Question

Find five rational numbers between 3 and 4.

Answer 1

Hint: Convert the two numbers in the equivalent fraction by multiplying and dividing each number by \[(n+1)\] where n is the number of required rational numbers between the two given numbers.

Complete step-by-step solution -
In the question, we have to find the five rational numbers between 3 and 4.
So here we will divide and multiply 3 with \[(5+1=6)\] to get the fraction given as:
\[\begin{align}
  & \Rightarrow 3=\dfrac{3\times 6}{6} \\
 & \Rightarrow 3=\dfrac{18}{6} \\
\end{align}\]
Next, we will do the same with number 4, and we get:
\[\begin{align}
  & \Rightarrow 4=\dfrac{4\times 6}{6} \\
 & \Rightarrow 4=\dfrac{24}{6} \\
\end{align}\]
Now, we can take any five fractions between \[\dfrac{18}{6}\] and \[\dfrac{24}{6}\] , as follows:
\[\dfrac{19}{6},\dfrac{20}{6},\dfrac{21}{6},\dfrac{22}{6}\]and \[\dfrac{23}{6}\].
All these are rational number, as they are of the form \[\dfrac{p}{q}\], where \[q\ne 0\] So all the above numbers \[\dfrac{19}{6},\dfrac{20}{6},\dfrac{21}{6},\dfrac{22}{6}\]and \[\dfrac{23}{6}\].
 are the rational numbers. Also, when the numbers are not in the form of \[\dfrac{p}{q}\], then it will be an irrational number. An example of irrational number are\[\sqrt{2},\sqrt{3},\sqrt{5},\,\,\text{etc}\text{.}\]
Now, the rational numbers \[\dfrac{19}{6},\dfrac{20}{6},\dfrac{21}{6},\dfrac{22}{6}\]and \[\dfrac{23}{6}\] lies between \[\dfrac{18}{6}=3\] and \[\dfrac{24}{6}=4\]
So, the five required rational numbers between 3 and 4 are:
\[\dfrac{19}{6},\dfrac{20}{6},\dfrac{21}{6},\dfrac{22}{6}\] and \[\dfrac{23}{6}\].

Note: The rational number is in the form of \[\dfrac{p}{q}\], where \[q\ne 0\]. The denominator has to be the same for the two equivalent fractions of the two numbers so formed. The alternate method to find the rational number between two numbers will be finding the average of the two numbers. This way we will get the one rational between two given numbers. Then again taking the average of the first number and the rational number formed above to get the second rational number between them and so on.