Find 10 rational numbers between `(-3)/(4) and (5)/(6)` .

Answer

To find 10 rational numbers between \(-\frac{3}{4}\) and \(\frac{5}{6}\), we will follow these steps:

### Step 1: Find a common denominator
The denominators of the two fractions are 4 and 6. The least common multiple (LCM) of 4 and 6 is 12. This will be our common denominator.

### Step 2: Convert the fractions to have the common denominator
- Convert \(-\frac{3}{4}\) to a fraction with a denominator of 12:
\[ -\frac{3}{4} = -\frac{3 \times 3}{4 \times 3} = -\frac{9}{12} \]

- Convert \(\frac{5}{6}\) to a fraction with a denominator of 12:
\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \]

### Step 3: List the rational numbers between \(-\frac{9}{12}\) and \(\frac{10}{12}\)
Now that we have the two fractions as \(-\frac{9}{12}\) and \(\frac{10}{12}\), we can find rational numbers between them. The rational numbers will be of the form \(\frac{n}{12}\) where \(n\) is an integer between -9 and 10.

The integers between -9 and 10 are:
- -8
- -7
- -6
- -5
- -4
- -3
- -2
- -1
- 0
- 1
- 2
- 3
- 4
- 5
- 6
- 7
- 8
- 9

### Step 4: Write the rational numbers
The 10 rational numbers between \(-\frac{9}{12}\) and \(\frac{10}{12}\) are:
1. \(-\frac{8}{12}\)
2. \(-\frac{7}{12}\)
3. \(-\frac{6}{12}\)
4. \(-\frac{5}{12}\)
5. \(-\frac{4}{12}\)
6. \(-\frac{3}{12}\)
7. \(-\frac{2}{12}\)
8. \(-\frac{1}{12}\)
9. \(0\) (which can be written as \(\frac{0}{12}\))
10. \(\frac{1}{12}\)

### Final Answer
The 10 rational numbers between \(-\frac{3}{4}\) and \(\frac{5}{6}\) are:
\[ -\frac{8}{12}, -\frac{7}{12}, -\frac{6}{12}, -\frac{5}{12}, -\frac{4}{12}, -\frac{3}{12}, -\frac{2}{12}, -\frac{1}{12}, 0, \frac{1}{12} \]