Question

How many rational numbers are there between \[\dfrac{1}{6}\] and 1?

Answer 1

Hint: To solve this question first we assume that the endpoints of the numbers are included and then we define what are the rational numbers and then we increase the value of \[q\] by one and then make all the possibilities on \[p\] and write all the rational number and eliminating repeated number.

Complete step-by-step answer:
Here we have to find all the rational numbers between two numbers \[\dfrac{1}{6}\] and 1.
A rational number is a number that can be expressed in the form of \[\dfrac{p}{q}\], here \[p,q\] are the integers and \[q\] is not equal to zero.
In question a word is given “between '' that does not tell us whether we will include endpoints or not. If we include those numbers then the interval is closed interval and expressed as \[\left[ {\dfrac{1}{6},1} \right]\] and if we not include those numbers then that is in open interval and expressed as \[\left( {\dfrac{1}{6},1} \right)\].
In this question, there are infinite numbers of rational numbers between the given numbers in the question.
To write the rational number between both the numbers we start to increase the count of \[q\] by one like \[q = 1\] and all the possibilities of \[p\], then we again increase \[q\] by one and list all the possibilities.
All the numbers are distant and rational numbers between both the numbers.
\[1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{2}{3},\dfrac{1}{4},\dfrac{3}{4},\dfrac{1}{5},\dfrac{2}{5},\dfrac{3}{5},\dfrac{4}{5},\dfrac{1}{6},\dfrac{5}{6},\dfrac{2}{7},\dfrac{3}{7},\dfrac{4}{7},\dfrac{5}{7},\dfrac{6}{7},\dfrac{3}{8},\dfrac{5}{8},\dfrac{7}{8},\dfrac{2}{9},\dfrac{4}{9},...\]
There are infinite numbers of rational numbers between both the given numbers but these are some rational numbers.

Note: Irrational numbers are those numbers that are not represented in the form of \[\dfrac{p}{q}\], these types of numbers are also of two types like terminating and nonterminating numbers. To solve these types of questions students must know all the terms like these.