Discussion
By which number should \(\left(-\dfrac{1}{4}\right)^{-1}\) be multiplied so that the product is the reciprocal of \(\left(-\dfrac{7}{4}\right)^{-1}\) ?
Answer: A
The number is 7/16.
\[ \begin{aligned} \left(-\tfrac{1}{4}\right)^{-1} &= -4 \\ \left(-\tfrac{7}{4}\right)^{-1} &= -\tfrac{4}{7} \\ \text{its reciprocal} &= -\tfrac{7}{4} \\ -4 \times k &= -\tfrac{7}{4} \Rightarrow k = \tfrac{7}{16} \end{aligned} \]
A check: \(-4 \times \tfrac{7}{16} = -\tfrac{7}{4}\). The answer is positive because a negative number must be multiplied by a positive one to give another negative number, which rules out both negative options. 16/7 is the reciprocal of the correct answer.
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