Question

Find a rational number between $\dfrac{1}{4}$ and $\dfrac{1}{3}$ ?

Answer 1

Hint: In this question, we have to find one rational number lying between the $2$ rational numbers already given to us. Hence, we first find the equivalent rational numbers of both the numbers between which one rational number is to be found with the denominator so that the denominator is same for both the rational numbers. We can find equivalent rational numbers by multiplying or dividing the numerator and denominator of the rational by the same number. Then, we find the rational numbers between the two numbers by choosing the numerator accordingly.

Complete step by step answer:
We are given the rational numbers $\dfrac{1}{4}$ and $\dfrac{1}{3}$ 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. We know that the LCM of the denominators of the two rational numbers is $12$.So, we get,
$\dfrac{1}{4} \times \dfrac{{30}}{{30}} = \dfrac{{30}}{{120}}$ and $\dfrac{1}{3} \times \dfrac{{40}}{{40}} = \dfrac{{40}}{{120}}$
Now, we can easily find a rational numbers between $\dfrac{{30}}{{120}}$ and \[\dfrac{{40}}{{120}}\] by choosing the numerators accordingly.

So, we can choose the numerator of the rational numbers between $30$ and $40$ with denominator as $120$ so that the rational numbers lie between $\dfrac{{30}}{{120}}$ and \[\dfrac{{40}}{{120}}\]. Now, we know that $31$ lie between $30$ and $40$. So, the rational number $\dfrac{{31}}{{120}}$ lie between $\dfrac{{30}}{{120}}$ and \[\dfrac{{40}}{{120}}\]. Also, these rational numbers don’t have any common factor in numerator and denominator.So, they are in their simplest forms.

Hence, a rational numbers between $\dfrac{1}{4}$ and $\dfrac{1}{3}$ is $\dfrac{{31}}{{120}}$.

Note:Equivalent rational numbers are the rational numbers that have different numerator and denominator but are equal to the same value. Following the information and the steps mentioned in the above solution, we can solve similar questions. Care should be taken while doing calculations in order to be sure of the answer. Also, there are infinite rational numbers between any two given rational numbers. So, the answer to the given problem may vary from person to person.