The sum of two rational numbers is `(-1)/(2)`. If one of them is `(-9)/(10)`, find the other.

Answer

To find the other rational number when the sum of two rational numbers is \(-\frac{1}{2}\) and one of them is \(-\frac{9}{10}\), we can follow these steps:

### Step-by-Step Solution:

1. **Set Up the Equation**:
We know that the sum of two rational numbers \(x\) and \(y\) is given by:
\[ x + y = -\frac{1}{2} \]
Here, we are given that \(y = -\frac{9}{10}\).

2. **Substitute the Known Value**:
Substitute \(y\) into the equation:
\[ x - \frac{9}{10} = -\frac{1}{2} \]

3. **Isolate \(x\)**:
To find \(x\), we need to add \(\frac{9}{10}\) to both sides of the equation:
\[ x = -\frac{1}{2} + \frac{9}{10} \]

4. **Find a Common Denominator**:
The denominators of the fractions are 2 and 10. The least common denominator (LCD) is 10. We need to convert \(-\frac{1}{2}\) to a fraction with a denominator of 10:
\[ -\frac{1}{2} = -\frac{5}{10} \]

5. **Perform the Addition**:
Now we can add the two fractions:
\[ x = -\frac{5}{10} + \frac{9}{10} = \frac{9 - 5}{10} = \frac{4}{10} \]

6. **Simplify the Fraction**:
Simplifying \(\frac{4}{10}\) gives us:
\[ x = \frac{2}{5} \]

7. **Conclusion**:
Therefore, the other rational number is:
\[ \frac{2}{5} \]