Find 5 rational numbers between `-(1)/(6)and(5)/(21)`.

Explanation

To find 5 rational numbers between \(-\frac{1}{6}\) and \(\frac{5}{21}\), we can follow these steps:

### Step 1: Find the LCM of the denominators.
The denominators are 6 and 21. We need to find the least common multiple (LCM) of these two numbers.

- The prime factorization of 6 is \(2 \times 3\).
- The prime factorization of 21 is \(3 \times 7\).

The LCM is found by taking the highest power of each prime factor:
- LCM = \(2^1 \times 3^1 \times 7^1 = 42\).

### Step 2: Convert both fractions to have the same denominator.
Now we convert \(-\frac{1}{6}\) and \(\frac{5}{21}\) to have a denominator of 42.

1. For \(-\frac{1}{6}\):
\[ -\frac{1}{6} = -\frac{1 \times 7}{6 \times 7} = -\frac{7}{42} \]

2. For \(\frac{5}{21}\):
\[ \frac{5}{21} = \frac{5 \times 2}{21 \times 2} = \frac{10}{42} \]

### Step 3: Identify the range of rational numbers.
Now we have the two fractions:
- \(-\frac{7}{42}\) and \(\frac{10}{42}\).

### Step 4: Find 5 rational numbers between \(-\frac{7}{42}\) and \(\frac{10}{42}\).
We can choose any rational numbers between these two fractions. Here are five examples:

1. \(-\frac{6}{42}\)
2. \(-\frac{5}{42}\)
3. \(-\frac{4}{42}\)
4. \(-\frac{3}{42}\)
5. \(-\frac{2}{42}\)

### Final Answer:
The 5 rational numbers between \(-\frac{1}{6}\) and \(\frac{5}{21}\) are:
\(-\frac{6}{42}, -\frac{5}{42}, -\frac{4}{42}, -\frac{3}{42}, -\frac{2}{42}\).

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