question_answer

Let a, b, c, be the three rational numbers where a \[a=\frac{2}{3},b=\frac{4}{5}\] and \[c=-\frac{5}{6}.\]

Verify:

(a) \[a+\left( b+c \right)=\left( a+b \right)+\text{ }c\](Associative property of addition).

(b) \[a\times \left( b\times c \right)=\left( a\times b \right)\text{ }\times c\](Associative property of multiplication).

Answer:

(a)  L.H.S \[=a+\left( b+c \right)\]

\[=\frac{2}{3}+\left[ \frac{4}{5}+\left( \frac{-5}{6} \right) \right]\]

\[=\frac{2}{3}+\left[ \frac{24-25}{30} \right]\]

\[=\frac{2}{3}+\left( \frac{-1}{30} \right)\]

\[=\frac{20-1}{30}=\frac{19}{30}\]

R.H.S. of (i)    =(a+b)+c

\[=\left( \frac{2}{3}+\frac{4}{5} \right)+\left( \frac{-5}{6} \right)\]

\[=\left( \frac{10+12}{15} \right)+\left( \frac{-5}{6} \right)\]

\[=\frac{22}{15}-\frac{5}{6}=\frac{44-25}{30}=\frac{19}{30}\]

So,       \[\frac{2}{3}+\left[ \frac{4}{5}+\left( \frac{-5}{2} \right) \right]=\left( \frac{2}{3}+\frac{4}{5} \right)+\left( \frac{-5}{6} \right)\]

Hence Verified

(b) L.H.S           \[=a\times \left( b\times c \right)\]

\[=\frac{2}{3}\times \left[ \frac{4}{5}\times \left( \frac{-5}{6} \right) \right]\]

\[=\frac{2}{3}\times \left( \frac{-20}{30} \right)=\frac{2}{3}\times \left( \frac{-2}{3} \right)\]

\[=\frac{2\times (-2)}{3\times 3}=\frac{-4}{9}\]

R, H.S.                       \[=\text{ }\left( a\times b \right)\times \text{ }c\]

\[=\left( \frac{2}{3}\times \frac{4}{5} \right)\times \left( \frac{-5}{6} \right)\]

\[=\frac{2\times 4}{3\times 5}\times \frac{-5}{6}\]

\[=\frac{8}{15}\times \left( \frac{-5}{6} \right)\]

\[=\frac{8\times (-5)}{15\times 6}=\frac{-40}{90}=\frac{-4}{9}\]

So,       \[\frac{2}{3}\times \left[ \frac{4}{5}\times \left( \frac{-5}{6} \right) \right]=\left[ \frac{2}{3}\times \frac{4}{5} \right]\times \left( \frac{-5}{6} \right)\]

Hence Verified