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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.
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