Question
Find three rational numbers between $\frac{2}{3}$ and $\frac{3}{4}$.
✓
Answer
First rational number between $\frac{2}{3}$ and $\frac{3}{4}$ $=\frac{1}{2}\Big[\frac{2}{3}+\frac{3}{4}\Big]$ $=\frac{1}{2}\Big[\frac{8+9}{12}\Big]$ $=\frac{1}{2}\times\frac{17}{12}$ $=\frac{17}{24}$ $\therefore\frac{2}{3}<\frac{17}{24}<\frac{3}{4}$Second rational number between $\frac{2}{3}$ and $\frac{17}{24}$
$=\frac{1}{2}\Big[\frac{2}{3}+\frac{17}{24}\Big]$ $=\frac{1}{2}\Big[\frac{16+17}{24}\Big]$ $=\frac{1}{2}\times\frac{33}{24}$ $=\frac{33}{48}$and third rational number between $\frac{17}{24}$ and $\frac{3}{4}$
$=\frac{1}{2}\Big[\frac{17}{24}+\frac{3}{4}\Big]$ $=\frac{1}{2}\Big[\frac{17+18}{24}\Big]$ $=\frac{1}{2}\times\frac{35}{24}$ $=\frac{35}{48}$ $\therefore\frac{2}{3}<\frac{33}{48}<\frac{17}{24}<\frac{35}{48}<\frac{3}{4}$ $\therefore$ Required three rational numbers $\frac{33}{48},\frac{17}{24},\frac{35}{48}$
$=\frac{1}{2}\Big[\frac{2}{3}+\frac{17}{24}\Big]$ $=\frac{1}{2}\Big[\frac{16+17}{24}\Big]$ $=\frac{1}{2}\times\frac{33}{24}$ $=\frac{33}{48}$and third rational number between $\frac{17}{24}$ and $\frac{3}{4}$
$=\frac{1}{2}\Big[\frac{17}{24}+\frac{3}{4}\Big]$ $=\frac{1}{2}\Big[\frac{17+18}{24}\Big]$ $=\frac{1}{2}\times\frac{35}{24}$ $=\frac{35}{48}$ $\therefore\frac{2}{3}<\frac{33}{48}<\frac{17}{24}<\frac{35}{48}<\frac{3}{4}$ $\therefore$ Required three rational numbers $\frac{33}{48},\frac{17}{24},\frac{35}{48}$
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