Find a rational number between `(-1)/(3) and (1)/(2)`.

Answer

To find a rational number between \(-\frac{1}{3}\) and \(\frac{1}{2}\), we can use the method of finding the mean (average) of the two numbers. Here are the steps:

### Step-by-Step Solution:

1. **Identify the two rational numbers:**
- \( x = -\frac{1}{3} \)
- \( y = \frac{1}{2} \)

2. **Calculate the mean (average) of the two numbers:**
\[ \text{Mean} = \frac{x + y}{2} \]

3. **Substitute the values of \(x\) and \(y\):**
\[ \text{Mean} = \frac{-\frac{1}{3} + \frac{1}{2}}{2} \]

4. **Find a common denominator for the fractions \(-\frac{1}{3}\) and \(\frac{1}{2}\):**
- The least common multiple (LCM) of 3 and 2 is 6.

5. **Convert the fractions to have the common denominator 6:**
\[ -\frac{1}{3} = -\frac{2}{6} \]
\[ \frac{1}{2} = \frac{3}{6} \]

6. **Add the fractions:**
\[ \text{Sum} = -\frac{2}{6} + \frac{3}{6} = \frac{1}{6} \]

7. **Divide the sum by 2 to find the mean:**
\[ \text{Mean} = \frac{\frac{1}{6}}{2} = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \]

8. **Verify that \(\frac{1}{12}\) is between \(-\frac{1}{3}\) and \(\frac{1}{2}\):**
- Convert \(-\frac{1}{3}\) and \(\frac{1}{2}\) to have the denominator 12 for comparison:
\[ -\frac{1}{3} = -\frac{4}{12} \]
\[ \frac{1}{2} = \frac{6}{12} \]
- Check the inequality:
\[ -\frac{4}{12} < \frac{1}{12} < \frac{6}{12} \]

Since \(\frac{1}{12}\) is indeed between \(-\frac{1}{3}\) and \(\frac{1}{2}\), it is a valid rational number between the given numbers.

### Final Answer:
The rational number between \(-\frac{1}{3}\) and \(\frac{1}{2}\) is \(\frac{1}{12}\).