Question

Find the five rational numbers between 3 and 4.

Answer 1

Hint: The rational number between a and b is $\dfrac{{(a + b)}}{2}$, use this equation to find the rational numbers between the given two numbers. Rational numbers are the numbers which can be written as $\dfrac{p}{q}$ where p and q are integers and q≠0.

Complete step-by-step solution -
Given, to find 5 rational numbers between 3 and 4.
As, we know, the rational number between a and b is given by, $\dfrac{{(a + b)}}{2}$
Hence, the rational number between 3 and 4 is $\dfrac{{(3 + 4)}}{2} = \dfrac{7}{2} = 3.5$
Rational number between 3 and 7/2 is $\left[ {\dfrac{{3 + \left( {\dfrac{7}{2}} \right)}}{2}} \right] = \dfrac{{13}}{4} = 3.25$
Rational number between 3 and 13/4 is $\left[ {\dfrac{{3 + \left( {\dfrac{{13}}{4}} \right)}}{2}} \right] = \dfrac{{25}}{8} = 3.125$
Rational number between 3 and 25/8 is $\left[ {\dfrac{{3 + \left( {\dfrac{{25}}{8}} \right)}}{2}} \right] = \dfrac{{49}}{{16}} = 3.0625$
Rational number between 25/8 and 2 is $\left[ {\dfrac{{\left( {\dfrac{{25}}{8}} \right) + 4}}{2}} \right] = \dfrac{{57}}{{16}} = 3.5625$
Therefore, five rational numbers between 3 and 4 are 7/2, 13/4, 25/8, 49/16 and 57/16.

Note: Whenever such a type of question appears, i.e., it is asked to find a rational number between any two rational numbers, suppose x and y, the simplest approach is to divide their sum by 2, as mentioned in the question.This is called as mean method, Proceed step by step, obtain the value of (sum of the two no.)/2, then find the next rational number by dividing the sum of the first rational no. obtained in step 1 with the first number in the range, repeat it till you get all the required rational numbers.