Find 20 rational numbers between `(-5)/(6) and (5)/(8)`.

Answer

To find 20 rational numbers between \(-\frac{5}{6}\) and \(\frac{5}{8}\), we can follow these steps:

### Step 1: Convert the fractions to have a common denominator
We need to find a common denominator for \(-\frac{5}{6}\) and \(\frac{5}{8}\). The least common multiple (LCM) of 6 and 8 is 24.

### Step 2: Convert \(-\frac{5}{6}\) to a fraction with a denominator of 24
To convert \(-\frac{5}{6}\) to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 4:
\[ -\frac{5}{6} = -\frac{5 \times 4}{6 \times 4} = -\frac{20}{24} \]

### Step 3: Convert \(\frac{5}{8}\) to a fraction with a denominator of 24
To convert \(\frac{5}{8}\) to a fraction with a denominator of 24, we multiply both the numerator and the denominator by 3:
\[ \frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24} \]

### Step 4: Identify the range of rational numbers
Now we have the two fractions:
- Lower limit: \(-\frac{20}{24}\)
- Upper limit: \(\frac{15}{24}\)

### Step 5: List the rational numbers between \(-\frac{20}{24}\) and \(\frac{15}{24}\)
We can find rational numbers by taking fractions with the same denominator (24) between -20 and 15. The rational numbers are:
- \(-\frac{19}{24}\)
- \(-\frac{18}{24}\)
- \(-\frac{17}{24}\)
- \(-\frac{16}{24}\)
- \(-\frac{15}{24}\)
- \(-\frac{14}{24}\)
- \(-\frac{13}{24}\)
- \(-\frac{12}{24}\)
- \(-\frac{11}{24}\)
- \(-\frac{10}{24}\)
- \(-\frac{9}{24}\)
- \(-\frac{8}{24}\)
- \(-\frac{7}{24}\)
- \(-\frac{6}{24}\)
- \(-\frac{5}{24}\)
- \(-\frac{4}{24}\)
- \(-\frac{3}{24}\)
- \(-\frac{2}{24}\)
- \(-\frac{1}{24}\)
- \(0\)
- \(\frac{1}{24}\)
- \(\frac{2}{24}\)
- \(\frac{3}{24}\)
- \(\frac{4}{24}\)
- \(\frac{5}{24}\)
- \(\frac{6}{24}\)
- \(\frac{7}{24}\)
- \(\frac{8}{24}\)
- \(\frac{9}{24}\)
- \(\frac{10}{24}\)
- \(\frac{11}{24}\)
- \(\frac{12}{24}\)
- \(\frac{13}{24}\)
- \(\frac{14}{24}\)
- \(\frac{15}{24}\)

We can select any 20 of these rational numbers to satisfy the requirement.