Question

If a number is multiplied by ${{\left( -7 \right)}^{-1}}$, the product obtained is ${{\left( \dfrac{6}{7} \right)}^{-1}}$. Find the number.

Answer 1

Hint: Assume that the number is x. Use the fact that x multiplied by ${{\left( -7 \right)}^{-1}}$ is equal to ${{\left( \dfrac{6}{7} \right)}^{-1}}$ and hence form an equation in x. Multiply both sides of the equation by 7 and simplify. Again, multiply both sides of the equation by $\dfrac{6}{7}$ and simplify. Cross multiply and hence find the value of x. Verify your solution.

Complete step-by-step answer:
Let the number is x.
Since x multiplied by ${{\left( -7 \right)}^{-1}}$ is equal to ${{\left( \dfrac{6}{7} \right)}^{-1}}$, we get
$x\times {{\left(- 7 \right)}^{-1}}={{\left( \dfrac{6}{7} \right)}^{-1}}$
Multiplying both sides by 7, we get
$-x={{\left( \dfrac{6}{7} \right)}^{-1}}\times 7$
Again multiplying both sides by $\left( \dfrac{6}{7} \right)$, we get
$-\dfrac{6x}{7}=7$
Again multiplying both sides by 7, we get
$-6x=7\times 7=49$
Dividing both sides by 6, we get
$-x=\dfrac{49}{6}\Rightarrow x=\dfrac{-49}{6}$
Hence the number is $-\dfrac{49}{6}$

Note:Verification:
$-\dfrac{49}{6}$ multiplied by ${{\left( -7 \right)}^{-1}}$ is equal to $\dfrac{49}{6}\times {{7}^{-1}}=\dfrac{{{7}^{2}}\left( {{7}^{-1}} \right)}{6}$
We know that ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
Hence, we have
$-\dfrac{49}{6}\times {{\left( -7 \right)}^{-1}}=\dfrac{7}{6}$
We know that ${{a}^{m}}={{\left( \dfrac{1}{a} \right)}^{-m}}$
Hence, we have
$-\dfrac{49}{6}\times {{\left( -7 \right)}^{-1}}={{\left( \dfrac{6}{7} \right)}^{-1}}$
Hence our solution is verified to be correct.