By which number should (-$\frac{1}{4})^{-1}$ be multiplied so that the product is the reciprocal of (-$\frac{7}{4})^{-1}$ ?

Question

Accepted Answer

To solve the problem, we need to determine by which number $(- \frac{1}{4})^{-1}$ should be multiplied so that the product is the reciprocal of $(- \frac{7}{4})^{-1}$.

### Step-by-Step Explanation:

1. **Calculate $(- \frac{1}{4})^{-1}$:**

The expression $(- \frac{1}{4})^{-1}$ represents the reciprocal of $- \frac{1}{4}$. The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. Therefore, the reciprocal of $- \frac{1}{4}$ is $-4$.

2. **Calculate the reciprocal of $(- \frac{7}{4})^{-1}$:**

First, find $(- \frac{7}{4})^{-1}$, which is the reciprocal of $- \frac{7}{4}$. The reciprocal of $- \frac{7}{4}$ is $- \frac{4}{7}$.

Now, we need the reciprocal of $- \frac{4}{7}$, which is $- \frac{7}{4}$.

3. **Determine the number to multiply by:**

We need to find a number $x$ such that:

$$
   (-4) \times x = - \frac{7}{4}
   $$

Solving for $x$:

$$
   x = \frac{- \frac{7}{4}}{-4} = \frac{7}{16}
   $$

Therefore, the number that satisfies this equation is $\frac{7}{16}$.

### Explanation of Why Each Option is Correct or Incorrect:

- **Option 1: $\frac{7}{16}$**

This is the correct answer because, as shown in the calculation above, multiplying $-4$ by $\frac{7}{16}$ results in $- \frac{7}{4}$, which is the reciprocal of $(- \frac{7}{4})^{-1}$.

- **Option 2: $\frac{16}{7}$**

This option is incorrect because multiplying $-4$ by $\frac{16}{7}$ does not yield $- \frac{7}{4}$. Instead, it would result in $- \frac{64}{7}$, which is not the desired product.

- **Option 3: $-\frac{16}{7}$**

This option is incorrect because multiplying $-4$ by $-\frac{16}{7}$ results in $\frac{64}{7}$, which is not equal to $- \frac{7}{4}$.

- **Option 4: $-\frac{7}{16}$**

This option is incorrect because multiplying $-4$ by $-\frac{7}{16}$ results in $\frac{7}{4}$, which is not the reciprocal of $(- \frac{7}{4})^{-1}$.

In conclusion, Option 1 ($\frac{7}{16}$) is the correct answer as it satisfies the condition given in the problem.