`{:("Column A", "n is a positive integer and 0 < x < 1","Column B"),((n^2)/x," ",n^2):}`

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 problem, we need to compare the two columns given:

**Column A:** \(\frac{n^2}{x}\)
**Column B:** \(n^2\)

Where \(n\) is a positive integer and \(0 < x < 1\).

### Step 1: Understanding the Variables
- \(n\) is any positive integer (1, 2, 3, ...).
- \(x\) is a positive fraction that lies between 0 and 1 (for example, 0.1, 0.5, etc.).

### Step 2: Analyzing Column A
Column A is \(\frac{n^2}{x}\). Since \(x\) is a positive fraction less than 1, dividing by \(x\) will make the value of \(\frac{n^2}{x}\) larger than \(n^2\).

### Step 3: Analyzing Column B
Column B is simply \(n^2\).

### Step 4: Comparing the Two Columns
To compare the two columns, we can express the comparison mathematically:
- We need to determine if \(\frac{n^2}{x} > n^2\).

This can be rewritten as:
\[ \frac{n^2}{x} > n^2 \]

### Step 5: Simplifying the Inequality
To simplify the inequality, we can multiply both sides by \(x\) (since \(x > 0\)):
\[ n^2 > n^2 \cdot x \]

### Step 6: Rearranging the Inequality
Now, we can rearrange the inequality:
\[ n^2 - n^2 \cdot x > 0 \]
\[ n^2(1 - x) > 0 \]

### Step 7: Analyzing the Result
Since \(n^2\) is always positive (as \(n\) is a positive integer) and \(1 - x\) is also positive (because \(0 < x < 1\)), we conclude that:
\[ n^2(1 - x) > 0 \]

This means that \(\frac{n^2}{x} > n^2\).

### Conclusion
Thus, Column A is always greater than Column B for any positive integer \(n\) and any positive fraction \(x\) where \(0 < x < 1\).

**Final Answer:** Column A is larger.