Question

Find three rational numbers between $\frac{2}{3}$ and $\frac{3}{4}.$

Answer 1

First rational number between $\frac{2}{3}$ and $\frac{3}{4}$

$=\frac{1}{2}[\frac{2}{3}+\frac{3}{4}]=\frac{1}{2}\left[\frac{8+9}{12}\right]=\frac{1}{2}\times \frac{17}{12}=\frac{17}{24}$

$\therefore \frac{2}{3}<\frac{17}{24}<\frac{3}{4}$

Second rational number between $\frac{2}{3}$ and $\frac{17}{24}$

$=\frac{1}{2}[\frac{2}{3}+\frac{17}{24}]=\frac{1}{2}\left[\frac{16+17}{24}\right]=\frac{1}{2}\times \frac{33}{24}=\frac{33}{48}$

Third rational number between $\frac{17}{24}$ and $\frac{3}{4}$

$=\frac{1}{2}[\frac{17}{24}+\frac{3}{4}]=\frac{1}{2}\left[\frac{17+18}{24}\right]=\frac{1}{2}\times \frac{35}{24}=\frac{35}{48}$

$\because \frac{2}{3}<\frac{33}{48}<\frac{3}{4}$

$\therefore$ Required rational numbers

$=\frac{33}{48},\frac{17}{24},\frac{35}{48}$