Answer:
(i) We have, \[-\frac{3}{5}=-\frac{3\times 1}{5\times 1},-\frac{6}{10}=-\frac{3\times 2}{5\times 2}\] \[-\frac{9}{15}=-\frac{3\times 3}{5\times 3},-\frac{12}{20}=-\frac{3\times 4}{5\times 4}\] Thus, we observe a pattern in these numbers. Four more rational numbers would be \[-\frac{3\times 5}{5\times 5}=-\frac{15}{25},-\frac{3\times 6}{5\times 6}=-\frac{18}{30},\] \[-\frac{3\times 7}{5\times 7}=-\frac{21}{35}\] and \[-\frac{3\times 8}{5\times 8}=-\frac{24}{40}\] (ii) We have, \[-\frac{1}{4}=-\frac{1\times 1}{4\times 1},-\frac{2}{8}=-\frac{1\,\times 2}{4\times 2},-\frac{3}{12}=-\frac{1\times 3}{4\times 3}\] Thus, we observe a patterns in the numbers. Four more rational numbers would be \[-\frac{1\times 4}{4\times 4}=-\frac{4}{16},\frac{-1\times 5}{4\times 5}=-\frac{5}{20},\]\[\frac{-1\times 6}{4\times 6}=-\frac{6}{24}\]and \[\frac{-1\times 7}{4\times 7}=-\frac{7}{28}\] (iii) We have, \[\frac{2}{-12}=\frac{-1\times -2}{6\times -2},\frac{3}{-18}=\frac{-1\times -3}{6\times -3},\frac{4}{-24}=\frac{-1\times -4}{6\times -4}\] Thus, we observe a pattern in these numbers. Four more rational numbers would be \[\frac{-1\times -5}{6\times -5}=\frac{5}{-30},\frac{-1\times -6}{6\times -6}=\frac{6}{-36},\] \[\frac{-1\times -7}{6\times -7}=\frac{7}{-42}\]and \[\frac{-1\times -8}{6\times -8}=\frac{8}{-48}\] (iv) we have, \[\frac{2}{-3}=\frac{-2\times -1}{3\times -1},\frac{4}{-6}=\frac{-2\times -2}{3\times -2},\frac{6}{-9}=\frac{-2\times -3}{3\times -3}\] Thus, we observe a pattern in these numbers. Four more rational numbers would be \[\frac{-2\times -4}{3\times -4}=\frac{8}{-12},\frac{-2\times -5}{3\times -5}=\frac{10}{-15},\] \[\frac{-2\times -6}{3\times -6}=\frac{12}{-18}\]and \[\frac{-2\times -7}{3\times -7}=\frac{14}{-21}\].
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