Question

Find five rational numbers between $\dfrac{3}{5}{\text{ and }}\dfrac{4}{5}.$

Hint- As we know that rational numbers are represented as $\dfrac{{\text{p}}}{q}$ . And we have to find more rational numbers between $\dfrac{3}{5}{\text{ and }}\dfrac{4}{5}.$ So, we multiply the numerator and denominator by the same number.

Given numbers are $\dfrac{3}{5}{\text{ and }}\dfrac{4}{5}.$
So, we have to find five numbers, we will multiply the given numbers by $\dfrac{6}{6}$
Let the number ${\text{A = }}\dfrac{3}{5}{\text{ and B = }}\dfrac{4}{5}$
Now, multiply A by $\dfrac{6}{6}$ , we obtain
${\text{A = }}\dfrac{3}{5} \times \dfrac{6}{6} = \dfrac{{18}}{{30}}$
And, multiply B by $\dfrac{6}{6}$ , we obtain
${\text{B = }}\dfrac{4}{5} \times \dfrac{6}{6} = \dfrac{{24}}{{30}}$
So, between $\dfrac{{18}}{{30}}{\text{ and }}\dfrac{{24}}{{30}}$ , we have to find rational numbers
Here, $\dfrac{{18}}{{30}} > \dfrac{{19}}{{30}} > \dfrac{{20}}{{30}} > \dfrac{{21}}{{30}} > \dfrac{{22}}{{30}} > \dfrac{{23}}{{30}} > \dfrac{{24}}{{30}}$
Hence five rational numbers between ${\text{A = }}\dfrac{3}{5}{\text{ and B = }}\dfrac{4}{5}$ are
$\dfrac{{19}}{{30}},\dfrac{{20}}{{30}},\dfrac{{21}}{{30}},\dfrac{{22}}{{30}},\dfrac{{23}}{{30}}$

Note- To solve these types of questions, basic definitions of numbers, their properties must be remembered. Some definitions such as Irrational numbers have decimal expansion that neither terminate nor periodic and cannot be expressed as fraction for any integers. This question can also be done by continuous finding the average of the given number first and then the average of numbers obtained.