Find six rational numbers between `3` and `4` .

Answer

To find six rational numbers between 3 and 4, we can follow these steps:

### Step 1: Understand the range
We need to find rational numbers that lie between the integers 3 and 4.

### Step 2: Convert the integers to fractions
We can express 3 and 4 as fractions:
- \( 3 = \frac{3}{1} \)
- \( 4 = \frac{4}{1} \)

### Step 3: Find a common denominator
To find rational numbers between these two fractions, we can convert them to have a common denominator. A simple way to do this is to multiply both fractions by a number that gives us a larger denominator. Let's multiply both by 6:
- \( 3 = \frac{3 \times 6}{1 \times 6} = \frac{18}{6} \)
- \( 4 = \frac{4 \times 6}{1 \times 6} = \frac{24}{6} \)

### Step 4: Identify the range of numerators
Now we have:
- \( \frac{18}{6} \) and \( \frac{24}{6} \)

The numerators between 18 and 24 are 19, 20, 21, 22, and 23.

### Step 5: Write the rational numbers
Now we can write the rational numbers using the common denominator:
1. \( \frac{19}{6} \)
2. \( \frac{20}{6} \)
3. \( \frac{21}{6} \)
4. \( \frac{22}{6} \)
5. \( \frac{23}{6} \)

### Step 6: Add one more rational number
To find a sixth rational number, we can take the average of 3 and 4, which is \( \frac{3.5}{1} \) or \( \frac{7}{2} \). We can convert this to have a denominator of 6:
- \( \frac{7}{2} = \frac{7 \times 3}{2 \times 3} = \frac{21}{6} \) (but this is already included)

Instead, we can choose another rational number:
- \( \frac{24}{6} \) is not included, so we can take \( \frac{25}{6} \).

### Final List of Rational Numbers
Thus, the six rational numbers between 3 and 4 are:
1. \( \frac{19}{6} \)
2. \( \frac{20}{6} \)
3. \( \frac{21}{6} \)
4. \( \frac{22}{6} \)
5. \( \frac{23}{6} \)
6. \( \frac{25}{6} \)