The sum of three rational numbers is `(-6)/(5).` If two of the numbers are `(3)/(10) and (-9)/(15),` then find the third.

Options

A. `(9)/(10)`
B. `(-9)/(10)`
C. `(-21)/(10)`
D. `(21)/(10)`

Answer

To find the third rational number when the sum of three rational numbers is given, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- The sum of three rational numbers is \(-\frac{6}{5}\).
- The two known rational numbers are \(\frac{3}{10}\) and \(-\frac{9}{15}\).

2. **Convert the Second Number to a Simplified Form**:
- The second number \(-\frac{9}{15}\) can be simplified.
- \(-\frac{9}{15} = -\frac{3}{5}\) (by dividing both the numerator and denominator by 3).

3. **Set Up the Equation**:
- Let the third rational number be \(z\).
- The equation representing the sum of the three numbers is:
\[ \frac{3}{10} - \frac{3}{5} + z = -\frac{6}{5} \]

4. **Find a Common Denominator**:
- The denominators are 10 and 5. The least common multiple (LCM) of 10 and 5 is 10.
- Rewrite \(-\frac{3}{5}\) with a denominator of 10:
\[ -\frac{3}{5} = -\frac{6}{10} \]

5. **Substitute Back into the Equation**:
- Now, substitute the values into the equation:
\[ \frac{3}{10} - \frac{6}{10} + z = -\frac{6}{5} \]

6. **Combine Like Terms**:
- Combine \(\frac{3}{10}\) and \(-\frac{6}{10}\):
\[ -\frac{3}{10} + z = -\frac{6}{5} \]

7. **Isolate \(z\)**:
- To isolate \(z\), add \(\frac{3}{10}\) to both sides:
\[ z = -\frac{6}{5} + \frac{3}{10} \]

8. **Convert \(-\frac{6}{5}\) to a Common Denominator**:
- Convert \(-\frac{6}{5}\) to a denominator of 10:
\[ -\frac{6}{5} = -\frac{12}{10} \]

9. **Perform the Addition**:
- Now, substitute and perform the addition:
\[ z = -\frac{12}{10} + \frac{3}{10} = -\frac{12 - 3}{10} = -\frac{9}{10} \]

10. **Conclusion**:
- The third rational number \(z\) is \(-\frac{9}{10}\).

### Final Answer:
The third rational number is \(-\frac{9}{10}\).