Question

Find two rational numbers between the following using average method:
a) -2 and 3
b) $\dfrac{2}{3}$and $\dfrac{{13}}{{14}}$

Answer 1

Hint: In order to find the number of rational numbers between any two numbers, add the given two numbers and divide the sum of two numbers by 2. Repeat this again adding the average of previous two rational numbers with the third number.

Complete step-by-step solution:
Here, in first part of question, we have
a) -2 and 3
As we know, the simplest method to find a rational number between any two rational numbers a and b is to divide their sum by 2.
Here, we have a= -2 and b= 3
rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{ - 2 + 3}}{2}$= $\dfrac{1}{2}$= 0.5
Hence, we need to find two rational numbers between them,
Now we find rational number between -2 and 0.5 (which is rational numbers between -2 and 3)
Here, we have a= -2 and b= 0.5
Rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{ - 2 + 0.5}}{2}$= $\dfrac{{ - 1.5}}{2}$= -0.75
Therefore, two rational numbers between -2 and 3 are $\dfrac{1}{2}$ and $\dfrac{{ - 1.5}}{2}$

b) $\dfrac{2}{3}$ and $\dfrac{{13}}{{14}}$
Now, In Second part of question-
Here, we have a= $\dfrac{2}{3}$ and b= $\dfrac{{13}}{{14}}$
rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{\dfrac{2}{3} + \dfrac{{13}}{{14}}}}{2}$= $\dfrac{{\dfrac{{28 + 39}}{{42}}}}{2}$= $\dfrac{{\dfrac{{67}}{{42}}}}{2}$= $\dfrac{{67}}{{84}}$
Hence, we need to find two rational numbers between them,
Now we find rational number between $\dfrac{2}{3}$ and $\dfrac{{67}}{{84}}$ (which is rational numbers between $\dfrac{2}{3}$ and $\dfrac{{13}}{{14}}$)
Here, we have a= $\dfrac{2}{3}$ and b= $\dfrac{{67}}{{84}}$
rational numbers between them = $\dfrac{{{\text{a + b}}}}{{\text{2}}}$= $\dfrac{{\dfrac{2}{3} + \dfrac{{67}}{{84}}}}{2} = \dfrac{{\dfrac{{56 + 67}}{{84}}}}{2} = \dfrac{{\dfrac{{123}}{{84}}}}{2} = \dfrac{{123}}{{168}}$
Therefore, two rational numbers between $\dfrac{2}{3}$and $\dfrac{{13}}{{14}}$ is $\dfrac{{67}}{{84}}$ and $\dfrac{{123}}{{168}}$.

Note: Always remember that a rational number is a number which can be written in the form of $\dfrac{{\text{p}}}{{\text{q}}}$(ratio) where the denominator(q) is not equal to 0. This means it can be represented in the form of a fraction. Therefore, we say every rational number has a numerator and a denominator, that is, one integer divided by another integer, where the denominator is not equal to zero.