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Verify the following:

(i) \((\frac{3}{4}+\frac{-2}{5})+\frac{-7}{10}= \frac{3}{4}+ (\frac{-2}{5}+\frac{-7}{10})\)

(ii) \((\frac{-7}{11}+\frac{2}{-5}) + \frac{-13}{22} = \frac{-7}{11}+(\frac{2}{-5}+\frac{-13}{22})\)

(iii) \(-1 +(\frac{-2}{3}+\frac{-3}{4}) = (-1+\frac{-2}{3}) +\frac{-3}{4}\)

1 Answer

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 (i)

\((\frac{3}{4}+\frac{-2}{5})+\frac{-7}{10}= \frac{3}{4}+ (\frac{-2}{5}+\frac{-7}{10})\)

LHS\((\frac{3}{4}+\frac{-2}{5})\) \(+\frac{-7}{10}\)

\((\frac{3\times5+(-2)\times4)}{20})+\frac{-7}{10}\)

\((\frac{15-8}{20}) + \frac{-7}{10}\)

\(\frac{7}{20}+\frac{-7}{10}\)

\(\frac{7\times1+(-7)\times2}{20}\)

\(\frac{7-14}{20}\)

\(\frac{-7}{20}\)

RHS \(\frac{3}{4}+(\frac{-2}{5}+\frac{-7}{10})\)

\(\frac{3}{4}+ (\frac{(-2)\times2+(-7)\times1}{10})\)

\(\frac{3}{4}\) + \((\frac{-4-7}{10})\)

\(\frac{3}{4}+\frac{-11}{10}\)

\(\frac{3\times5+(-11)\times2}{20}\)

\(\frac{15-22}{20}\)

\(\frac{-7}{20}\)

RHS = LHS 

(ii)

\((\frac{-7}{11}+\frac{2}{-5})+\frac{-13}{22}= \frac{-7}{11}+ (\frac{2}{-5}+\frac{-13}{22})\)

LHS =   \((\frac{-7}{11}+\frac{2}{-5})\) \(+\frac{-13}{22}\)

\((\frac{-7}{11}+\frac{2}{-5}) +\frac{-13}{22}\)

\((\frac{-7\times5+(-2)\times11)}{55})+\frac{-13}{22}\)

\((\frac{-35-22}{55}) +\frac{-13}{22}\)

\(\frac{-57}{55}+\frac{-13}{22}\)

\(\frac{-57\times2+(-13)\times5}{110}\)

\(\frac{-144-65}{110}\)

\(\frac{-179}{110}\)

RHS\(\frac{-7}{11}+(\frac{2}{-5}+\frac{-13}{22})\)

\(\frac{-7}{11}+(\frac{-2}{5}+\frac{-13}{22})\)

\(\frac{-7}{11}+(\frac{(-2)\times22+(-13)\times5}{-110})\)

\(\frac{-7}{11}+(\frac{-44-65}{-110})\)

\(\frac{-7}{11}+\frac{-109}{110}\)

\(\frac{-7\times10+(-109)\times1}{110}\)

\(\frac{-70-109}{20}\)

\(\frac{-179}{110}\)

RHS = LHS

(iii)

\(-1+(\frac{-2}{3}+\frac{-3}{4}) = (-1 + \frac{-2}{3}+\frac{-3}{4})\)

LHS\(-1+(\frac{-2}{3}+\frac{-3}{4})\)

\(\frac{-1}{1} +(\frac{-2}{3}+\frac{-3}{4})\)

\(\frac{-1}{1}+(\frac{-2\times4+(-3)\times3}{12})\)

\(\frac{-1}{1}+(\frac{-8-9}{12})\)

\(\frac{-1}{1}+\frac{-17}{12}\)

\(\frac{-1\times12+(-17)\times1}{12}\)

\(\frac{-12-17}{12}\)

\(\frac{-29}{12}\)

RHS\((-1 +\frac{-2}{3}) +\frac{-3}{4}\)

\((\frac{-1}{1}+\frac{-2}{3})+\frac{-3}{4}\)

\((\frac{-1\times3+(-2)\times1}{3})+\frac{-3}{4}\)

\((\frac{-3-2}{3})+\frac{-3}{4}\)

\(\frac{-5}{3}+\frac{-3}{4}\)

\(\frac{-5\times4+(-3)\times3}{12}\)

\(\frac{-20-9}{20}\)

\(\frac{-29}{12}\)

RHS = LHS 

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