Question

Write three rational numbers that lie between the two given numbers $\dfrac{4}{5},\dfrac{2}{3}$.

Answer 1

Hint: Here we need to insert three numbers between them so first we make both denominators equal so that we can compare quantities. After that take the average of the numbers by $\dfrac{{a + b}}{2}$ to find the rational number between them.

Complete step-by-step answer:
Take LCM of denominators. Now, Divide LCM by each numerator then multiply and divide by the following quantity into fractions.
Therefore, the rational number between two numbers say a and b can be obtained by doing the operation $\dfrac{{a + b}}{2}$.
The rational number between $\dfrac{2}{3}$ and $\dfrac{4}{5}$ is,
$ \Rightarrow \dfrac{{\dfrac{2}{3} + \dfrac{4}{5}}}{2}$
Take LCM of the rational number in the denominator,
$ \Rightarrow \dfrac{{\dfrac{{10 + 12}}{{15}}}}{2}$
Move 15 to the denominator and simplify the expression,
$ \Rightarrow \dfrac{{22}}{{30}}$
Cancel out the common factors,
$\therefore \dfrac{{11}}{{15}}$
Now, the rational number between $\dfrac{2}{3}$ and $\dfrac{{11}}{{15}}$ is,
$ \Rightarrow \dfrac{{\dfrac{2}{3} + \dfrac{{11}}{{15}}}}{2}$
Take LCM of the rational number in the denominator,
$ \Rightarrow \dfrac{{\dfrac{{10 + 11}}{{15}}}}{2}$
Move 15 to the denominator and simplify the expression,
$ \Rightarrow \dfrac{{21}}{{30}}$
Cancel out the common factors,
$\therefore \dfrac{7}{{10}}$
Now, the rational number between $\dfrac{7}{{10}}$ and $\dfrac{4}{5}$ is,
$ \Rightarrow \dfrac{{\dfrac{7}{{10}} + \dfrac{4}{5}}}{2}$
Take LCM of the rational number in the denominator,
$ \Rightarrow \dfrac{{\dfrac{{7 + 8}}{{10}}}}{2}$
Move 10 to the denominator and simplify the expression,
$ \Rightarrow \dfrac{{15}}{{20}}$
Cancel out the common factors,
$\therefore \dfrac{3}{4}$

Hence, the three rational numbers that lie between the two given numbers $\dfrac{4}{5},\dfrac{2}{3}$ are $\dfrac{7}{{10}},\dfrac{{11}}{{15}}$ and $\dfrac{3}{4}$.

Note: Numbers which can be written as p/q form where p and q are integer and there is no common factor between p and q called rational number (q can’t be equal to 0). The numbers which are not rational are called irrational numbers ($\pi ,\sqrt 3 ,\sqrt 2 $ are irrational numbers). Here we need to insert three numbers between them so first we make both denominators equal so that we can compare quantities.