The product of two rational numbers is `(128)/(45).` If one of the numbers is `(-7)/(15),` then find the other rational number.

Options

A. `-128//21`
B. `128//21`
C. `-36//73`
D. `41//70`

Answer

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

1. **Identify the known values**: We know that the product of two rational numbers is \( \frac{128}{45} \) and one of the numbers is \( \frac{-7}{15} \).

2. **Set up the equation**: Let the unknown rational number be \( x \). According to the problem, we can write the equation:
\[ x \times \left( \frac{-7}{15} \right) = \frac{128}{45} \]

3. **Isolate \( x \)**: To find \( x \), we need to isolate it on one side of the equation. We can do this by multiplying both sides of the equation by the reciprocal of \( \frac{-7}{15} \):
\[ x = \frac{128}{45} \div \left( \frac{-7}{15} \right) \]
This is equivalent to:
\[ x = \frac{128}{45} \times \left( \frac{-15}{7} \right) \]

4. **Multiply the fractions**: Now we can multiply the fractions:
\[ x = \frac{128 \times -15}{45 \times 7} \]

5. **Simplify the expression**: Before multiplying, we can simplify:
- The denominator \( 45 \) can be factored as \( 3 \times 15 \).
- Thus, we can cancel \( 15 \) in the numerator and denominator:
\[ x = \frac{128 \times -1}{3 \times 7} \]

6. **Calculate the final value**: Now we can compute the final value:
\[ x = \frac{-128}{21} \]

So, the other rational number is:
\[ \boxed{\frac{-128}{21}} \]