question_answer 1)

List five rational numbers between (i) ? 1 and 0                         (ii) ? 2 and ? 1 (iii) \[-\frac{4}{5}\] and \[-\frac{2}{3}\]   (iv) \[-\frac{1}{2}\]and \[\frac{2}{3}\].

Answer:

                (i) We have,  \[-1=\frac{-1}{1}=\frac{-1\times 10}{1\times 10}=\frac{-10}{10},\]                 \[0=\frac{0}{1}=\frac{0\times 10}{1\times 10}=\frac{0}{10}\]                 So,  \[\frac{-10}{10}<-\frac{9}{10}<-\frac{8}{10}<-\frac{7}{10}<-\frac{6}{10}<-\frac{5}{10}<\frac{0}{10}\] \[\Rightarrow \]               \[-1<-\frac{\mathbf{9}}{\mathbf{10}}<-\frac{\mathbf{8}}{\mathbf{10}}<-\frac{\mathbf{7}}{\mathbf{10}}<-\frac{\mathbf{6}}{\mathbf{10}}<-\frac{\mathbf{5}}{\mathbf{10}}<0\] So, five rational numbers between \[-1\] and 0 are \[-\frac{9}{10},-\frac{8}{10},-\frac{7}{10},-\frac{6}{10}\]and \[-\frac{5}{10}\]. (ii) We have, \[-2=-\frac{2}{1}=\frac{-2\times 10}{1\times 10}=-\frac{20}{10}\] \[-1=-\frac{1}{1}=\frac{-1\times 10}{1\times 10}=\frac{-10}{10}\] So,          \[-\frac{20}{10}<-\frac{19}{10}<-\frac{18}{10}<-\frac{17}{10}<-\frac{16}{10}<-\frac{15}{10}<-\frac{10}{10}\] \[\Rightarrow \]               \[-\mathbf{2}<-\frac{\mathbf{19}}{\mathbf{10}}<\frac{\mathbf{-18}}{\mathbf{10}}<\frac{\mathbf{-17}}{\mathbf{10}}<\frac{\mathbf{-16}}{\mathbf{10}}<-\frac{\mathbf{15}}{\mathbf{10}}>-\mathbf{1}\] So, five rational numbers between ? 2 and ? 1 are \[-\frac{19}{10},-\frac{18}{10},-\frac{17}{10},-\frac{16}{10}\]and \[-\frac{15}{10}\] (iii) We have, \[-\frac{4}{5}=-\frac{4\times 3}{5\times 3}=\frac{-12}{15}\] \[\frac{-2}{5}=-\frac{2\times 3}{5\times 3}=-\frac{6}{15}\] So,          \[-\frac{12}{15}<-\frac{11}{15}<-\frac{10}{15}<-\frac{9}{15}<-\frac{8}{15}<-\frac{7}{15}<-\frac{6}{15}\] \[\Rightarrow \]               \[-\frac{4}{5}<-\frac{11}{15}<-\frac{10}{15}<-\frac{9}{15}<-\frac{8}{15}<-\frac{7}{15}<-\frac{2}{3}\] So, five rational numbers between \[-\frac{4}{5}\]and \[-\frac{2}{3}\]are \[-\frac{11}{15},-\frac{10}{15},-\frac{9}{15},-\frac{8}{15}\]and \[-\frac{7}{15}\].                 (iv) We have,                 \[-\frac{1}{2}=-\frac{1\times 6}{2\times 6}=-\frac{6}{12}\]                 \[\frac{2}{3}=\frac{2\times 4}{3\times 4}=\frac{8}{12}\]                 So, \[-\frac{6}{12}<-\frac{4}{12}<-\frac{3}{12}<0<\frac{4}{12}<\frac{6}{12}<\frac{8}{12}\] \[\Rightarrow \]               \[-\frac{1}{2}<-\frac{1}{3}<-\frac{1}{4}<0<\frac{1}{3}<\frac{1}{2}<\frac{2}{3}\] So, five rational numbers between \[-\frac{1}{2}\] and \[\frac{2}{3}\] are \[-\frac{1}{3},-\frac{1}{4},0,\frac{1}{3}\] and \[\frac{1}{2}\].