Find `10` rational numbers between `-(1)/(9)` and `(4)/(9)` .

Answer

To find 10 rational numbers between \(-\frac{1}{9}\) and \(\frac{4}{9}\), we can follow these steps:

### Step 1: Identify the rational numbers
We start with the two rational numbers:
- Lower limit: \(-\frac{1}{9}\)
- Upper limit: \(\frac{4}{9}\)

### Step 2: Convert to a common denominator
To make it easier to find numbers between them, we can convert both fractions to have a common denominator. The denominators are both 9, so we can multiply both fractions by 3 to convert them into a common denominator of 27.

\[ -\frac{1}{9} = -\frac{1 \times 3}{9 \times 3} = -\frac{3}{27} \]
\[ \frac{4}{9} = \frac{4 \times 3}{9 \times 3} = \frac{12}{27} \]

### Step 3: List the rational numbers between the two fractions
Now we need to find rational numbers between \(-\frac{3}{27}\) and \(\frac{12}{27}\). We can list the fractions with the denominator of 27.

The numbers between \(-3\) and \(12\) (numerators) are:
- \(-2\)
- \(-1\)
- \(0\)
- \(1\)
- \(2\)
- \(3\)
- \(4\)
- \(5\)
- \(6\)
- \(7\)
- \(8\)
- \(9\)
- \(10\)
- \(11\)

### Step 4: Write the rational numbers
Now we convert these numerators back to fractions with a denominator of 27:

1. \(-\frac{2}{27}\)
2. \(-\frac{1}{27}\)
3. \(\frac{0}{27}\)
4. \(\frac{1}{27}\)
5. \(\frac{2}{27}\)
6. \(\frac{3}{27}\)
7. \(\frac{4}{27}\)
8. \(\frac{5}{27}\)
9. \(\frac{6}{27}\)
10. \(\frac{7}{27}\)

### Final Answer
The 10 rational numbers between \(-\frac{1}{9}\) and \(\frac{4}{9}\) are:
- \(-\frac{2}{27}, -\frac{1}{27}, 0, \frac{1}{27}, \frac{2}{27}, \frac{3}{27}, \frac{4}{27}, \frac{5}{27}, \frac{6}{27}, \frac{7}{27}\)