Q 6.     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)   $\frac{1}{3}\ and\ \frac{-1}{9}$

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

Answers (1)

To compare we multiply both numbers with denominators:

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

$\Rightarrow \frac{-7\times9}{21\times9} = \frac{-63}{189}$

$\Rightarrow \frac{3\times21}{9\times21} = \frac{63}{189}$

$\Rightarrow \frac{-63}{189} \neq \frac{63}{189}$

Here, they are equal but are in opposite signs hence, $\frac{-7}{21}\ and\ \frac{3}{9}$ do not represent the same rational numbers.

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

$\Rightarrow \frac{-16\times-25}{20\times-25} = \frac{400}{-500}$

$\Rightarrow \frac{20\times20}{-25\times20} = \frac{400}{-500}$

$\Rightarrow \frac{400}{-500} = \frac{400}{-500}$

So, they represent the same rational number.

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

Here, Both represents the same number as these minus signs on both numerator and denominator of $\frac{-2}{-3} = \frac{2}{3}$ will cancel out and gives the positive value.

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

$\Rightarrow \frac{-3\times20}{5\times20} = \frac{-60}{100}$

$\Rightarrow \frac{-12\times5}{20\times5} = \frac{-60}{100}$

$\Rightarrow \frac{-60}{100} = \frac{-60}{100}$

So, they represent the same rational number.

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

$\Rightarrow \frac{8\times15}{-5\times15} = \frac{120}{-75}$

$\Rightarrow \frac{-24\times-5}{15\times-5} = \frac{120}{-75}$

$\Rightarrow \frac{120}{-75} = \frac{120}{-75}$

So, they represent the same rational number.

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

$\Rightarrow \frac{1\times9}{3\times9} = \frac{9}{27}$

$\Rightarrow \frac{-1\times3}{9\times3} = \frac{-3}{27}$

$\Rightarrow \frac{9}{27} \neq \frac{-3}{27}$

So, They do not represent the same rational number.

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

Here, the denominators of both are the same but $-5 \neq 5$ .

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