Question

Find five rational numbers between 3 and 4 by mean method.

Answer 1

Hint: If a and b are two rational numbers then the rational number between these two numbers is given by\[\dfrac{{a + b}}{2}\], and the rational number can be written in the form \[\dfrac{p}{q}\], where p and q are integers and \[q \ne 0\].

Complete step-by-step answer:
Given numbers are 3 and 4, and now we have to find the rational numbers between these two numbers.
We know that the rational number between numbers a and b is given by\[\dfrac{{a + b}}{2}\],
So, here a=3, and b=4,
Now rational number between 3 and 4\[ = \dfrac{{3 + 4}}{2} = \dfrac{7}{2}\],
Here the values of a=3 and b=\[\dfrac{7}{2}\],
Now again rational number between 3 and\[\dfrac{7}{2}\]\[ = \dfrac{{3 + \dfrac{7}{2}}}{2} = \dfrac{{6 + 7}}{4} = \dfrac{{13}}{4}\],
Here the values of a=3 and b=\[\dfrac{{13}}{4}\],
The rational number between 3 and\[\dfrac{{13}}{4}\]\[ = \dfrac{{3 + \dfrac{{13}}{4}}}{2} = \dfrac{{12 + 13}}{8} = \dfrac{{25}}{8}\],
Here the values of a=4 and b=\[\dfrac{{25}}{8}\],
The rational number between 4 and\[\dfrac{{25}}{8}\]\[ = \dfrac{{4 + \dfrac{{25}}{8}}}{2} = \dfrac{{32 + 25}}{{16}} = \dfrac{{57}}{{16}}\],
Here the values of a=4 and b=\[\dfrac{{57}}{{16}}\],
The rational number between 4 and\[\dfrac{{57}}{{16}}\]\[ = \dfrac{{4 + \dfrac{{57}}{{16}}}}{2} = \dfrac{{64 + 57}}{{32}} = \dfrac{{121}}{{32}}\],
The five rational numbers between 3 and 4 are\[\dfrac{7}{2},\dfrac{{13}}{4},\dfrac{{25}}{8},\dfrac{{57}}{{16}},\dfrac{{121}}{{32}}\].

\[\therefore \]The five rational numbers between 3 and 4 using mean method are\[\dfrac{7}{2},\dfrac{{13}}{4},\dfrac{{25}}{8},\dfrac{{57}}{{16}},\dfrac{{121}}{{32}}\].

Note:
A rational number can be written in the form \[\dfrac{p}{q}\], where p and q are integers and \[q \ne 0\], There are always definite amount of numbers between two natural/whole numbers or integers. But, there can be an indefinite amount of numbers between two rational numbers. Between any 2 numbers, it is not necessary that there will be an integer or a whole number but there is always a rational number.
Example, there are no integer or whole or natural numbers between 1 and 2, but there are rational numbers like, \[\dfrac{1}{2},\dfrac{1}{3},\dfrac{2}{3},\dfrac{1}{4},\dfrac{3}{4},\]…..Etc.