List three rational numbers between `4/5 and 5/6`.

Answer

To find three rational numbers between \( \frac{4}{5} \) and \( \frac{5}{6} \), we can follow these steps:

### Step 1: Identify the two rational numbers
We have the two rational numbers:
- \( \frac{4}{5} \)
- \( \frac{5}{6} \)

### Step 2: Find a common denominator
The denominators of the two fractions are 5 and 6. We need to find the least common multiple (LCM) of these two numbers to convert them to equivalent fractions with a common denominator.

The LCM of 5 and 6 is 30.

### Step 3: Convert the fractions to have the common denominator
Now, we will convert both fractions to have a denominator of 30.

For \( \frac{4}{5} \):
\[ \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30} \]

For \( \frac{5}{6} \):
\[ \frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30} \]

Now we have:
- \( \frac{4}{5} = \frac{24}{30} \)
- \( \frac{5}{6} = \frac{25}{30} \)

### Step 4: Identify the range between the two fractions
Now we need to find three rational numbers between \( \frac{24}{30} \) and \( \frac{25}{30} \). The numerators of these fractions are 24 and 25, respectively.

### Step 5: Choose numerators between 24 and 25
The integers between 24 and 25 are 25, which is not between them. However, we can multiply both the numerator and denominator of our fractions by 4 to create a larger range.

\[ \frac{24}{30} = \frac{24 \times 4}{30 \times 4} = \frac{96}{120} \]
\[ \frac{25}{30} = \frac{25 \times 4}{30 \times 4} = \frac{100}{120} \]

Now we have:
- \( \frac{96}{120} \)
- \( \frac{100}{120} \)

### Step 6: List three rational numbers between 96 and 100
We can now find three rational numbers between 96 and 100. The integers between 96 and 100 are 97, 98, and 99.

Thus, the three rational numbers are:
- \( \frac{97}{120} \)
- \( \frac{98}{120} \)
- \( \frac{99}{120} \)

### Final Answer
The three rational numbers between \( \frac{4}{5} \) and \( \frac{5}{6} \) are:
- \( \frac{97}{120} \)
- \( \frac{98}{120} \)
- \( \frac{99}{120} \)

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