Find three rational numbers between

Solution:

Rational number between $\frac{2}{3}$ and $\frac{3}{4}$ :

$\frac{1}{2}\left(\frac{2}{3}+\frac{3}{4}\right)$

$=\frac{1}{2}\left(\frac{8+9}{12}\right)$

$=\frac{17}{24}$

We know,

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

Rational number between $\frac{2}{3}$ and $\frac{17}{24}$ :

$\frac{1}{2}\left(\frac{2}{3}+\frac{17}{24}\right)$

$=\frac{1}{2}\left(\frac{16+17}{24}\right)$

$=\frac{1}{2}\left(\frac{33}{24}\right)$

$=\frac{33}{48}=\frac{33 \div 3}{48 \div 3}=\frac{11}{16}$

Rational number between $\frac{17}{24}$ and $\frac{3}{4}$ :

$\frac{1}{2}\left(\frac{17}{24}+\frac{3}{4}\right)$

$=\frac{1}{2}\left(\frac{17+18}{24}\right)$

$=\frac{1}{2}\left(\frac{35}{24}\right)$

$=\frac{35}{48}$

We know,

$\frac{2}{3}<\frac{11}{16}<\frac{17}{24}<\frac{35}{48}<\frac{3}{4}$

Thus, the three rational numbers are $\frac{11}{16}, \frac{17}{24}$ and $\frac{35}{48}$.