The product of three rational numbes is `( - (4)/(11))` If two of the numbers are `(- (2)/(3)) and (- (1)/(11)),` then find the third number.

Options

A. `-6`
B. 6
C. 11
D. `-11`

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Information:**
We know that the product of three rational numbers \( x, y, z \) is given as:
\[ x \cdot y \cdot z = -\frac{4}{11} \]
Two of the numbers are:
\[ x = -\frac{2}{3}, \quad y = -\frac{1}{11} \]
We need to find \( z \).

2. **Substitute the Known Values:**
Substitute the values of \( x \) and \( y \) into the equation:
\[ -\frac{2}{3} \cdot -\frac{1}{11} \cdot z = -\frac{4}{11} \]

3. **Calculate the Product of \( x \) and \( y \):**
First, calculate the product of \( x \) and \( y \):
\[ -\frac{2}{3} \cdot -\frac{1}{11} = \frac{2}{3 \cdot 11} = \frac{2}{33} \]

4. **Set Up the Equation:**
Now, substitute this product back into the equation:
\[ \frac{2}{33} \cdot z = -\frac{4}{11} \]

5. **Isolate \( z \):**
To find \( z \), multiply both sides of the equation by the reciprocal of \( \frac{2}{33} \):
\[ z = -\frac{4}{11} \cdot \frac{33}{2} \]

6. **Simplify the Right Side:**
Now, perform the multiplication:
\[ z = -\frac{4 \cdot 33}{11 \cdot 2} \]
Simplifying this gives:
\[ z = -\frac{132}{22} = -6 \]

7. **Conclusion:**
Therefore, the third rational number \( z \) is:
\[ z = -6 \]