Ex.9.1 Q1 Rational-Numbers Solution - NCERT Maths Class 7

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 \(\begin{align}\frac{2}{3}\end{align}\)

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

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