Simplify: \[\left( {\dfrac{{ - 1}}{{10}}} \right) \div \left( {\dfrac{{ - 8}}{5}} \right)\]

Hint: In general if we have a division of two fraction or two rational numbers, that is \[\left( {\dfrac{a}{b}} \right) \div \left( {\dfrac{c}{d}} \right)\]. Where ‘a’, ‘b’, ‘c’, and ‘d’ are integers. To solve this we take the reciprocal of the second fraction that is \[\dfrac{d}{c}\] and we multiply it to the first fraction. That is we have \[\left( {\dfrac{a}{b}} \right) \times \left( {\dfrac{d}{c}} \right)\]. We know how to multiply this. After multiplying this we will get the required result.

Complete step-by-step solution:
If we have multiplication instead of division operators, we would have solved this easily. So we convert the division operator into the multiplication operative by taking the reciprocal of a second fraction or rational number.
Given \[\left( {\dfrac{{ - 1}}{{10}}} \right) \div \left( {\dfrac{{ - 8}}{5}} \right)\].
In this the both numbers are rational numbers. Because we have integers. If we have only natural numbers it would become a fraction.
Now taking the reciprocal of the second rational number.
The reciprocal of \[\dfrac{{ - 8}}{5} \Rightarrow \dfrac{{ - 5}}{8}\].
Then above becomes,
\[\left( {\dfrac{{ - 1}}{{10}}} \right) \times \left( {\dfrac{{ - 5}}{8}} \right)\]
 We know that the product of two rational numbers is the product of their numerator divided by the product of their denominator.
Then we have,
\[\left( {\dfrac{{ - 1}}{{10}}} \right) \times \left( {\dfrac{{ - 5}}{8}} \right) = \left( {\dfrac{{ - 1 \times - 5}}{{10 \times 8}}} \right)\]
We know that the product of two negative numbers gives us a positive number.
\[ \Rightarrow = \left( {\dfrac{5}{{80}}} \right)\]
Dividing we have,
\[ \Rightarrow = \dfrac{1}{{16}}\]is the required answer.

Hence the correct answer is \[= \dfrac{1}{{16}}\]is the required answer.

Note: Don’t get confused with \[\dfrac{{ - 5}}{8}\] and \[ - \dfrac{5}{8}\] both are the same. Follow the same procedure if they give two fractions. That is without negative signs. In above we mention that it is a rational number for our understanding purpose only. We also know that the product of a positive number and a negative number gives us a negative number. Similarly for products of negative numbers and a positive number gives us a negative number.