The product of two rational numbers is `-2/3`. If one of them is `16/39`, find the other

Answer

To find the other rational number when the product of two rational numbers is given, we can set up the equation based on the information provided.

### Step-by-Step Solution:

1. **Set Up the Equation**:
We know that the product of two rational numbers is given as:
\[ \text{First Rational Number} \times \text{Second Rational Number} = -\frac{2}{3} \]
Let the first rational number be \( \frac{16}{39} \) and the second rational number be \( x \). Therefore, we can write:
\[ \frac{16}{39} \times x = -\frac{2}{3} \]

2. **Isolate \( x \)**:
To find \( x \), we can isolate it by dividing both sides of the equation by \( \frac{16}{39} \):
\[ x = -\frac{2}{3} \div \frac{16}{39} \]

3. **Dividing by a Fraction**:
Dividing by a fraction is the same as multiplying by its reciprocal. Thus, we can rewrite the equation as:
\[ x = -\frac{2}{3} \times \frac{39}{16} \]

4. **Multiply the Fractions**:
Now, we multiply the fractions:
\[ x = -\frac{2 \times 39}{3 \times 16} \]

5. **Simplify the Expression**:
Calculate the numerator and the denominator:
\[ x = -\frac{78}{48} \]

6. **Reduce the Fraction**:
We can simplify \( -\frac{78}{48} \) by finding the greatest common divisor (GCD) of 78 and 48, which is 6:
\[ x = -\frac{78 \div 6}{48 \div 6} = -\frac{13}{8} \]

### Final Answer:
The other rational number is:
\[ x = -\frac{13}{8} \]