Question

Find four rational numbers between $\dfrac{1}{2}$ and $\dfrac{3}{4}$ ?

Answer 1

Hint: In this question, we have to find four rational numbers lying between the $2$ rational numbers already given to us. Hence, we first find the equivalent rational numbers of both the numbers between which three rational numbers are to be found with a larger denominator so that the rational numbers can be found without any problem. We can find equivalent rational numbers by multiplying or dividing the numerator and denominator of the rational by the same number. Then, we find four rational numbers between the two numbers by choosing the numerator accordingly.

Complete step by step answer:
So, we are given the rational numbers $\dfrac{1}{2}$ and $\dfrac{3}{4}$ in the questions itself. So, we first find out the equivalent rational numbers of these. Hence, we multiply the numerator and denominator by the same number so that the value does not change. So, multiplying the numerator and denominator by $10$, we get the$\dfrac{1}{2}$ as,
$\dfrac{1}{2} \times \dfrac{{10}}{{10}} = \dfrac{{10}}{{20}}$
Similarly, multiplying the numerator and denominator by $5$, we get $\dfrac{3}{4}$ as,
$\dfrac{3}{4} \times \dfrac{5}{5} = \dfrac{{15}}{{20}}$
Now, we can easily find four rational numbers between $\dfrac{{10}}{{20}}$ and $\dfrac{{15}}{{20}}$ by choosing the numerators accordingly.

So, we can choose the numerator of the rational numbers between $10$ and $15$ with denominator as $20$ so that the rational numbers lie between $\dfrac{{10}}{{20}}$ and $\dfrac{{15}}{{20}}$.Now, we know that $11$, $12$, $13$ and $14$ lie between $10$ and $15$. So, the rational numbers $\dfrac{{11}}{{20}}$, $\dfrac{{12}}{{20}}$, $\dfrac{{13}}{{20}}$ and $\dfrac{{14}}{{20}}$ lie between $\dfrac{{10}}{{20}}$ and $\dfrac{{15}}{{20}}$. Now, we cancel the common factors between the common factors in numerator and denominator. So, we get the rational numbers as $\dfrac{{11}}{{20}}$, $\dfrac{3}{5}$, $\dfrac{{13}}{{20}}$ and $\dfrac{7}{{10}}$. Now, the rational numbers are in the simplest form.

Hence, four rational numbers between $\dfrac{1}{2}$ and $\dfrac{3}{4}$ are: $\dfrac{{11}}{{20}}$, $\dfrac{3}{5}$, $\dfrac{{13}}{{20}}$ and $\dfrac{7}{{10}}$.

Note: Equivalent rational numbers are the rational numbers that have different numerator and denominator but are equal to the same value. Also, there are infinite rational numbers between any two given rational numbers. We must simplify the rational numbers by cancelling out the common factors in numerator and denominator. We must take care of the calculations in order to be sure of the answer.