By what number should \[{{\left( -25 \right)}^{-1}}\] be divided so that the quotient may be \[{{\left( 5 \right)}^{-1}}\] ?
(a) \[-\dfrac{1}{5}\]
(b) \[\dfrac{1}{5}\]
(c) \[\dfrac{1}{25}\]
(d) \[-\dfrac{1}{25}\]

Hint: Assuming the number which divides \[{{\left( -25 \right)}^{-1}}\] to be x and changing the given numbers into fractions and then proceeding with details given in the question.

Complete step-by-step answer:

Before proceeding with the question, we should know about dividend, divisor and quotient.
So the number which we divide is called the dividend, and the number by which we divide the dividend is called the divisor and the result which we get is called the quotient. If any number which is left over is called the remainder.

Here from the details given in the question, dividend is \[{{\left( -25 \right)}^{-1}}\], divisor is x and quotient is \[{{\left( 5 \right)}^{-1}}\]. Now simplifying the given numbers into simple fractions we get,

Dividend \[=-\dfrac{1}{25}\]

Divisor \[=x\]

Quotient \[=\dfrac{1}{5}\]

Now forming an equation with all the above details we get,

 \[\Rightarrow \dfrac{\dfrac{-1}{25}}{x}=\dfrac{1}{5}........(1)\]

Rearranging equation (1) we get,

 \[\Rightarrow \dfrac{-1}{25x}=\dfrac{1}{5}........(2)\]

Now cancelling similar terms and cross multiplying we get,

 \[\Rightarrow 25x=-5........(3)\]

Now dividing equation (3) by 15 on both sides to get x,

 \[\Rightarrow x=\dfrac{-5}{25}=-\dfrac{1}{5}\]

Hence \[-\dfrac{1}{5}\] is the number which divides \[{{\left( -25 \right)}^{-1}}\] to yield quotient which is equal to \[{{\left( 5 \right)}^{-1}}\].

Hence option (a) is the right answer.

Note: Understanding the concept of dividend, divisor and quotient is the key here. To avoid any mistakes, we transform the given numbers into simple fractions. Chances of mistakes are when we are forming the equation from the given details and in a hurry we may write x in numerator and \[{{\left( -25 \right)}^{-1}}\] in denominator.