0).

### Step 2: Plug in \( x = 1 \), \( y = 1 \)
- Column A: \( \frac{1}{1} = 1 \)
- Column B: \( 1 \times 1 = 1 \)

**Comparison:** Both columns are equal (1 = 1).

### Step 3: Plug in \( x = 2 \), \( y = 1 \)
- Column A: \( \frac{2}{1} = 2 \)
- Column B: \( 2 \times 1 = 2 \)

**Comparison:** Both columns are equal (2 = 2).

### Step 4: Plug in \( x = 2 \), \( y = 2 \)
- Column A: \( \frac{2}{2} = 1 \)
- Column B: \( 2 \times 2 = 4 \)

**Comparison:** Column A is 1 and Column B is 4 (1 < 4).

### Step 5: Plug in \( x = -2 \), \( y = 1 \)
- Column A: \( \frac{-2}{1} = -2 \)
- Column B: \( -2 \times 1 = -2 \)

**Comparison:** Both columns are equal (-2 = -2).

### Step 6: Plug in \( x = \frac{1}{2} \), \( y = 1 \)
- Column A: \( \frac{\frac{1}{2}}{1} = \frac{1}{2} \)
- Column B: \( \frac{1}{2} \times 1 = \frac{1}{2} \)

**Comparison:** Both columns are equal (\(\frac{1}{2} = \frac{1}{2}\)).

### Conclusion:
From the evaluations, we can see that:
- For \( x = 0, 1, 2, -2, \frac{1}{2} \) with \( y = 1 \) or \( y = 2 \), we have both columns equal in some cases and unequal in others.

Thus, we conclude that there is not enough information to decide which column is larger overall.