To solve the problem of determining the number which [tex]\(\left(-\frac{1}{4}\right)^{-1}\)[/tex] should be multiplied by to result in the reciprocal of [tex]\(\left(-\frac{7}{4}\right)^{-1}\)[/tex], let's break down the steps methodically.
### Step 1: Compute [tex]\(\left(-\frac{1}{4}\right)^{-1}\)[/tex]
The expression [tex]\(\left(-\frac{1}{4}\right)^{-1}\)[/tex] is the reciprocal of [tex]\(-\frac{1}{4}\)[/tex]. The reciprocal of a fraction [tex]\(\frac{a}{b}\)[/tex] is [tex]\(\frac{b}{a}\)[/tex].
[tex]\[
\left(-\frac{1}{4}\right)^{-1} = -4
\][/tex]
### Step 2: Compute [tex]\(\left(-\frac{7}{4}\right)^{-1}\)[/tex]
Similarly, [tex]\(\left(-\frac{7}{4}\right)^{-1}\)[/tex] is the reciprocal of [tex]\(-\frac{7}{4}\)[/tex].
[tex]\[
\left(-\frac{7}{4}\right)^{-1} = -\frac{4}{7}
\][/tex]
### Step 3: Find the reciprocal of [tex]\(\left(-\frac{7}{4}\right)^{-1}\)[/tex]
The reciprocal of [tex]\(\left(-\frac{7}{4}\right)^{-1}\)[/tex] or [tex]\(-\frac{4}{7}\)[/tex] is:
[tex]\[
\frac{1}{\left(-\frac{4}{7}\right)} = -\frac{7}{4}
\][/tex]
### Step 4: Determine the multiplier
We need to find the number [tex]\(x\)[/tex] such that multiplying [tex]\(\left(-\frac{1}{4}\right)^{-1}\)[/tex] by [tex]\(x\)[/tex] equals the reciprocal of [tex]\(\left(-\frac{7}{4}\right)^{-1}\)[/tex].
[tex]\[
\left(-\frac{1}{4}\right)^{-1} \times x = -\frac{7}{4}
\][/tex]
Substitute [tex]\(\left(-\frac{1}{4}\right)^{-1}\)[/tex] with [tex]\(-4\)[/tex]:
[tex]\[
-4 \times x = -\frac{7}{4}
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-\frac{7}{4}}{-4} = \frac{7}{16} = 0.4375
\][/tex]
### Final Result
Thus, the number by which [tex]\(\left(-\frac{1}{4}\right)^{-1}\)[/tex] should be multiplied to obtain the reciprocal of [tex]\(\left(-\frac{7}{4}\right)^{-1}\)[/tex] is:
[tex]\[
\boxed{0.4375}
\][/tex]
