Question

For each set of rational number, given below, verify the associative property of addition of rational number:

(iv) ${\displaystyle -1,\frac{5}{6}\text{and}\frac{-2}{3}}$ 

Answer 1

Show that :

This verifies associative  property  of the addition of rational numbers.

${\displaystyle \frac{-1}{1}+(\frac{5}{6}+\frac{-2}{3})=(\frac{-1}{1}+\frac{5}{6})+\frac{-2}{3}}$

∴${\displaystyle \frac{-1}{1}+(\frac{5}{6}+\frac{-2}{3})}$   

2	6, 3
3	3, 3
	1, 1

∴ LCM of 6 and 3 = 6

=${\displaystyle \frac{-1}{1}+(\frac{5\times 1}{6\times 1}+\frac{-2\times 2}{3\times 2})}$    (∴LCM of 6 and 3 = 6)

=${\displaystyle \frac{-1}{1}+\left(\frac{5-4}{6}\right)}$

=${\displaystyle \frac{-1}{1}+\frac{1}{6}}$

=${\displaystyle \frac{-1\times 6}{1\times 6}+\frac{1\times 1}{6\times 1}}$    (∴ LCM of 1 and 6 = 1)

=${\displaystyle \frac{-6+1}{6}=\frac{-5}{6}}$

And, ${\displaystyle (\frac{-1}{1}+\frac{5}{6})+\frac{-2}{3}}$

=${\displaystyle (\frac{-1\times 6}{1\times 6}+\frac{5\times 1}{6\times 1})+\frac{-2}{3}}$  (∴LCM of 1 and 6 =6) 

=${\displaystyle \left(\frac{-6+5}{6}\right)+\frac{-2}{3}}$    

=${\displaystyle \frac{-1}{6}+\frac{-2}{3}}$           (∴ LCM of 1 and 6 = 6)

=${\displaystyle \frac{-1\times 1}{6\times 1}+\frac{-2\times 2}{3\times 2}}$

=${\displaystyle \frac{-1-4}{6}=\frac{-5}{6}}$  (∴LCM of 6 and 3 = 6)

∴${\displaystyle \frac{-1}{1}+(\frac{5}{6}+\frac{-2}{3})=(\frac{-1}{1}+\frac{5}{6})+\frac{-2}{3}}$