The product of two rational numbers is `(-14)/(45).` If one numbers is `(-2)/(9) ,` find the other number.

Answer

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

### Step 1: Set up the equation
Let the other number be \( x \). According to the problem, the product of the two numbers is given by:
\[ \left(-\frac{2}{9}\right) \times x = -\frac{14}{45} \]

### Step 2: Isolate \( x \)
To find \( x \), we can rearrange the equation:
\[ x = \frac{-\frac{14}{45}}{-\frac{2}{9}} \]

### Step 3: Simplify the right side
When dividing by a fraction, we multiply by its reciprocal:
\[ x = \frac{-\frac{14}{45}}{-\frac{2}{9}} = \frac{14}{45} \times \frac{9}{2} \]

### Step 4: Multiply the fractions
Now we can multiply the fractions:
\[ x = \frac{14 \times 9}{45 \times 2} \]

### Step 5: Calculate the numerator and denominator
Calculating the numerator:
\[ 14 \times 9 = 126 \]
Calculating the denominator:
\[ 45 \times 2 = 90 \]
So, we have:
\[ x = \frac{126}{90} \]

### Step 6: Simplify the fraction
Now we need to simplify \( \frac{126}{90} \):
- The greatest common divisor (GCD) of 126 and 90 is 18.
- Dividing both the numerator and denominator by 18:
\[ x = \frac{126 \div 18}{90 \div 18} = \frac{7}{5} \]

### Final Answer
Thus, the other number is:
\[ \frac{7}{5} \]

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