Write five rational numbers between `(-1)/(2)and(-1)/(3)`.

Answer

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

### Step 1: Compare the two rational numbers
First, we need to compare \(-\frac{1}{2}\) and \(-\frac{1}{3}\). To do this, we can convert them to have a common denominator.

### Step 2: Find a common denominator
The denominators of \(-\frac{1}{2}\) and \(-\frac{1}{3}\) are 2 and 3. The least common multiple (LCM) of 2 and 3 is 6.

### Step 3: Convert both fractions
Now, we convert both fractions to have the common denominator of 6:
- For \(-\frac{1}{2}\):
\[ -\frac{1}{2} = -\frac{1 \times 3}{2 \times 3} = -\frac{3}{6} \]
- For \(-\frac{1}{3}\):
\[ -\frac{1}{3} = -\frac{1 \times 2}{3 \times 2} = -\frac{2}{6} \]

### Step 4: List the rational numbers between the two fractions
Now we have:
\[ -\frac{3}{6} \quad \text{and} \quad -\frac{2}{6} \]
We need to find five rational numbers between \(-\frac{3}{6}\) and \(-\frac{2}{6}\).

### Step 5: Choose a finer denominator
To find five rational numbers, we can use a finer denominator. Let's multiply both fractions by 6:
- For \(-\frac{3}{6}\):
\[ -\frac{3}{6} = -\frac{3 \times 6}{6 \times 6} = -\frac{18}{36} \]
- For \(-\frac{2}{6}\):
\[ -\frac{2}{6} = -\frac{2 \times 6}{6 \times 6} = -\frac{12}{36} \]

### Step 6: Identify five rational numbers
Now we can find five rational numbers between \(-\frac{18}{36}\) and \(-\frac{12}{36}\):
1. \(-\frac{17}{36}\)
2. \(-\frac{16}{36}\)
3. \(-\frac{15}{36}\)
4. \(-\frac{14}{36}\)
5. \(-\frac{13}{36}\)

### Final Answer
The five rational numbers between \(-\frac{1}{2}\) and \(-\frac{1}{3}\) are:
1. \(-\frac{17}{36}\)
2. \(-\frac{16}{36}\)
3. \(-\frac{15}{36}\)
4. \(-\frac{14}{36}\)
5. \(-\frac{13}{36}\)

---