If it is true that `1//55 < x < 1//22 and 1//33 < x < 1//11`, then which of the following numbers could x equal?
I. `1//54`
II. `1//23`
III. `1//12`

Options

A. I only
B. II only
C. III only
D. I and II only

Answer

To solve the problem, we need to analyze the given inequalities and determine which of the provided fractions could be a value for \( x \).

### Step-by-Step Solution:

1. **Identify the inequalities**:
We have two sets of inequalities:
- \( \frac{1}{55} < x < \frac{1}{22} \)
- \( \frac{1}{33} < x < \frac{1}{11} \)

2. **Determine the range of \( x \)**:
We need to find the intersection of the two ranges defined by the inequalities.

- From the first inequality \( \frac{1}{55} < x < \frac{1}{22} \):
- The lower bound is \( \frac{1}{55} \) and the upper bound is \( \frac{1}{22} \).

- From the second inequality \( \frac{1}{33} < x < \frac{1}{11} \):
- The lower bound is \( \frac{1}{33} \) and the upper bound is \( \frac{1}{11} \).

3. **Compare the bounds**:
To find the intersection:
- The lower bound for \( x \) will be the larger of \( \frac{1}{55} \) and \( \frac{1}{33} \).
- The upper bound for \( x \) will be the smaller of \( \frac{1}{22} \) and \( \frac{1}{11} \).

- Since \( \frac{1}{33} > \frac{1}{55} \), the lower bound is \( \frac{1}{33} \).
- Since \( \frac{1}{22} < \frac{1}{11} \), the upper bound is \( \frac{1}{22} \).

Therefore, we have:
\[ \frac{1}{33} < x < \frac{1}{22} \]

4. **Evaluate the options**:
Now we check which of the given options fall within this range.

- **Option I: \( \frac{1}{54} \)**
- Since \( 54 > 33 \), \( \frac{1}{54} < \frac{1}{33} \). Thus, \( \frac{1}{54} \) is not in the range.

- **Option II: \( \frac{1}{23} \)**
- Since \( 23 > 22 \), \( \frac{1}{23} < \frac{1}{22} \) and \( \frac{1}{23} > \frac{1}{33} \). Thus, \( \frac{1}{23} \) is in the range.

- **Option III: \( \frac{1}{12} \)**
- Since \( 12 < 22 \), \( \frac{1}{12} > \frac{1}{22} \). Thus, \( \frac{1}{12} \) is not in the range.

5. **Conclusion**:
The only option that satisfies the inequalities is:
- **Option II: \( \frac{1}{23} \)**

### Final Answer:
The correct answer is **Option II only**.

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