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.

 

 

 

 

 

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