By what number should `((1)/(2)) ^(-1)` be multiplied so that the product is `((-5)/(4)) ^(-1)` ?

Options

A. `((-5)/(4)) `
B. `((-2)/(5)) `
C. `((4)/(5)) `
D. `((-1)/(4)) `

Answer

To solve the problem step by step, we need to find the number \( x \) that should be multiplied by \( \left(\frac{1}{2}\right)^{-1} \) so that the product equals \( \left(-\frac{5}{4}\right)^{-1} \).

### Step-by-Step Solution:

1. **Set up the equation:**
We start with the equation:
\[ x \cdot \left(\frac{1}{2}\right)^{-1} = \left(-\frac{5}{4}\right)^{-1} \]

2. **Simplify \( \left(\frac{1}{2}\right)^{-1} \):**
Using the property of exponents that states \( a^{-n} = \frac{1}{a^n} \), we can simplify:
\[ \left(\frac{1}{2}\right)^{-1} = 2 \]
Therefore, our equation becomes:
\[ x \cdot 2 = \left(-\frac{5}{4}\right)^{-1} \]

3. **Simplify \( \left(-\frac{5}{4}\right)^{-1} \):**
Similarly, we can simplify:
\[ \left(-\frac{5}{4}\right)^{-1} = -\frac{4}{5} \]
Now our equation is:
\[ x \cdot 2 = -\frac{4}{5} \]

4. **Solve for \( x \):**
To find \( x \), we divide both sides by 2:
\[ x = -\frac{4}{5} \div 2 \]
Dividing by 2 is the same as multiplying by \( \frac{1}{2} \):
\[ x = -\frac{4}{5} \cdot \frac{1}{2} = -\frac{4 \cdot 1}{5 \cdot 2} = -\frac{4}{10} \]

5. **Simplify \( -\frac{4}{10} \):**
We can simplify \( -\frac{4}{10} \) by dividing both the numerator and the denominator by 2:
\[ x = -\frac{2}{5} \]

### Final Answer:
The number that should be multiplied is:
\[ \boxed{-\frac{2}{5}} \]