The product of two rational numbers is `(8)/(9)`. If one of them is `-(5)/(6)`, find the other.

Answer

To find the other rational number when the product of two rational numbers is \( \frac{8}{9} \) and one of them is \( -\frac{5}{6} \), we can follow these steps:

### Step 1: Set up the equation
Let the other rational number be \( x \). According to the problem, we have:
\[ -\frac{5}{6} \times x = \frac{8}{9} \]

### Step 2: Isolate \( x \)
To find \( x \), we need to isolate it. We can do this by dividing both sides of the equation by \( -\frac{5}{6} \):
\[ x = \frac{8}{9} \div -\frac{5}{6} \]

### Step 3: Convert division to multiplication
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the equation as:
\[ x = \frac{8}{9} \times -\frac{6}{5} \]

### Step 4: Multiply the fractions
Now, we can multiply the fractions:
\[ x = \frac{8 \times -6}{9 \times 5} \]
Calculating the numerator and denominator:
\[ x = \frac{-48}{45} \]

### Step 5: Simplify the fraction
Next, we simplify \( \frac{-48}{45} \). The greatest common divisor (GCD) of 48 and 45 is 3. Therefore, we divide both the numerator and the denominator by 3:
\[ x = \frac{-16}{15} \]

### Conclusion
Thus, the other rational number is:
\[ x = -\frac{16}{15} \]

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