Question

What are three rational numbers between \[ - 2{\text{ and }} - {\text{1}}\]?
A. \[\dfrac{{ - 1}}{2},\dfrac{{ - 1}}{3},\dfrac{{ - 1}}{5}\]
B. \[\dfrac{{ - 3}}{2},\dfrac{{ - 7}}{4},\dfrac{{ - 5}}{4}\]
C. \[\dfrac{{ - 12}}{5},\dfrac{{ - 22}}{5},\dfrac{{12}}{5}\]
D. \[\dfrac{3}{2},\dfrac{7}{4},\dfrac{5}{4}\]

Answer 1

Hint: We can write an infinite number of rational numbers between any two integers. So, here go for option verification as they have given options. All the rational numbers given in an must lie in the given integers to be the correct option. So, use this method to reach the solution of the given problem.

Complete step-by-step answer:

As we can write infinite rational numbers between \[ - 2{\text{ and }} - {\text{1}}\], we will go for option verification.
In the option verification, if all the rational numbers in a given option lie in between \[ - 2{\text{ and }} - {\text{1}}\] the option is correct otherwise wrong.
consider option A. \[\dfrac{{ - 1}}{2},\dfrac{{ - 1}}{3},\dfrac{{ - 1}}{5}\]
\[ \Rightarrow \dfrac{{ - 1}}{2} = - 0.5\] which does not lie between \[ - 2{\text{ and }} - {\text{1}}\]
\[ \Rightarrow \dfrac{{ - 1}}{3} = - 0.3333333\] which does not lie between \[ - 2{\text{ and }} - {\text{1}}\]
\[ \Rightarrow \dfrac{{ - 1}}{5} = - 0.2\] which does not lie between \[ - 2{\text{ and }} - {\text{1}}\]
Thus, option A is wrong
Now consider option B. \[\dfrac{{ - 3}}{2},\dfrac{{ - 7}}{4},\dfrac{{ - 5}}{4}\]
\[ \Rightarrow \dfrac{{ - 3}}{2} = - 1.5\] which lie between \[ - 2{\text{ and }} - {\text{1}}\]
\[ \Rightarrow \dfrac{{ - 7}}{4} = - 1.75\] which lie between \[ - 2{\text{ and }} - {\text{1}}\]
\[ \Rightarrow \dfrac{{ - 5}}{4} = - 1.25\] which lie between \[ - 2{\text{ and }} - {\text{1}}\]
Thus, option B is correct. \[\dfrac{{ - 12}}{5},\dfrac{{ - 22}}{5},\dfrac{{12}}{5}\]
\[ \Rightarrow \dfrac{{ - 12}}{5} = - 2.4\] which does not lie between \[ - 2{\text{ and }} - {\text{1}}\]
\[ \Rightarrow \dfrac{{ - 22}}{5} = - 4.4\] which does not lie between \[ - 2{\text{ and }} - {\text{1}}\]
\[ \Rightarrow \dfrac{{12}}{5} = 2.4\] which does not lie between \[ - 2{\text{ and }} - {\text{1}}\]
Thus, option C is incorrect or wrong.
Next consider option C.
Then consider option D. \[\dfrac{3}{2},\dfrac{7}{4},\dfrac{5}{4}\]
\[ \Rightarrow \dfrac{3}{2} = 1.5\] which does not lie between \[ - 2{\text{ and }} - {\text{1}}\]
\[ \Rightarrow \dfrac{7}{4} = 1.75\] which does not lie between \[ - 2{\text{ and }} - {\text{1}}\]
\[ \Rightarrow \dfrac{5}{4} = 1.25\] which does not lie between \[ - 2{\text{ and }} - {\text{1}}\]
Thus, option D is incorrect or wrong.
Therefore, the only correct option is B. \[\dfrac{{ - 12}}{5},\dfrac{{ - 22}}{5},\dfrac{{12}}{5}\]

Note: A rational number is a number that can be expressed as the quotient or dfraction p/q of two integers, a numerator of p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number.