A rational number between `(5)/(7)` and `(9)/(11)` is .

Options

A. `(59)/(77)`
B. `(31)/(77)`
C. `(23)/(11)`
D. `(7)/(9)`

Answer

To find a rational number between \( \frac{5}{7} \) and \( \frac{9}{11} \), we can follow these steps:

### Step 1: Identify the two fractions
The two fractions we need to compare are:
- \( \frac{5}{7} \)
- \( \frac{9}{11} \)

### Step 2: Convert the fractions to a common denominator
To compare these fractions easily, we can convert them to a common denominator. The least common multiple (LCM) of the denominators 7 and 11 is 77.

#### Convert \( \frac{5}{7} \):
\[ \frac{5}{7} = \frac{5 \times 11}{7 \times 11} = \frac{55}{77} \]

#### Convert \( \frac{9}{11} \):
\[ \frac{9}{11} = \frac{9 \times 7}{11 \times 7} = \frac{63}{77} \]

### Step 3: Determine the range
Now we have:
- \( \frac{5}{7} = \frac{55}{77} \)
- \( \frac{9}{11} = \frac{63}{77} \)

We need to find a rational number \( x \) such that:
\[ \frac{55}{77} < x < \frac{63}{77} \]

### Step 4: Choose a rational number
A simple rational number that lies between \( \frac{55}{77} \) and \( \frac{63}{77} \) can be found by taking the average of the two fractions or simply choosing a number that fits the criteria.

For example, we can choose:
\[ x = \frac{59}{77} \]

### Step 5: Verify the chosen number
Now we need to verify that \( \frac{59}{77} \) is indeed between \( \frac{5}{7} \) and \( \frac{9}{11} \).

1. Check if \( \frac{55}{77} < \frac{59}{77} \):
- This is true since \( 55 < 59 \).

2. Check if \( \frac{59}{77} < \frac{63}{77} \):
- This is true since \( 59 < 63 \).

Thus, \( \frac{59}{77} \) is a rational number that lies between \( \frac{5}{7} \) and \( \frac{9}{11} \).

### Conclusion
A rational number between \( \frac{5}{7} \) and \( \frac{9}{11} \) is:
\[ \frac{59}{77} \]