List five rational numbers between:
`- (1)/(2) and (2)/(3)`

Answer

To find five rational numbers between \(-\frac{1}{2}\) and \(\frac{2}{3}\), we can follow these steps:

### Step 1: Write the given rational numbers
We start with the two rational numbers:
\[ -\frac{1}{2} \quad \text{and} \quad \frac{2}{3} \]

### Step 2: Find a common denominator
To compare these two fractions, we need a common denominator. The denominators are 2 and 3. The least common multiple (LCM) of 2 and 3 is 6.

### Step 3: Convert both fractions to have the same denominator
Now, we convert both fractions to have a denominator of 6:

- For \(-\frac{1}{2}\):
\[ -\frac{1}{2} = -\frac{1 \times 3}{2 \times 3} = -\frac{3}{6} \]

- For \(\frac{2}{3}\):
\[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \]

Now we have:
\[ -\frac{3}{6} \quad \text{and} \quad \frac{4}{6} \]

### Step 4: Identify the range between the two fractions
The rational numbers we need to find will be between \(-\frac{3}{6}\) and \(\frac{4}{6}\).

### Step 5: Choose five rational numbers between \(-\frac{3}{6}\) and \(\frac{4}{6}\)
We can select any five rational numbers between these two fractions. Here are five examples:

1. \(-\frac{2}{6}\) (which simplifies to \(-\frac{1}{3}\))
2. \(-\frac{1}{6}\)
3. \(0\) (which can be written as \(\frac{0}{6}\))
4. \(\frac{1}{6}\)
5. \(\frac{2}{6}\) (which simplifies to \(\frac{1}{3}\))

### Final List of Rational Numbers
Thus, the five rational numbers between \(-\frac{1}{2}\) and \(\frac{2}{3}\) are:
\[ -\frac{1}{3}, -\frac{1}{6}, 0, \frac{1}{6}, \frac{1}{3} \]

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