Question

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

(i) $\frac{1}{2},\frac{2}{3},-\frac{1}{6}$

Answer 1

Solving $\frac{1}{2}+(\frac{2}{3}+(-\frac{1}{6}))$

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

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

$=\frac{1}{2}+(\frac{4}{6}-\frac{1}{6})$

$=\frac{1}{2}+\frac{4-1}{6}$

$=\frac{1}{2}+\frac{3}{6}$

$=\frac{1}{2}+\frac{1}{2}$

$=\frac{1+1}{2}$

$=\frac{2}{2}$

$=1$ (equation 1)

Solving $(\frac{1}{2}+\frac{2}{3})+\frac{-1}{6}$

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

(LCM of 2 and 3 is 6)

$=(\frac{3}{6}+\frac{4}{6})-\frac{1}{6}$

$=\frac{3+4}{6}-\frac{1}{6}$

$=\frac{7}{6}-\frac{1}{6}$

$=\frac{7-1}{6}$

$=\frac{6}{6}$

$=1$ (equation 2)

From (1) and (2), $\frac{1}{2}+(\frac{2}{3}+(-\frac{1}{6}))=(\frac{1}{2}+\frac{2}{3})+\frac{-1}{6}$

Hence, the associative property of addition of rational numbers is verified.