Ex.9.1 Q6 Rational-Numbers Solution - NCERT Maths Class 7

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Question

Which of the following pairs represent the same rational number?

(i) \(\begin{align}\frac{{ - 7}}{{21}}\end{align}\) and \(\begin{align}\frac{3}{9} \end{align}\) (ii) \(\begin{align}\frac{{ - 16}}{{20}}\end{align}\) and \(\begin{align}\frac{{20}}{{ - 25}}\end{align}\) (iii) \(\begin{align}\frac{{ - 2}}{{ - 3}}\end{align}\) and \(\begin{align}\frac{2}{3}\end{align}\)
(iv) \(\begin{align}\frac{-3}{5}\end{align}\) and \(\begin{align}\frac{{ - 12}}{{20}}\end{align}\) (v) \(\begin{align}\frac{8}{{ -5}}\end{align}\) and \(\begin{align}\frac{{ - 24}}{{15}}\end{align}\) (vi) \(\begin{align}\frac{1}{3}\end{align}\) and \(\begin{align}\frac{{ - 1}}{9}\end{align}\)
(vii) \(\begin{align}\frac{{ - {\rm{5}}}}{{ - {\rm{9}}}}\end{align}\) and \(\begin{align}\frac{{\rm{5}}}{{ - {\rm{9}}}}\end{align}\)    

Text Solution

What is known?

Two pair of rational numbers.

What is unknown?

Which pair represent the same rational number.

Reasoning:

In such type of questions, reduce the rational numbers to the lowest or simplest form. By reducing them to the simplest form you can easily get out the same rational numbers.

Steps:

(i) \(\begin{align}\frac{{ - {\rm{7}}}}{{{\rm{21}}}}\end{align}\) and \(\begin{align}\frac{{\rm{3}}}{{\rm{9}}}\end{align}\)

On reducing them to the simplest form, we get

\(\begin{align}\frac{{ - 7}}{{21}}\end{align}\) \(\begin{align}=\end{align}\)\(\begin{align}\frac{{ - 1}}{3}\,\end{align}\)and \(\begin{align}\frac{3}{9}\end{align}\) \(\begin{align}=\end{align}\) \(\begin{align}\frac{1}{3}\end{align}\)

Since, \(\begin{align}\frac{{ - 1}}{3}\; \ne \frac{1}{3}\end{align}\)

Therefore,  \(\begin{align}\frac{{ - 7}}{{21}}\end{align}\) and \(\begin{align}\frac{3}{9} \end{align}\)and does not represent the pair of same rational numbers.

(ii)  \(\begin{align}\frac{{ - 16}}{{20}}\end{align}\) and \(\begin{align}\frac{{20}}{{ - 25}}\end{align}\)

On reducing them to the simplest form, we get

\(\begin{align}\frac{{ - 16}}{{20}} = \frac{{ - 4}}{5}\end{align}\) and \(\begin{align}\frac{{20}}{{ - 25}} = \frac{4}{{ - 5}}\end{align}\)

since, \(\begin{align}\frac{{ - 4}}{5} = \frac{4}{{ - 5}}\end{align}\)

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

(iii) \(\begin{align}\frac{{ - 2}}{{ - 3}}\end{align}\) and \(\begin{align}\frac{2}{3}\end{align}\)

On reducing them to the simplest form, we get

\(\begin{align}\frac{{ - 2}}{{ - 3}} = \frac{2}{3}\end{align}\) and \(\begin{align}\frac{2}{3} = \frac{2}{3}\end{align}\)

since, \(\begin{align}\frac{-2}{-3} = \frac{2}{3}\end{align}\)  

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

(iv) \(\begin{align}\frac{{ - 3}}{5}{\rm{and}}\frac{{ - 12}}{{20}}\end{align}\) 

On reducing them to the simplest form, we get

\(\begin{align}\frac{{ - 3}}{5} = \frac{{ - 3}}{5}\end{align}\) and \(\begin{align}\frac{{ - 12}}{{20}} = \frac{{ - 3}}{5}\end{align}\)

since, \(\begin{align}\frac{{ - 3}}{5} = \frac{{ - 3}}{5}\end{align}\)  

Therefore, \(\begin{align}\frac{{ - 3}}{5}\end{align}\) and \(\begin{align}\frac{{ - 12}}{{20}}\end{align}\) represent the pair of same rational numbers.

(v) \(\begin{align}\frac{8}{{ -5}}\end{align}\) and \(\begin{align}\frac{{ - 24}}{{15}}\end{align}\)

On reducing them to the simplest form, we get

\(\begin{align}\frac{8}{{ - 5}} = \frac{{ - 8}}{{ - 5}}\end{align}\) and \(\begin{align}\frac{{ - 24}}{{15}} = \frac{{ - 8}}{5}\end{align}\)

since, \(\begin{align}\frac{8}{{ - 5}} = \frac{{ - 8}}{5}\end{align}\)

Therefore,\(\begin{align}\frac{8}{{ - 5}}\end{align}\) and \(\begin{align}\frac{{ - 24}}{{15}}\end{align}\)  represent the pair of same rational numbers.

(vi) \(\begin{align}\frac{1}{3}\end{align}\) and \(\begin{align}\frac{{ - 1}}{9}\end{align}\)

On reducing them to the simplest form, we get

\(\begin{align}\frac{1}{3} = \frac{1}{3}\end{align}\) and \(\begin{align}\frac{{ - 1}}{9} = \frac{{ - 1}}{9}\end{align}\)

since, \(\begin{align}\frac{1}{3} \ne \frac{{ - 1}}{9}\end{align}\)

Therefore,\(\begin{align}\frac{1}{3}\end{align}\) and \(\begin{align}\frac{-1}{9}\end{align}\)   represent the pair of same rational numbers

(vii) \(\begin{align}\frac{{ - {\rm{5}}}}{{ - {\rm{9}}}}\end{align}\) and \(\begin{align}\frac{{\rm{5}}}{{ - {\rm{9}}}}\end{align}\)

On reducing them to the simplest form, we get

\(\begin{align}\frac{{ - 5}}{{ - 9}} = \frac{5}{9}\end{align}\) and \(\begin{align}{\mkern 1mu} \frac{5}{{ - 9}} = \frac{{ - 5}}{9}\end{align}\)

since, \(\begin{align}\frac{5}{9} \ne \frac{{ - 5}}{9}\end{align}\)

Therefore, \(\begin{align}\frac{{ - 5}}{{ - 9}}\end{align}\)  and \(\begin{align}\frac{{ 5}}{{ - 9}}\end{align}\)  does not represent the pair of same rational numbers.

 

  
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