Question

Insert 4 rational numbers between $\dfrac{3}{4}$ and $1$ without using $\left( \dfrac{a+b}{2} \right)$ formula.

Answer 1

Hint: For solving this question first we will see the definition of rational and irrational numbers. After that, we will multiply the numerator and denominator of $\dfrac{3}{4}$ by $5$ to write $\dfrac{3}{4}=\dfrac{3\times 5}{4\times 5}=\dfrac{15}{20}$. Then, we can answer this question correctly without any doubt.

Complete step-by-step solution -
Given:
We have to insert 4 rational numbers between $\dfrac{3}{4}$ and $1$ without using $\left( \dfrac{a+b}{2} \right)$ formula.
First, we should know the definition of rational and irrational numbers.
For a number to be rational when we express a number in the form of $\dfrac{p}{q}$ , where $p,q$ are integers such that $q\ne 0$ . For example: $2,\dfrac{4}{3}$ and any integer are rational numbers. But when any number cannot be expressed in such form then, that number will be irrational. For example: $\pi ,e,\sqrt{2}$ are irrational numbers. Few properties related to rational and irrational numbers are as follows:
1. Product of a rational and irrational number is an irrational number
2. Sum and difference of a rational and irrational number is always an irrational number
3. Division of any rational and irrational number is always an irrational number
Now, to insert 4 rational numbers between $\dfrac{3}{4}$ and $1$ . Then, just follow the following steps:
i). First multiply the numerator and denominator of $\dfrac{3}{4}$ by $5$.
ii). Multiplying the numerator and the denominator of $\dfrac{3}{4}$ by 5 we get,
 $\dfrac{3}{4}=\dfrac{3\times 5}{4\times 5}=\dfrac{15}{20}$
The other rational numbers after $\dfrac{15}{20}$ are $\dfrac{16}{20},\dfrac{17}{20},\dfrac{18}{20},\dfrac{19}{20}$ which lies between $\dfrac{3}{4}$ and $1$
Thus, required 4 rational numbers are $\dfrac{16}{20},\dfrac{17}{20},\dfrac{18}{20},\dfrac{19}{20}$ .

Note: You might have thought why we haven’t taken $\dfrac{14}{20}$ or the numbers less than $\dfrac{14}{20}$ because $\dfrac{14}{20}$ is 0.7 which is less than $\dfrac{3}{4}$ and we are asked to find the rational numbers between $\dfrac{3}{4}$ and $1$ so $\dfrac{14}{20}$ could not be possible and we have taken numbers greater than $\dfrac{15}{20}$.
We are stopped on $\dfrac{20}{20}$ because its value is 1 and we have to find the rational numbers between $\dfrac{3}{4}$ and $1$.