`{:("Column A",x = 1/(1 + 1/(1 + 1/2)),"Column B"),(x,,1):}`

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 simplify the expression for \( x \) given in Column A and then compare it with the value in Column B.

### Step-by-Step Solution:

1. **Start with the expression for \( x \):**
\[ x = \frac{1}{1 + \frac{1}{1 + \frac{1}{2}}} \]

2. **Simplify the innermost fraction:**
- Calculate \( 1 + \frac{1}{2} \):
\[ 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \]

3. **Substitute back into the expression:**
- Now substitute \( \frac{3}{2} \) into the expression:
\[ x = \frac{1}{1 + \frac{1}{\frac{3}{2}}} \]

4. **Simplify the next fraction:**
- Calculate \( \frac{1}{\frac{3}{2}} \):
\[ \frac{1}{\frac{3}{2}} = \frac{2}{3} \]

5. **Substitute back again:**
- Now substitute \( \frac{2}{3} \) into the expression:
\[ x = \frac{1}{1 + \frac{2}{3}} \]

6. **Combine the terms in the denominator:**
- Calculate \( 1 + \frac{2}{3} \):
\[ 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \]

7. **Final calculation for \( x \):**
- Substitute \( \frac{5}{3} \) back into the expression for \( x \):
\[ x = \frac{1}{\frac{5}{3}} = \frac{3}{5} \]

8. **Convert \( x \) to decimal:**
- Calculate \( \frac{3}{5} \):
\[ \frac{3}{5} = 0.6 \]

9. **Compare with Column B:**
- Column A: \( x = 0.6 \)
- Column B: \( 1 \)

10. **Conclusion:**
- Since \( 0.6 < 1 \), we conclude that Column B is greater than Column A.

### Final Answer:
Column B is greater than Column A.