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
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