Ch. - 9 Rational Numbers

Ch. - 9 Rational Numbers

Chapter 9 Ex.9.1 Question 1

List five rational numbers between: 

(i) \(–1\) and \(0\)

(ii) \(–2\) and \(-1\)

(iii) \(\begin{align}\frac{{ - 4}}{5}\end{align}\) and \(\begin{align}\frac{{ - 2}}{3}\end{align}\)

 (iv) \(\begin{align} - \frac{1}{2}\end{align}\) and \(\begin{align}\frac{2}{3}\end{align}\)

 

Solution

Video Solution

 

What is known?  

Two integers.

What is unknown?

Five rational numbers between the given two integers.

Reasoning:

These questions can be solved easily with the concept of like fractions, First make the fractions like by making their denominator equal. You can make denominator equal either by taking \(L.C.M\) of denominator or by multiplying both numerator and denominator by same integer. By applying these methods, you can get the like fractions and can easily find out the rational numbers between the given numbers. 

Steps:

(i) \(–1\) and \(\)\(0\)
Multiplying both numerator and denominator by \(6,\) we get 

\[\begin{align}\frac{{ - 1 \times 6}}{{1 \times 6}} &= \frac{{ - 6}}{6},\\\frac{{0 \times 6}}{{1 \times 6}} &= \frac{0}{6}\end{align}\]

Five rational numbers between \(-1 \)and \(0\) are,

\[\begin{align}\frac{{ - 6}}{6} \!<\! \frac{{ - 5}}{6} \!<\! \frac{{ - 4}}{6} \!<\! \frac{{ - 3}}{6} \!<\! \frac{{ - 2}}{6} \!<\! \frac{{ - 1}}{6} \!<\! \frac{0}{6}\\-1 \!<\! \frac{{ - 5}}{6} \!<\! \frac{{ - 4}}{6} \!<\! \frac{{ - 3}}{6} \!<\! \frac{{ - 2}}{6} \!<\! \frac{{ - 1}}{6} \!<\! 0 \end{align}\]

Thus, the five rational numbers between\( -1\) and \(0 \) are

\[\frac{{ - 5}}{6},\frac{{ - 2}}{3},\frac{{ - 1}}{2},\frac{{ - 1}}{3},\frac{{ - 1}}{6}\]

(ii) \(–2\) and \(-1\)
Multiplying both numerator and denominator by \(6\), we get

\[\begin{align}\frac{{ - 2{\times}6}}{{1{\times}6}} &= \frac{{ - 12}}{6},\\\frac{{ - 1{\times}6}}{{1{\times}6}} &= \frac{{ - 6}}{6}\end{align}\]

Five rational numbers between \(-2\) and \(-1\) are,

\[\frac{{ - 12}}{6} \!<\! \frac{{ - 11}}{6} \!<\! \frac{{ - 10}}{6} \!<\! \frac{{ - 9}}{6} \!<\! \frac{{ - 8}}{6} \!<\! \frac{{ - 7}}{6} \!<\! \frac{{ - 6}}{6}\]

\[ - 2 < \frac{{ - 11}}{6} \!<\! \frac{{ - 10}}{6} \!<\! \frac{{ - 9}}{6} \!<\! \frac{{ - 8}}{6} \!<\! \frac{{ - 7}}{6} \!<\! - 1\]

Thus, the five rational numbers between \(-2 \) and \(-1 \) are

\[\frac{{ - 11}}{6},\frac{{ - 5}}{3},\frac{{ - 3}}{2},\frac{{ - 4}}{3},\frac{{ - 7}}{6}\]

\({\rm{(iii)}}\,\,\begin{align}\frac{{ - 4}}{5}\,{\rm{and}}\,\,\frac{{ - 2}}{3}\end{align}\)

Converting \(\begin{align}\frac{{ - 4}}{5} \end{align}\) and \(\begin{align}\frac{{ - 2}}{3} \end{align}\) into like fractions, we get

\[\begin{align}\frac{{ - 4}}{5} = \frac{{ - {4 \times 9}}}{{5\times 9}} = \frac{{ - 36}}{{45}}\\\frac{\begin{align}
\\- 2\end{align}}{3} = \frac{{ - 2{\times 15}}}{{3 \times 15}} = \frac{{ - 30}}{{45}}\end{align}\]

Five rational numbers between \(\begin{align}\frac{{ - 4}}{5} \end{align}\) and \(\begin{align}\frac{{ - 2}}{3}\,\text{are,}\end{align}\)

\[\frac{{ - 36}}{{45}} \!<\! \frac{{ - 35}}{{45}} \!<\! \frac{{ - 34}}{{45}} \!<\! \frac{{ - 33}}{{45}} \!<\! \frac{{ - 32}}{{45}} \!<\! \frac{{ - 31}}{{45}} \!<\! \frac{{ - 30}}{{45}}\]

\[\frac{{ - 4}}{5} \!<\! \frac{{ - 35}}{{45}} \!<\! \frac{{ - 34}}{{45}} \!<\! \frac{{ - 33}}{{45}} \!<\! \frac{{ - 32}}{{45}} \!<\! \frac{{ - 31}}{{45}} \!<\! \frac{{ - 2}}{3}\]

Thus, the five rational numbers between \(\begin{align}\frac{{ - 4}}{5} \end{align}\) and \(\begin{align}\frac{{ - 2}}{3} \,\text{are,}\end{align}\)

\[\frac{{ - 7}}{9},\frac{{ - 34}}{{45}},\frac{{ - 11}}{{15}},\frac{{ - 32}}{{45}},\frac{{ - 31}}{{45}}\]

\(\begin{align}{\rm{(iv)}} - \frac{1}{2}\,\,{\rm{and}}\,\,\frac{2}{3}\end{align}\)

Converting \(\begin{align} - \frac{1}{2}\end{align}\) and \(\begin{align}\frac{2}{3} \end{align}\) into like fractions, we get

\[\begin{align}\frac{{ - 1}}{2} &= \frac{{ - 1 \times 3}}{{2\times3}}\\&= \frac{{ - 3}}{6}\end{align}\]

\[\begin{align}\frac{2}{3} &= \frac{{2 \times 2}}{{3 \times 2}}\\&= \frac{4}{6}\end{align}\]

Five rational numbers between \(\begin{align}\frac{-1}{2} \,\rm{and} \,\frac{2}{3}\,\text{are,}\end{align}\)

\[\begin{align}\frac{{ - 3}}{6} \!<\! \frac{{ - 2}}{6} \!<\! \frac{{ - 1}}{6} \!<\! 0 \!<\! \frac{2}{6} \!<\! \frac{3}{6} \!<\! \frac{4}{6}\end{align}\]

Therefore, the five rational numbers between\(\begin{align}\frac{-1}{2} \,\rm{and} \,\frac{2}{3}\,\text{are,}\end{align}\)

\[\begin{align}\frac{{ - 1}}{3},\,\,\frac{{ - 1}}{6},\,\,\,0,\,\,\frac{1}{3}\,,\,\,\frac{1}{2}\end{align}\]

Chapter 9 Ex.9.1 Question 2

Write four more rational numbers in each of the following patterns:

(i) \(\begin{align}\frac{{ - 3}}{5},\frac{{ - 6}}{5},\frac{{ - 9}}{5},\frac{{ - 12}}{5}\end{align}\)

(iii)\(\begin{align}\frac{{ - 1}}{4},\frac{{ - 2}}{8},\frac{{ - 3}}{{12}}\end{align}\)

(iii)\(\begin{align}\frac{{ - 1}}{6},\frac{2}{{ - 12}},\frac{3}{{ - 18}},\frac{4}{{ - 24}}\end{align}\)

(iv) \(\begin{align}\frac{{ - 2}}{3},\frac{2}{{ - 3}},\frac{4}{{ - 6}},\frac{6}{{ - 9}}\end{align}\)

 

Solution

Video Solution

 

What is known?

Patterns of the numbers.

What is unknown?

Four more rational numbers in each of the given patterns.

Reasoning:

This question is based on a definite pattern, while solving such type of questions observe the numerator and denominator carefully. Here both numerator and denominator are following a definite pattern, observe this pattern and follow the same pattern, you can easily find out the next four rational numbers.

Steps:

(i) \(\begin{align}\frac{{ - 3}}{5},\frac{{ - 6}}{{10}},\frac{{ - 9}}{{15}},\frac{{ - 12}}{{20}}\end{align}\)

\[\begin{align}\frac{{ - 3 \! \times \! 1}}{{5 \! \times \! 1}},\frac{{ - 3 \! \times \! 2}}{{5 \! \times \! 2}},\frac{{ - 3 \! \times \! 3}}{{5 \! \times \! 3}},\frac{{ - 3 \! \times \! 4}}{{5 \! \times \! 4}} \ldots \ldots\end{align} \]

Next four rational numbers in the same pattern,

\[\begin{align}\frac{{ - 3 \! \times \! 5}}{{5 \! \times \! 5}},\frac{{ - 3 \! \times \! 6}}{{5 \! \times \! 6}},\frac{{ - 3 \! \times \! 7}}{{5 \! \times \! 7}},\frac{{ - 3 \! \times \! 8}}{{5 \! \times \! 8}}\end{align}\]

Therefore, the numbers are \(\begin{align}\frac{{ - 15}}{{25}},\frac{{ - 18}}{{30}},\frac{{ - 21}}{{35}},\;\frac{{ - 24}}{{40}}\end{align}\)

\(\begin{align} {\rm{(ii)}} \,\,& \frac{{ - 1}}{4},\frac{{ - 2}}{8},\frac{{ - 3}}{{12}} \end{align}\)

\(\begin{align} \frac{{ - 1 \times 1}}{{4 \times 1}},\frac{{ - 1 \times 2}}{{4 \times 2}},\frac{{ - 1 \times 3}}{{4 \times 3}} \ldots \ldots \end{align}\)

Next four rational numbers in the same pattern,

\[\begin{align}\frac{{ - 1 \times 4}}{{4 \times 4}},\frac{{ - 1 \times 5}}{{4 \times 5}},\frac{{ - 1 \times 6}}{{4 \times 6}},\frac{{ - 1 \times 7}}{{4 \times 7}}\end{align}\]

Therefore, the numbers are \(\begin{align}\frac{{ - 4}}{{16}},\frac{{ - 5}}{{20}},\frac{{ - 6}}{{24}},\frac{{ - 7}}{{28}} \cdots\end{align}\)

\(\begin{align} {\rm{(iii)}} & \frac{{ - 1}}{6},\frac{2}{{ - 12}},\frac{3}{{ - 18}},\frac{4}{{ - 24}}\\ & \frac{{1 \times 1}}{{6{\times1}}},\frac{{1\times2}}{{6{\times 2}}},\frac{{1{\times3}}}{{6\times 3}},\frac{{1{\times4}}}{{6{\times 4}}} \ldots \end{align}\)

Next four rational numbers in the same pattern,

\[\begin{align}\frac{{1 \times 5}}{{-6 \times5}},\frac{{1 \times 6}}{{-6 \times 6}},\frac{{1 \times 7}}{{-6 \times7}},\frac{{1 \times 8}}{{-6 \times8}}\end{align}\]

Therefore, the numbers are\(\begin{align}\frac{5}{{ - 30}},\frac{6}{{ - 36}},\frac{7}{{ - 42}},\frac{8}{{ - 48}} \ldots \ldots\end{align}\)

\(\begin{align} {\rm{(iv)}} & \frac{{ - 2}}{3},\frac{2}{{ - 3}},\frac{4}{{ - 6}},\frac{6}{{ - 9}} \end{align}\)

\(\begin{align} \frac{{-2 \times1}}{{3 \times 1}},\frac{{2 \times 1}}{{-3 \times1}},\frac{{2 \times 2}}{{-3 \times 2}},\frac{{2 \times 3}}{{-3 \times 3}} \ldots \end{align}\)

Next four rational numbers in the same pattern,

\[\begin{align}\frac{{2 \times 4}}{{-3 \times4}},\frac{{2 \times 5}}{{-3 \times5}},\frac{{2 \times 6}}{{-3 \times6}},\frac{{2 \times 7}}{{-3 \times7}}\end{align}\]

Therefore, the numbers are\(\begin{align}\frac{8}{{ - 12}},\frac{{10}}{{ - 15}},\frac{{12}}{{ - 18}},\;\frac{{14}}{{ - 21}}\end{align}\)

Chapter 9 Ex.9.1 Question 3

Give four rational numbers equivalent to:

\(\begin{align}{{\rm{ (i) }}\;\;\frac{{ - 2}}{7}}\end{align}\)

\(\begin{align}{{\rm{(ii) }}\;\;\frac{5}{{ - 3}}}\end{align}\)

\(\begin{align}{\rm{(iii)}}\,\,\,\frac{4}{9}\end{align}\)

 

Solution

Video Solution

 

What is known?

Three rational numbers.

What is unknown?

Four rational numbers equivalent to each of the given rational number.

Reasoning:

To find out the equivalent fraction of any rational number, multiply the numerator and the denominator of the given number by the same numbers. Remember here it is asked for four equivalent rational numbers that means you have to multiply four different numbers, one by one in both numerator and denominator of the given number.

Steps:

(i) \(\begin{align}\frac{{ - 2}}{7}\end{align}\)

Multilying both numerator and denominator with the same number, we get

\[\begin{align}  \frac{-2\times 2}{7\times 2}&=\frac{-4}{14}, \\  \frac{-2\times 3}{7\times 3}&=\frac{-6}{21}, \\  \frac{-2\times 4}{7\times 4}&=\frac{-8}{28}, \\  \frac{-2\times 5}{7\times 5}&=\frac{-10}{35} \\ \end{align}\]

Therefore, the equivalent fractions to the number \(\begin{align}\frac{{ - 2\;}}{7}\end{align}\)are,

\[\begin{align}\frac{{ - 4}}{{14}},\frac{{ - 6}}{{21}},\frac{{ - 8}}{{28}},\frac{{ - 10}}{{35}}\end{align}\]

 (ii)  \(\begin{align}\frac{5}{{ - 3}}\end{align}\)

Multiplying both numerator and denominator with the same number, we get

\[\begin{align}  & \frac{5\times 2}{-3\times 2}=\frac{10}{-6}, \\  & \frac{5\times 3}{-3\times 3}=\frac{15}{-9}, \\  & \frac{5\times 4}{-3\times 4}=\frac{20}{-12}, \\  & \frac{5\times 5}{-3\times 5}=\frac{25}{-15} \\ \end{align}\]

Therefore, the equivalent fractions to the number \(\begin{align}\frac{5}{{ - 3}}\end{align}\)are,

\[\begin{align}\frac{{10}}{{ - 6}},\frac{{15}}{{ - 9}},\frac{{20}}{{ - 12}},\frac{{25}}{{ - 15}}\end{align}\]

(iii) \(\begin{align} \frac{4}{9}\end{align}\)

Multiplying both numerator and denominator with the same number, we get

\[\begin{align}  \frac{4\times 2}{9\times 2}&=\frac{8}{18}, \\  \frac{4\times 3}{9\times 3}&=\frac{12}{27}, \\  \frac{4\times 4}{9\times 4}&=\frac{16}{36}, \\  \frac{4\times 5}{9\times 5}&=\frac{20}{45} \\ \end{align}\]

Therefore, the equivalent fractions to the number \(\begin{align}\frac{4}{9}\end{align}\)are,

\[\begin{align}\frac{8}{{18}},\frac{{12}}{{27}},\frac{{16}}{{36}}\,{\rm{and}}\,\frac{{20}}{{45}}\end{align}\]

Chapter 9 Ex.9.1 Question 4

Draw the number line and represent the following rational numbers on it:

(i) \(\begin{align}\frac{3}{4}\end{align}\)

(ii) \(\begin{align}{\frac{{ - 5}}{8}}\end{align}\)

(iii) \(\begin{align}\frac{{ - 7}}{4}\end{align}\)

(iv) \(\begin{align}\frac{7}{8}\end{align}\)

 

Solution

Video Solution

 

Steps:

 

Chapter 9 Ex.9.1 Question 5

The points \(P, Q, R, S, T, U, A \) and \(B\) on the number line are such that, \(TR = RS = SU\) and \(AP = PQ = QB\). Name the rational numbers represented by \(P, Q, R\) and \(S\).

 

Solution

Video Solution

 

Steps:

Distance between \(U\) and \(T = 1\) unit. It is divided into three equal parts

\[\begin{align}{\rm{TR}} = {\rm{RS}} = {\rm{SU}} = \frac{1}{3}\\{\rm{R}} =  - 1 - \frac{1}{3} = \frac{{ - 4}}{3}\\{\rm{S}} =  - 1 - \frac{2}{3} = \frac{{ - 5}}{3}\end{align}\]

Similarly, \(AB = 1\) unit

It is divided into three equal parts

\[\begin{align}{\rm{AP}} = {\rm{PQ}} &= {\rm{QB}} = \frac{1}{3}\\{\rm{P}} = 2 + \frac{1}{3} &= \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\\{\rm{Q}} = 2 + \frac{2}{3} &= \frac{6}{3} + \frac{2}{3} = \frac{8}{3}\end{align}\]

Thus, the rational number \(P, Q, R \) and \(S\) are

\[\begin{align} \frac{7}{3},\frac{8}{3},\frac{{ - 4}}{3}\,\,{\rm{and}}\,\frac{{ - 5}}{3}\end{align}\]

Chapter 9 Ex.9.1 Question 6

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

 

Solution

Video 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.

 

Chapter 9 Ex.9.1 Question 7

Rewrite the following rational numbers in the simplest form:

(i) \(\begin{align}\frac{ - 8}{6}\end{align}\) 

(ii) \(\begin{align}\frac{25}{45}\end{align}\)

(iii) \(\begin{align}\frac{ -44}{72}\end{align}\)

(iv)  \(\begin{align}\frac{ -8}{10}\end{align}\)

 

Solution

Video Solution

 

What is known?

Rational numbers.

What is unknown?

Simplest form of the given rational numbers.

Reasoning:

While solving such type of questions find the \(\begin{align}H.C.F\end{align}\) of numerator and denominator and then divide both numerator and denominator by the \(\begin{align}H.C.F\end{align}\). After dividing it you will get the simplest form of the rational number.
 

Steps:

(i) \(\begin{align}\frac{ - 8}{6}\end{align}\) 
\(\begin{align}H.C.F\end{align}\).of \(\begin{align}8\end{align}\) and \(\begin{align}6\end{align}\) is two. Dividing the numerator and denominator by \(\begin{align}H.C.F\end{align}\)., we get

\[\begin{align}\frac{{ - 8 \div 2}}{{6 \div 2}} = \frac{{ - 4}}{3}\end{align}\]

 

(ii) \(\begin{align}\frac{25}{45}\end{align}\)
\(\begin{align}H.C.F\end{align}\) of \(\begin{align}25\end{align}\) and \(\begin{align}45\end{align}\) is \(\begin{align} 5\end{align}\). Dividing the numerator and denominator by \(\begin{align}H.C.F\end{align}\)., we get,

\[\begin{align}\frac{{25 \div 5}}{{45 \div 25}} = \frac{5}{9}\end{align}\]

 

(iii) \(\begin{align}\frac{ -44}{72}\end{align}\)
\(\begin{align}H.C.F\end{align}\) of \(\begin{align}44\end{align}\) and \(\begin{align}72\end{align}\) is \(\begin{align}4\end{align}\). Dividing the numerator and denominator by \(\begin{align}H.C.F\end{align}\)., we get,

\[\begin{align}\frac{{ - 44 \div 4}}{{72 \div 4}} = \frac{{ - 11}}{{18}}\end{align}\]

(iv)  \(\begin{align}\frac{ -8}{10}\end{align}\)
\(\begin{align}H.C.F\end{align}\) of \(\begin{align}8\end{align}\) and \(\begin{align}10\end{align}\) is \(\begin{align}2\end{align}\). Dividing the numerator and denominator by \(\begin{align}H.C.F\end{align}\)., we get,

\[\begin{align}\frac{{ - 8 \div 2}}{{10 \div 2}} = \frac{{ - 4}}{5}\end{align}\]

 

 

Chapter 9 Ex.9.1 Question 8

Fill in the boxes with the correct symbol out of \(>\), \(<\), and\( =\).

(i) \(\begin{align} \frac{{ - 5}}{7} \quad\boxed{\;\;}\quad \frac{2}{3} \end{align}\)

(ii) \(\begin{align} \frac{{ - 4}}{5} \quad\boxed{\;\;}\quad \frac{{ - 5}}{7}\end{align}\) 

(iii) \(\begin{align} \frac{{ - 7}}{8} \quad\boxed{\;\;}\quad \frac{{14}}{{ - 16}}\end{align}\) 

(iv)\(\begin{align} \frac{{ - 8}}{5} \quad\boxed{\;\;}\quad \frac{{ - 7}}{4}\end{align}\)

(v)\(\begin{align}  \frac{1}{{ - 3}} \quad\boxed{\;\;}\quad \frac{{ - 1}}{4}\end{align}\)

(vi)\(\begin{align}  \frac{5}{{ - 11}} \quad\boxed{\;\;}\quad \frac{{ - 5}}{{11}}\end{align}\)

(vii)  \(\begin{align}  0 \quad\boxed{\;\;}\quad \frac{{ - 7}}{6}\end{align}\)

 

 

Solution

Video Solution

 

What is known?

Two rational numbers.

What is unknown?

Comparison of the two rational numbers i.e., which one is smaller and which one is greater or are they both equal.

Reasoning:

In this type of questions, first find the \(LCM\) of the denominators of both the rational numbers. Then make denominator of each rational number equal to \(LCM\) by multiplying numerator and denominator with the same number (convert them into like fractions). Then comparison of the two numbers can be easily made.

Steps:

(i) \(\begin{align} \frac{{ - 5}}{7} \quad\boxed{\;\;}\quad \frac{2}{3} \end{align}\)

\[\begin{align}   \frac{{ - 5 \times 3}}{{7 \times 3}} &\quad\boxed{\;\;}\quad \frac{{2 \times 7}}{{3 \times 7}}\\ \frac{{ - 15}}{{21}} &\quad\boxed{\;\;}\quad \frac{{14}}{{21}}\\ \frac{{ - 15}}{{21}} &\quad\boxed{<}\quad \frac{{14}}{{21}}\end{align}\]

(Positive number is greater than the negative number)

Therefore, \(\begin{align}\frac{{ - 5}}{7} < \frac{2}{3}\end{align}\)

(ii) \(\begin{align} \frac{{ - 4}}{5} \quad\boxed{\;\;}\quad \frac{{ - 5}}{7}\end{align}\) 

\[\begin{align}\frac{{ - 4 \times 7}}{{5 \times 7}}&\quad\boxed{\;\;}\quad \frac{{ - 5 \times 5}}{{7 \times 5}}\\ \frac{{ - 28}}{{35}}&\quad\boxed{\;\;}\quad \frac{{ - 25}}{{35}}\\ \frac{{ - 28}}{{35}}&\quad\boxed{<}\quad \frac{{ - 25}}{{35}}\end{align}\]

Therefore, \(\begin{align}\frac{{ - 4}}{5}\,\, < \frac{{ - 5}}{7}\end{align}\)

(iii) \(\begin{align} \frac{{ - 7}}{8} \quad\boxed{\;\;}\quad \frac{{14}}{{ - 16}}\end{align}\) 

\[\begin{align}\frac{{ - 7 \times 2}}{{8 \times 2}}&\quad\boxed{\;\;}\quad \frac{{14\times 1}}{{ - 16\times 1}}\\ \frac{{ - 14}}{{16}} & \quad\boxed{\;\;}\quad \frac{{14}}{{ - 16}}\\ \frac{{ - 14}}{{16}} & \quad\boxed{=}\quad \frac{{ - 14}}{{16}}\\\end{align}\]

Therefore, \(\begin{align}\frac{{ - 7}}{8}\,\,\,\, = \,\,\frac{{14}}{{ - 16}}\end{align}\)

(iv)\(\begin{align} \frac{{ - 8}}{5} \quad\boxed{\;\;}\quad \frac{{ - 7}}{4}\end{align}\)

\[\begin{align} \frac{{ - 8 \times 4}}{{5 \times 4}}& \quad\boxed{\;\;}\quad \frac{{ - 7 \times 5}}{{4 \times 5}}\\ \frac{{ - 32}}{{20}} &\quad\boxed{\;\;}\quad \frac{{ - 35}}{{20}}\\ \frac{{ - 8\;}}{5} &\quad\boxed{>}\quad \frac{{ - 7\;}}{4}\end{align}\]

Therefore, \(\begin{align}\frac{{ - 8}}{5}\,\,\, > \,\,\,\frac{{ - 7}}{4}\end{align}\)

(v)\(\begin{align}\frac{1}{{ - 3}} \quad\boxed{\;\;}\quad \frac{{ - 1}}{4}\end{align}\)

\[\begin{align} \frac{{1 \times 4}}{{ - 3 \times 4}}&\quad\boxed{\;\;}\quad \frac{{ - 1 \times 3}}{{4 \times 3}}\\\frac{4}{{ - 12}}&\quad\boxed{\;\;}\quad \frac{{3}}{{ - 12}}\\\frac{1}{{ - 3}}&\quad\boxed{<}\quad \frac{{ - 1}}{{4}}\end{align}\]

Therefore,\(\begin{align}\frac{1}{{ - 3}}&\quad\boxed{>}\quad \frac{{ - 1}}{{4}}\end{align}\)

(vi)\(\begin{align} \frac{5}{{ - 11}} \quad\boxed{\;\;}\quad \frac{{ - 5}}{{11}}\end{align}\)

\[\begin{align} \frac{{ - 5}}{{11}}&\quad\boxed{\;\;}\quad \frac{{ - 5}}{{11}}\\ \frac{{ - 5}}{{11}}&\quad\boxed{=}\quad \frac{{ - 5}}{{11}}\end{align}\]

Therefore, \(\begin{align}\frac{{ - 5}}{{11}}&\quad\boxed{=}\quad \frac{{ - 5}}{{11}}\end{align}\)

(vi)  \(\begin{align}     0 \quad\boxed{\;\;}\quad \frac{{ - 7}}{6}\end{align}\)

\[\begin{align}&0\quad\boxed{\;\;}\quad\frac{{ - 7}}{6} \\&0\quad\boxed{>}\quad\frac{{ - 7}}{6}\end{align}\]

(\(0\) is always greater than negative integer)

Chapter 9 Ex.9.1 Question 9

Which is greater in each of the following:

(i) \(\begin{align}{\frac{2}{3},\frac{5}{2}}\end{align}\) 

(ii) \(\begin{align}{\frac{{ - 5}}{6},\frac{{ - 4}}{3}}\end{align}\)

(iii) \(\begin{align}\frac{{ - 3}}{4},\frac{2}{{ - 3}}\end{align}\)

iv) \(\begin{align}\frac{{ - 1}}{6},\frac{2}{{ - 12}}\end{align} \)

(v) \( \begin{align}- 3\frac{2}{7}, - 3\frac{4}{5}\end{align}\)

 

Solution

Video Solution

 

What is known?

Two rational numbers.

What is unknown?

Greater rational number of the two given rational numbers.

Reasoning:

In such type of questions take the \(L.C.M\) of denominator of both the rational numbers or convert them into like fractions. After converting them into like fractions comparison will be easy.

Steps:

(i)\(\begin{align}{\frac{2}{3},\frac{5}{2}}\end{align}\)

L.C.M of \(3\) and \( 2\) is \(6\)

\[\begin{align}{\frac{{2 \times 2}}{{3 \times 2}} = \frac{6}{9}}\end{align}\]

and

\[\begin{align}\frac{{5 \times 3}}{{2 \times 3}} = \frac{{15}}{6}\end{align}\]
Since,\(\begin{align}{\frac{4}{6} < \frac{{15}}{6}}\end{align}\)
So,\(\begin{align}{\frac{2}{3} < \frac{5}{2}}\end{align}\)

(ii) \(\begin{align}{\frac{{ - 5}}{6},\frac{{ - 4}}{3}}\end{align}\)

L.C.M of \(6\) and \(3\) is \(6\)

\[\begin{align}{\frac{{ - 5\times1}}{6\times1} = \frac{{ - 5}}{6}}\end{align}\]

and

\[\begin{align}\frac{{ - 4 \times 2}}{{3 \times 2}} = \frac{{ - 8}}{6}\end{align}\]
Since,\(\begin{align}{\frac{{ - 5}}{6} > \frac{{ - 8}}{6}}\end{align}\)
So,\(\begin{align}{\frac{ - 5}{6} > \frac{{ - 4}}{3}}\end{align}\)

(iii) \(\begin{align}\frac{{ - 3}}{4},\frac{2}{{ - 3}}\end{align}\)

L.C.M of \(4\) and \(3\) is \(12\)

\[\begin{align}\frac{ - 3}{4} = \frac{ - 3 \times 3}{4{\times3}} = \frac{ - 9}{12}\end{align}\]

and

\[\begin{align}\frac{2}{ - 3} = \frac{2{\times4}}{ - 3 \times 4} = \frac{ - 8}{12}\end{align}\]
Since, \(\begin{align}\frac{ - 9}{12} < \frac{ - 8}{12}\end{align}\)
So, \(\begin{align}\frac{ - 3}{4} < \frac{2}{ - 3}\end{align}\)

(iv) \(\begin{align}\frac{{ - 1}}{4},\frac{1}{{ 4}}\end{align} \)

\(\begin{align}\frac{{ - 1}}{4} < \frac{1}{4}\end{align}\)

(Negative number is always smaller than the positive number)

(v) \( \begin{align}- 3\frac{2}{7}, - 3\frac{4}{5}\end{align}\)

\(\begin{align} - 3\frac{2}{7} = \frac{{ - 23}}{7}\;\;{\rm{and}}\;\;\;{\mkern 1mu} 3\frac{4}{5} = \frac{{ - 19}}{5}\end{align}\)

L.C.M of \(7\) and \(5\) is \(35\)

\[\begin{align}\frac{{ - 23}}{7} &= \frac{{ - 23 \times 5}}{{7 \times 5}}\\&= \frac{{ - 115}}{{35}}\\ \frac{{ - 19}}{7} &= \frac{{ - 19\times7}}{{7 \times 5}}\\&= \frac{{ - 133}}{{35}}\end{align}\]

Since, \(\begin{align}\frac{{ - 115}}{{35}} > \frac{{ - 133}}{{35}}\end{align}\)

So,\(\begin{align}- 3\frac{2}{7}\,\,\,\,\, > \,\, - 3\frac{4}{5}\end{align}\)

Chapter 9 Ex.9.1 Question 10

Write the following rational numbers in ascending order:

(i) \(\begin{align}\frac{{ - 3}}{5},\frac{{ - 2}}{5},\frac{{ - 1}}{5}\end{align}\)

(ii) \(\begin{align}\frac{{ - 1}}{3},\frac{{ - 2}}{9},\frac{{ - 4}}{3}\end{align}\)

(iii) \(\begin{align}\frac{{ - 3}}{7},\frac{{ - 3}}{2},\frac{{ - 3}}{4}\end{align}\)

 

Solution

Video Solution

 

What is known?

Three rational numbers.

What is unknown?

Ascending order of the given rational numbers.

Reasoning:

In such type of questions take the \(L.C.M\) of denominator of the rational numbers or convert them into like fractions. After converting them into like fractions comparison will be easy.

Steps:

(i) \(\begin{align}\frac{{ - 3}}{5},\frac{{ - 2}}{5},\frac{{ - 1}}{5}\end{align}\)

Since denominator is same in all the rational numbers, these can be easily arranges into ascending order \( - 3\,\, < - 2\,\, < - 1\)

\[\begin{align}\frac{{ - 3}}{5}\,\, < \frac{{ - 2}}{5}\,\, < \frac{{ - 1}}{5}\end{align}\]

(ii) \(\begin{align}\frac{{ - 1}}{3},\frac{{ - 2}}{9},\frac{{ - 4}}{3}\end{align}\)

\(L.C.M\) of \(3\),\(9\) and \(3\) is \(9\)

So,

\[\begin{align} \frac{{ - 1}}{3} &= \frac{{ - 1 \times 3}}{{3 \times 3}}\\&= \frac{{ - 3}}{9} \\ \frac{{ - 2}}{9} &= \frac{{ - 2 \times 1}}{{9 \times 1}}\\ &=\frac{{ - 2}}{9}\end{align}\]

and

\[\begin{align} {\rm{and}}\,\,\,\frac{{ - 4}}{3} &= \frac{{ - 4 \times 3}}{{3 \times 3}}\\&= \frac{{ - 12}}{9}\end{align}\]

Arranging them into ascending order we get,

\[\begin{align}&\frac{{ - 12}}{9} < \frac{{ - 3}}{9} < \frac{{ - 2}}{9}\\&{\rm{Or}}\;\;\;{\mkern 1mu} \frac{{ - 4}}{3} < \frac{{ - 1}}{3} < \frac{{ - 2}}{9}\end{align}\]

(iii) \(\begin{align}\frac{{ - 3}}{7},\frac{{ - 3}}{2},\frac{{ - 3}}{4}\end{align}\)

\(L.C.M\) of \(7\), \(2\) and \(4\) is \(28\)

\[\begin{align}\frac{{ - 3}}{7}&= \frac{{ - 3 \times 4}}{{7 \times 4}}\\&= \frac{{ - 12}}{{28}}\\\frac{{ - 3}}{2} &= \frac{{ - 3 \times 14}}{{2 \times 14}}\\&= \frac{{ - 42}}{{28}}\\\frac{{ - 3}}{4} &= \frac{{ - 3 \times 7}}{{4 \times 7}} \\&= \frac{{ - 21}}{{28}}\end{align}\]

Arranging them into ascending order we get,

\[\begin{align}\frac{{ - 42}}{{28}} < \frac{{ - 21}}{{28}} < \frac{{ - 12}}{{28}}\end{align}\]

Therefore,

\[\begin{align} \frac{{ - 3}}{2} < \frac{{ - 3}}{4} < \frac{{ - 3}}{7}\end{align}\]

The chapter 9 begins with an introduction to Rational Numbers by firstly explaining the need for rational numbers.Then the concept of rational numbers, numerator and denominator of the rational number and equivalent rational numbers are explained in detail.This is followed by the concept of positive and negative rational numbers.Then the representation of rational numbers on a number line is studied.Rational numbers in the  standard form and comparison of rational numbers and rational numbers between two rational numbers  is the next topic of discussion.Various operations on rational numbers such as addition, subtraction, multiplication and division is explained in the last section of the chapter.

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