Question

Insert three rational numbers between $\dfrac{2}{3}{\text{ and }}\dfrac{3}{5}.$

Answer 1

Hint: As we know that rational numbers are represented as $\dfrac{{\text{p}}}{q}$ . And we have to find more rational numbers between $\dfrac{2}{3}{\text{ and }}\dfrac{3}{5}.$ so, we multiply the numerator and denominator by the same number.

Complete step-by-step answer:
Given numbers are $\dfrac{2}{3}{\text{ and }}\dfrac{3}{5}.$
So, we have to find three numbers, we will multiply the given numbers by $\dfrac{{20}}{{20}}{\text{ and }}\dfrac{{12}}{{12}}$ respectively
Let the number ${\text{A = }}\dfrac{2}{3}{\text{ and B = }}\dfrac{3}{5}$
Now, multiply A by $\dfrac{{20}}{{20}}$ , we obtain
${\text{A = }}\dfrac{2}{3} \times \dfrac{{20}}{{20}} = \dfrac{{40}}{{60}}$
And, multiply B by $\dfrac{{12}}{{12}}$ , we obtain
${\text{B = }}\dfrac{3}{5} \times \dfrac{{12}}{{12}} = \dfrac{{36}}{{60}}$
So, between $\dfrac{{40}}{{60}}{\text{ and }}\dfrac{{36}}{{60}}$ , we have to find rational numbers
Here, $\dfrac{{40}}{{60}} > \dfrac{{39}}{{60}} > \dfrac{{38}}{{60}} > \dfrac{{37}}{{60}} > \dfrac{{36}}{{60}}$
Hence three rational numbers between ${\text{A = }}\dfrac{2}{3}{\text{ and B = }}\dfrac{3}{5}$ are
$\dfrac{{39}}{{60}},\dfrac{{38}}{{60}},\dfrac{{37}}{{60}}$

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.