Question

Find a rational number between $\frac{2}{3}$ and $\frac{3}{4}$.

Answer 1

We know that, Arithmetic mean (Average) of $a,b$ is $\frac{a+b}{2}$.

$\frac{a+b}{2}$ is always between $a$ and $b$ .

Now, Given rational numbers are $\frac{2}{3}$ and $\frac{3}{4}$.

Average of $\frac{2}{3}$ and $\frac{3}{4}$

$=\frac{1}{2}[\frac{2}{3}+\frac{3}{4}]$

$=\frac{1}{2}[\frac{2}{3}\times \frac{4}{4}+\frac{3}{4}\times \frac{3}{3}]$

$=\frac{1}{2}[\frac{8}{12}+\frac{9}{12}]$

$=\frac{1}{2}\left[\frac{17}{12}\right]$

$=\frac{17}{24}$

Therefore, $\frac{17}{24}$ is one of the rational number between $\frac{2}{3}$ and $\frac{3}{4}$.