Question

List five rational numbers between $\frac{1}{2}$ and $\frac{2}{3}$.

Answer 1

Given fractions $\frac{1}{2}$ & $\frac{2}{3}$
Let us make denominators are equals as follow
$\frac{1}{2}=\frac{1\times 3}{2\times 3}$ & $\frac{2}{3}=\frac{2\times 2}{3\times 2}$
$\Rightarrow \frac{1}{2}=\frac{3}{6}$ & $\frac{2}{3}=\frac{4}{6}$
The resultant numerators values are not sufficient to insert the the five numbers between them, so again multiply $\frac{6}{6}$ to the resultant fractions to make their denominators are equal as follow.

$\frac{3\times 6}{6\times 6}=\frac{18}{36}$ &
$\frac{4\times 6}{6\times 6}=\frac{24}{36}$
Now, insert $5$ integers between the numerators $18$ and $24$ as $18<19<20<21<22<23<24$
Dividing 6 on both sides, we get
$\Rightarrow \frac{18}{36}<\frac{19}{36}<\frac{20}{36}<\frac{20}{36}<\frac{21}{36}<\frac{22}{36}<\frac{23}{36}<\frac{24}{36}$
Therefore, the five rational numbers between the given numbers are $\frac{19}{36},\frac{20}{36},\frac{20}{36},\frac{21}{36},\frac{22}{36},\frac{23}{36}$