The product of two rational numbers is `(-16)/(9)`. If one of the numbers is `(-4)/(3)`, find the other.

Options

A. `(4)/(3)`
B. `(-4)/(9)`
C. `(4)/(9)`
D. `(-4)/(3)`

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Information**:
We know that the product of two rational numbers is \(-\frac{16}{9}\) and one of the numbers is \(-\frac{4}{3}\). We need to find the other number, which we will denote as \(x\).

2. **Set Up the Equation**:
According to the problem, we can express the relationship as:
\[ -\frac{16}{9} = -\frac{4}{3} \times x \]

3. **Isolate \(x\)**:
To find \(x\), we can rearrange the equation. We can do this by dividing both sides by \(-\frac{4}{3}\):
\[ x = \frac{-\frac{16}{9}}{-\frac{4}{3}} \]

4. **Simplify the Right Side**:
Dividing by a fraction is the same as multiplying by its reciprocal. Thus, we can rewrite the equation as:
\[ x = -\frac{16}{9} \times -\frac{3}{4} \]

5. **Multiply the Fractions**:
Now, we can multiply the fractions:
\[ x = \frac{16 \times 3}{9 \times 4} \]

6. **Calculate the Numerator and Denominator**:
- The numerator: \(16 \times 3 = 48\)
- The denominator: \(9 \times 4 = 36\)

Therefore, we have:
\[ x = \frac{48}{36} \]

7. **Simplify the Fraction**:
To simplify \(\frac{48}{36}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 12:
\[ x = \frac{48 \div 12}{36 \div 12} = \frac{4}{3} \]

8. **Final Result**:
Thus, the other rational number is:
\[ x = \frac{4}{3} \]

### Summary:
The other rational number is \(\frac{4}{3}\).