Which of the following pairs represent the same rational number?
$(i)$. $-\frac{7}{21}$ and $\frac{3}{9}$
$(ii)$. $-\frac{16}{20}$ and $\frac{20}{-25}$ ​
$(iii)$. $\frac{-2}{-3}$ and $\frac{2}{3}$
$(iv)$. $\frac{-3}{5}$ and $\frac{-12}{20}$
$(v)$. $\frac{8}{5}$ and $\frac{-24}{15}$
$(vi)$. $\frac{1}{3}$ and $\frac{-1}{9}$
$(viii)$ $\frac{-5}{-9}$ and $\frac{5}{-9}$

Given: Pairs of rational numbers:

$(i)$. $-\frac{7}{21}$ and $\frac{3}{9}$

$(ii)$. $-\frac{16}{20}$ and $\frac{20}{-25}$

$(iii)$. $\frac{-2}{-3}$ and  $\frac{2}{3}$

$(iv)$. $\frac{-3}{5}$ and $\frac{-12}{20}$

$(v)$. $\frac{8}{-5}$ and $\frac{-24}{15}$

$(vi)$. $\frac{1}{3}$ and $\frac{-1}{9}$

$(viii)$ $\frac{-5}{-9}$ and $\frac{5}{-9}$

To do: To find pairs that represent the same rational number.

Solution: $(i)$. $-\frac{7}{21}$ and $\frac{3}{9}$

Given pairs are: $(i)$. $-\frac{7}{21}$ and $\frac{3}{9}$

On reducing the given fractions to the simplest form:

$-\frac{7}{21}$

$= -\frac{1}{3}$

And $\frac{3}{9}$

$= \frac{1}{3}$

On comparing both fractions we have: $-\frac{1}{3} ≠ \frac{1}{3}$

Therefore, pairs $-\frac{7}{21}$ and $\frac{3}{9}$ do not represent the same rational numbers.

$(ii)$. $-\frac{16}{20}$ and $\frac{20}{(-25)}$

Given pair of rational numbers : $-\frac{16}{20}$ and $\frac{20}{(-25)}$

On reducing both rational numbers to their simplest form:

$-\frac{16}{20}$

$= -\frac{4}{5}$

And $\frac{20}{(-25)} = \frac{4}{(-5)}$

On comparing the simplest form of given pair of rational numbers we have:

$-\frac{4}{5} = \frac{4}{(-5)}$

Therefore, $-\frac{16}{20}$ and $\frac{20}{(-25)}$ represents the pair of same rational numbers.

$(iii)$. $-\frac{2}{(-3)}$ and $\frac{2}{3}$

Given pair of rational numbers : $-\frac{2}{(-3)}$ and $\frac{2}{3}$

On reducing both rational numbers to their simplest form:

$-\frac{2}{(-3)} = \frac{2}{3}$ and $\frac{2}{3} = \frac{2}{3}$

On comparing the simplest form of given pair of rational numbers we have:, $\frac{-2}{-3} = \frac{2}{3}$

Therefore, $-\frac{2}{(-3)}$ and $\frac{2}{3}$ represents the pair of same rational numbers.

$(iv)$. $-\frac{3}{5}$ and $-\frac{12}{20}$

Given pair of rational numbers : $-\frac{3}{5}$ and $-\frac{12}{20}$

On reducing both rational numbers to their simplest form:

$-\frac{3}{5} = -\frac{3}{5}$

And $-\frac{12}{20} = -\frac{3}{5}$

On comparing the simplest form of given pair of rational numbers we have: $-\frac{3}{5} = -\frac{3}{5}$

Therefore, $-\frac{3}{5}$ and $-\frac{12}{20}$ represent the pair of same rational numbers.

$(v)$. $\frac{8}{(-5)}$ and $-\frac{24}{15}$

Given pair of rational numbers: $\frac{8}{(-5)}$ and $-\frac{24}{15}$

On reducing both rational numbers to their simplest form:

$\frac{8}{(-5)} = -\frac{8}{5}$

And $-\frac{24}{15} = -\frac{8}{5}$

On comparing the simplest form of given pair of rational numbers we have: $\frac{8}{-5} = -\frac{8}{5}$

Therefore, $\frac{8}{-5}$ and $-\frac{24}{15}$ represent the pair of the same rational numbers.

$(vi)$. $\frac{1}{3}$ and $-\frac{1}{9}$

Given pair of rational numbers: $\frac{1}{3}$ and $-\frac{1}{9}$

On reducing both rational numbers to their simplest form:

$\frac{1}{3} = \frac{1}{3}$

And $-\frac{1}{9} = -\frac{1}{9}$

On comparing the simplest form of given pair of rational numbers we have: $\frac{1}{3} ≠ -\frac{1}{9}$

Therefore, $\frac{1}{3}$ and $-\frac{1}{9}$ do not represent the pair of the same rational numbers.

$(vii)$. $-\frac{5}{(-9)}$ and $\frac{5}{(-9)}$

Given pair of rational numbers: $-\frac{5}{(-9)}$ and $\frac{5}{(-9)}$

On reducing both rational numbers to their simplest form:

$-\frac{5}{(-9)} = \frac{5}{9}$ and $\frac{5}{(-9)} = -\frac{5}{9}$

On comparing the simplest form of given pair of rational numbers we have: $\frac{5}{9} ≠ -\frac{5}{9}$

Therefore, $\frac{-5}{-9}$ and $\frac{5}{-9}$ do not represent the pair of the same rational numbers.