Question

For each set of rational numbers, given below, verify the associative property of addition of rational numbers:
$\left(i\right)\frac{1}{2},\frac{2}{3}\text{and}-\frac{1}{6}\left(ii\right)\frac{-2}{5},\frac{4}{15}\text{and}\frac{-7}{10}\left(iii\right)\frac{-7}{9},\frac{2}{-3}\text{and}\frac{-5}{18}\left(iv\right)-1,\frac{5}{6}\text{~}and\frac{-2}{3}$

Answer 1

$\left(i\right)\frac{1}{2},\frac{2}{3}\text{and}-\frac{1}{6}\text{Show that}:\frac{1}{2}+(\frac{2}{3}+\frac{-1}{6})=(\frac{1}{2}+\frac{2}{3})\frac{-1}{6}\because \frac{1}{2}+(\frac{2}{3}+\frac{-1}{6})$

$\therefore \text{LCM of 3 and 6=6}=\frac{1}{2}+(\frac{2\times 2}{3\times 2}+\frac{-1\times 1}{6\times 1})=\frac{1}{2}+(\frac{4}{6}-\frac{1}{6})=\frac{1}{2}+\left(\frac{4-1}{6}\right)=\frac{1}{2}+\left(\frac{3}{6}\right)=\frac{1\times 3}{2\times 3}+\frac{3\times 1}{6\times 1}(\because \text{LCM of 2 and 6=3})=\frac{3+3}{6}=\frac{6}{6}=1\text{And,}(\frac{1}{2}+\frac{2}{3})+\frac{-1}{6}$

$\therefore \text{LCM of 2 and 3=6}=(\frac{1\times 3}{2\times 3}+\frac{2\times 2}{3\times 2})+\frac{-1}{6}=\frac{3+4}{6}+\frac{-1}{6}=\frac{7-1}{6}=\frac{6}{6}=1\therefore \frac{1}{2}+(\frac{2}{3}+\frac{-1}{6})+(\frac{1}{2}+\frac{2}{3})+\frac{-1}{6}\text{This verifies associative property of the addition of rational numbers.}\left(ii\right)\frac{-2}{5},\frac{4}{15}\text{and}\frac{-7}{10}\text{Sho that}:\frac{-2}{5}+(\frac{4}{15}+\frac{-7}{10})=(\frac{-2}{5}+\frac{4}{15})+\frac{-7}{10}\because \frac{-2}{5}+(\frac{4}{15}+\frac{-7}{10})$

$\therefore \text{LCM of}15,10=2\times 3\times 5=30=\frac{-2}{5}+(\frac{4\times 2}{15\times 2}+\frac{-7\times 3}{10\times 3})(\because \text{LCM of 15 and 10=30})=\frac{-2}{5}+\left(\frac{8-21}{30}\right)=\frac{-2}{5}-\frac{13}{30}=\frac{-2\times 6}{5\times 6}-\frac{13\times 1}{30\times 1}=\frac{-12-13}{30}=\frac{-25}{30}=\frac{-5}{6}\text{And},(\frac{-2}{5}+\frac{4}{15})+\frac{-7}{10}$

$\therefore \text{LCM of 5 and 15=15}=\frac{-6+4}{15}+\frac{-7}{10}=\frac{-2}{15}+\frac{-7}{10}=\frac{-2\times 2}{15\times 2}-\frac{7\times 3}{10\times 3}=\frac{-4}{30}-\frac{21}{30}=\frac{-25}{30}=\frac{-5}{6}\therefore \frac{-2}{5}+(\frac{4}{15}+\frac{-7}{10})=(\frac{-2}{5}+\frac{4}{15})+\frac{-7}{10}\text{This verifies associative property of the addition of rational numbers.}\left(iii\right)\frac{-7}{9},\frac{2}{-3}\text{and}\frac{-5}{18}\text{Show that}:\frac{-7}{9}+(\frac{2}{-3}+\frac{-5}{18})=(\frac{-7}{9}+\frac{2}{-3})+\frac{-5}{18}\because \frac{-7}{9}+(\frac{2}{-3}+\frac{-5}{18})$

$\therefore \text{LCM of 3 and 18}=2\times 3\times 3=18=\frac{-7}{9}+(\frac{-2\times 6}{3\times 6}+\frac{-5\times 1}{18\times 1})(\because \text{LCM of 3 and 18=18})=\frac{-7}{9}+\left(\frac{-12-5}{18}\right)=\frac{-7}{9}+\frac{-17}{18}=\frac{-7\times 2}{9\times 2}+\frac{17\times 1}{18\times 1}(\because \text{LCM of 9 and 18=18})=\frac{-14-17}{18}=\frac{-31}{18}\text{And ,}(\frac{-7}{9}+\frac{2}{-3})+\frac{-5}{18}$

$\therefore \text{LCM of 3 and 9=3}=(\frac{-7\times 1}{9\times 1}+\frac{-2\times 3}{3\times 3})+\frac{-5}{18}(\because \text{LCM=9 and 3=9})=\frac{-7-6}{9}+\frac{-5}{18}=\frac{-13}{9}+\frac{-5}{18}=\frac{-13\times 2}{9\times 2}+\frac{-5\times 1}{18\times 1}=\frac{-26-5}{18}=\frac{-31}{18}\therefore \frac{-7}{9}+(\frac{2}{-3}+\frac{-5}{18})=(\frac{-7}{9}+\frac{2}{-3})+\frac{-5}{18}\text{This verifies associatevie property of the addition of rational numbers.}\left(iv\right)-1,\frac{5}{6}\text{~}and\frac{-2}{3}\text{Show that}:\frac{-1}{1}+(\frac{5}{6}+\frac{-2}{3})=(\frac{-1}{1}+\frac{5}{6})+\frac{-2}{3}\because \frac{-1}{1}+(\frac{5}{6}+\frac{-2}{3})$

$\therefore \text{LCM of 6 and 3=6}=\frac{-1}{1}+(\frac{5\times 1}{6\times 1}+\frac{-2\times 2}{3\times 2})(\because \text{LCM of and 3=6})=\frac{-1}{1}+\left(\frac{5-4}{6}\right)=\frac{-1}{1}+\frac{1}{6}=\frac{-1\times 6}{1\times 6}+\frac{1\times 1}{6\times 1}(\because \text{LCM of 1 and 6=1})=\frac{-6+1}{6}=\frac{-5}{6}\text{And},(\frac{-1}{1}+\frac{5}{6})+\frac{-2}{3}=-(\frac{-1\times 6}{1\times 6}+\frac{5\times 1}{6\times 1})+\frac{-2}{3}(\because \text{LCM of 1 and 6=6})=\left(\frac{-6+5}{6}\right)+\frac{-2}{3}=\frac{-1}{6}+\frac{-2}{3}=\frac{-1\times 1}{6\times 1}+\frac{-2\times 2}{3\times 2}(\because \text{LCM of 6 and 3=6})=\frac{-1-4}{6}=\frac{-5}{6}\therefore \frac{-1}{1}(\frac{5}{6}+\frac{-2}{3})=(\frac{-1}{1}+\frac{5}{6})+\frac{-2}{3}\text{This verifies associative poperty of the addition of rational numbers.}$