Question

List five rational numbers between: $\frac{-4}{5}$ and $\frac{-2}{3}$.

Answer 1

Given rational numbers are $\frac{-4}{5},\frac{-2}{3}$

Lets equate the denominators are same,

$\Rightarrow \frac{-4}{5}\times \frac{3}{3},\frac{-2}{3}\times \frac{5}{5}$

$=\frac{-12}{15},\frac{-10}{15}$

Since, we require five rational numbers between given rational numbers.

Here the numerators are, $-12,-10$

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

Integers between $-24$ and $-20$ are $-23,-22,-21$

Here, we have only three integers.

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

Integers between $-36$ and $-30$ are $-35,-34,-33,-32,-31$

Here, we have five integers.

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

$\frac{-12}{15}\times \frac{3}{3},\frac{-10}{15}\times \frac{3}{3}$

$\frac{-36}{45},\frac{-30}{45}$

$\Rightarrow$ Five rational numbers between $\frac{-12}{15},\frac{-10}{15}$ are $\frac{-35}{45},\frac{-35}{45},\frac{-35}{45},\frac{-35}{45},\frac{-35}{45}$

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