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Verify associativity of addition of rational numbers i.e., (x+y)+z = x+(y+z), when:(i) x = \(\frac{1}{2}\), y = \(\frac{2}{3}\), z = \(\frac{-1}{5}\)(ii) x = \(\frac{-2}{5}\), y = \(\frac{4}{3}\), z = \(-\frac{-7}{10}\)(iii) x = \(\frac{-7}{11}\),y = \(\frac{2}{-5}\),z = \(-\frac{-3}{22}\)(iv) x = -2, y = \(\frac{3}{5}\),z = \(\frac{-4}{3}\) |
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Answer» (i) In order to verify this property, let us consider the following expressions: Verification: \(\frac{1}{2}+[\frac{2}{3}+(\frac{-1}{5})] = \frac{1}{2}+[\frac{10}{15}-\frac{3}{15}]\) = \(\frac{1}{2}+\frac{7}{15}\) = \(\frac{15+14}{30}\) = \(\frac{29}{30}\) And, \((\frac{1}{2}+\frac{2}{3})+(\frac{-1}{5})=(\frac{3}{6}+\frac{4}{6})-\frac{1}{5}\) = \(\frac{7}{6}-\frac{1}{5}\) = \(\frac{35-6}{30}\) = \(\frac{29}{30}\) Therefore, The associative property of additional of rational numbers has been verified (ii) In order to verify this property, let us consider the following expressions: Verification: = \(\frac{-12}{30}+\frac{19}{30}\) = \(\frac{-12+19}{30}\) = \(\frac{7}{30}\) And, \((\frac{-2}{5}+\frac{4}{3})+(\frac{-7}{10})=(\frac{-6}{15}+\frac{20}{15})-\frac{7}{10}\) = \(\frac{14}{15}-\frac{7}{10}\) = \(\frac{28-21}{30}\) = \(\frac{7}{30}\) Therefore, The associative property of additional of rational numbers has been verified (iii) In order to verify this property, let us consider the following expressions: Verification: \(\frac{-7}{11}+[\frac{2}{-5}+(\frac{-3}{22})]=\frac{-7}{11}+[\frac{44}{-110}-\frac{15}{110}]\): = \(\frac{-7}{11}-\frac{29}{110}\) = \(\frac{-70-29}{110}\) = \(\frac{-99}{110}\) And, \((\frac{-7}{11}+\frac{2}{-5})+(\frac{-3}{22})\) \(=(\frac{-35}{55}-\frac{22}{35})-\frac{3}{22}\) = \(\frac{-57}{55}-\frac{3}{22}\) = \(\frac{-114+15}{110}\) = \(\frac{-99}{110}\) Therefore, The associative property of additional of rational numbers has been verified (iv) In order to verify this property, let us consider the following expressions: Verification: \(-2+[\frac{3}{5}+(\frac{-4}{3})]\) = \(-2+[\frac{9}{15}-\frac{20}{15}]\) = \(-2-\frac{11}{15}\) = \(\frac{-30-11}{15}\) = \(\frac{-41}{15}\) And, \((-2+\frac{3}{5})+(\frac{-4}{3})\) = \(=(\frac{-10}{5}+\frac{3}{35})-\frac{4}{3}\) = \(\frac{-7}{5}-\frac{4}{3}\) = \(\frac{-21-20}{15}\) = \(\frac{-41}{15}\) Therefore, The associative property of additional of rational numbers has been verified |
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