Question

List five rational numbers between:
(a) -1 and 0
(b) -2 and -1
(c) \[ - \dfrac{4}{5}\] and \[ - \dfrac{2}{3}\]
(d) \[ - \dfrac{1}{2}\] and \[\dfrac{2}{3}\]

Answer 1

Hint: Here in this question, we have to find the five rational numbers which are present in between the given numbers. The given sub questions are in the form of whole numbers and fractions. When the given question is in the form of a whole number, first we have to write in the form of a fraction. Then we have to multiply both numerator and denominator 6 and then we write the rational numbers which are present in between the given numbers. Suppose if the number is in the form of fraction first we have to determine the LCM and then the same procedure is followed.

Complete step-by-step answer:
In mathematics we have different kinds of numbers namely, natural number, whole number, integers, rational numbers, irrational numbers and real numbers.
Natural numbers - Contain all counting numbers which start from 1.
Whole Numbers - Collection of zero and natural numbers.
Integers- The collective result of whole numbers and negative of all-natural numbers.
Rational Numbers- Numbers that can be written in the form of \[\dfrac{p}{q}\] where \[q \ne 0\]
Irrational Numbers- All the numbers which are not rational and cannot be written in the form of \[\dfrac{p}{q}\]
Real numbers: Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”.
Now we consider the given question
We have to write five rational numbers which are present in between the given numbers.
(a) -1 and 0
These numbers can be written in the form of fraction, so we have
\[ \Rightarrow - \dfrac{1}{1}\] and \[\dfrac{0}{1}\]
Since we have to write the 5 rational numbers. We multiply both numerator and denominator by 6. So, we have
\[ \Rightarrow - \dfrac{6}{6}\] and \[\dfrac{0}{6}\]
Therefore, the five rational numbers which is present between \[ - \dfrac{6}{6}\] and \[\dfrac{0}{6}\] is
\[ \Rightarrow - \dfrac{5}{6}, - \dfrac{4}{6}, - \dfrac{3}{6}, - \dfrac{2}{6}, - \dfrac{1}{6}\]
On simplifying the terms, it can be written as
\[ \Rightarrow - \dfrac{5}{6}, - \dfrac{2}{3}, - \dfrac{1}{2}, - \dfrac{1}{3}, - \dfrac{1}{6}\]

(b) -2 and -1
These numbers can be written in the form of fraction, so we have
\[ \Rightarrow - \dfrac{{ - 2}}{1}\] and \[\dfrac{{ - 1}}{1}\]
Since we have to write the 5 rational numbers. We multiply both numerator and denominator by 6. So, we have
\[ \Rightarrow - \dfrac{{12}}{6}\] and \[ - \dfrac{6}{6}\]
Therefore, the five rational numbers which is present between \[ - \dfrac{{12}}{6}\] and \[ - \dfrac{6}{6}\] is
\[ \Rightarrow - \dfrac{{11}}{6}, - \dfrac{{10}}{6}, - \dfrac{9}{6}, - \dfrac{8}{6}, - \dfrac{7}{6}\]
On simplifying the terms, it can be written as
\[ \Rightarrow - \dfrac{{11}}{6}, - \dfrac{5}{3}, - \dfrac{3}{2}, - \dfrac{4}{3}, - \dfrac{7}{6}\]

(c) \[ - \dfrac{4}{5}\] and \[ - \dfrac{2}{3}\]
First, we take LCM for the denominators. The LCM for 5 and 3 is 15.
Therefore, the given number is written as
\[ \Rightarrow - \dfrac{{12}}{{15}}\] and \[ - \dfrac{{10}}{{15}}\]
Since we have to write the 5 rational numbers. We multiply both numerator and denominator by 6. So, we have
\[ \Rightarrow - \dfrac{{72}}{{90}}\] and \[ - \dfrac{{60}}{{90}}\]
Therefore, the five rational numbers which is present between \[ - \dfrac{{72}}{{90}}\] and \[ - \dfrac{{60}}{{90}}\] is
\[ \Rightarrow - \dfrac{{71}}{{90}}, - \dfrac{{70}}{{90}}, - \dfrac{{69}}{{90}}, - \dfrac{{68}}{{90}}, - \dfrac{{67}}{{90}}\]

(d) \[\dfrac{1}{2}\] and \[\dfrac{2}{3}\]
First, we take LCM for the denominators. The LCM for 2 and 3 is 6.
Therefore, the given number is written as
\[ \Rightarrow - \dfrac{3}{6}\] and \[\dfrac{4}{6}\]
Since we have to write the 5 rational numbers. We multiply both numerator and denominator by 6. So, we have
\[ \Rightarrow - \dfrac{{18}}{{36}}\] and \[\dfrac{{24}}{{36}}\]
Therefore, the five rational numbers which is present between \[ - \dfrac{{18}}{{36}}\] and \[\dfrac{{24}}{{36}}\] is
\[ \Rightarrow - \dfrac{{17}}{{36}}, - \dfrac{{16}}{{36}}, - \dfrac{{15}}{{36}}, - \dfrac{{14}}{{36}}, - \dfrac{{13}}{{36}}\]

Note: There are so many rational numbers which will be present in between the given numbers. To determine the rational numbers we multiply both numerator and denominator by 6, because it will be easy to find the five rational numbers. Suppose if we want to find the 6 rational numbers then we are going to multiply both numerator and denominator by 7.