The product of two rational number is `(- 750)/(183).` If one of the numbers is `(7)/(12),` find the other number.

Options

A. `(-3000)/(421)`
B. `- (3001)/(427)`
C. `-7 (11)/(427)`
D. `-7 (12)/(427)`

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
We know that the product of two rational numbers is given as:
\[ \frac{-750}{183} \]
Let one of the rational numbers be \( x = \frac{7}{12} \) and the other number be \( y \). Therefore, we can write the equation:
\[ x \cdot y = \frac{-750}{183} \]
Substituting the value of \( x \):
\[ \frac{7}{12} \cdot y = \frac{-750}{183} \]

### Step 2: Isolate \( y \)
To find \( y \), we can isolate it by multiplying both sides of the equation by the reciprocal of \( \frac{7}{12} \):
\[ y = \frac{-750}{183} \cdot \frac{12}{7} \]

### Step 3: Simplify the right side
Now, we can simplify the right side of the equation. First, we can multiply the numerators and the denominators:
\[ y = \frac{-750 \cdot 12}{183 \cdot 7} \]

### Step 4: Calculate the numerator and denominator
Calculating the numerator:
\[ -750 \cdot 12 = -9000 \]
Calculating the denominator:
\[ 183 \cdot 7 = 1281 \]
So we have:
\[ y = \frac{-9000}{1281} \]

### Step 5: Simplify the fraction
Now we need to check if \( \frac{-9000}{1281} \) can be simplified. We can check for common factors. The greatest common divisor (GCD) of 9000 and 1281 is 3. Dividing both the numerator and denominator by 3:
\[ y = \frac{-9000 \div 3}{1281 \div 3} = \frac{-3000}{427} \]

### Final Answer
Thus, the other rational number is:
\[ y = \frac{-3000}{427} \]

---