Question

Find five rational numbers between $\frac{2}{3}$ and $\frac{4}{5}$

Answer 1

Let us make the denominator same

$\frac{2}{3}\times \frac{5}{5}=\frac{10}{15}$

$\frac{4}{5}\times \frac{3}{3}=\frac{12}{15}$

Since, in the numerators,

$10\times 2=20,12\times 2=24$

Between $20$ and $24$ only three integers are available.

So, lets take the next multiples

$10\times 3=30,12\times 3=36$

Here, Between $30$ and $36$ five integers are available.

So, lets multiply by $\frac{3}{3}$

$\frac{10}{15}\times \frac{3}{3}=\frac{30}{45}$

$\frac{12}{15}\times \frac{3}{3}=\frac{36}{45}$

$\therefore 5$ rational numbers are $\frac{31}{45},\frac{32}{45},\frac{33}{45},\frac{34}{45},\frac{35}{45}$.

Note: There exist infinite, many rational numbers between any two rational numbers.