Question

Using the method of mean, find three rational numbers between \[ - 1\] and \[ - 2\].

Answer 1

Hint: Here, we need to find 3 rational numbers between \[ - 1\] and \[ - 2\]. We will assume the three rational numbers between \[ - 1\] and \[ - 2\] to be some variable. We will arrange these numbers in increasing order. Then, we will use the formula for the mean of two numbers to find the three rational numbers between \[ - 1\] and \[ - 2\] using the method of mean.

Formula Used: We will use the formula of the mean of two numbers \[x\] and \[y\] which is given by the formula \[\dfrac{{x + y}}{2}\].

Complete step-by-step answer:
Let the three rational numbers between \[ - 1\] and \[ - 2\] be \[a\], \[b\], and \[c\].
Thus, arranging the numbers in increasing order, we get
\[ - 1,a,b,c, - 2\]
Now, we will use the method of mean to find the three rational numbers.
According to the method of mean, the third rational number in the series \[ - 1,a,b,c, - 2\] is the mean of the first and fifth term. Similarly, the second rational number is the mean of the first and the third term, and the fourth term is the mean of the third and the fifth term.
The mean of two numbers \[x\] and \[y\] is given by the formula \[\dfrac{{x + y}}{2}\].
The third rational number in the series \[ - 1,a,b,c, - 2\] is the mean of the first and fifth term.
Thus, we get
\[ \Rightarrow b = \dfrac{{ - 1 + \left( { - 2} \right)}}{2}\]
Simplifying the expression, we get
\[ \Rightarrow b = \dfrac{{ - 1 - 2}}{2}\]
Subtracting the terms in the numerator, we get
\[ \Rightarrow b = \dfrac{{ - 3}}{2}\]
Dividing the terms in the expression, we get
\[ \Rightarrow b = - 1.5\]
The second rational number is the mean of the first and the third term
Thus, we get
\[a = \dfrac{{ - 1 + b}}{2}\]
Substituting \[b = - 1.5\] in the equation, we get
\[ \Rightarrow a = \dfrac{{ - 1 + \left( { - 1.5} \right)}}{2}\]
Simplifying the expression, we get
\[ \Rightarrow a = \dfrac{{ - 1 - 1.5}}{2}\]
Subtracting the terms in the numerator, we get
\[ \Rightarrow a = \dfrac{{ - 2.5}}{2}\]
Dividing the terms in the expression, we get
\[ \Rightarrow a = - 1.25\]
The fourth rational number is the mean of the third and the fifth term.
Thus, we get
\[ \Rightarrow c = \dfrac{{b + \left( { - 2} \right)}}{2}\]
Substituting \[b = - 1.5\] in the equation, we get
\[ \Rightarrow c = \dfrac{{ - 1.5 + \left( { - 2} \right)}}{2}\]
Simplifying the expression, we get
\[ \Rightarrow c = \dfrac{{ - 1.5 - 2}}{2}\]
Subtracting the terms in the numerator, we get
\[ \Rightarrow c = \dfrac{{ - 3.5}}{2}\]
Dividing the terms in the expression, we get
\[ \Rightarrow c = - 1.75\]
Thus, the series \[ - 1,a,b,c, - 2\] becomes \[ - 1, - 1.25, - 1.5, - 1.75, - 2\].
Therefore, the three rational numbers between \[ - 1\] and \[ - 2\] are \[ - 1.25\], \[ - 1.5\], and \[ - 1.75\].

Note: We found three rational numbers between \[ - 1\] and \[ - 2\] using the method of mean. A rational number is a number which can be written in the form \[\dfrac{p}{q}\], where the denominator \[q \ne 0\]. For example, \[5,\dfrac{7}{2}, - \dfrac{{15}}{7},5.6\], etc. are rational numbers. Rational numbers include every integer, fraction, decimal. An integer can be a rational number but every rational cannot be an integer.