Question

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

(ii) $\frac{-2}{5},\frac{4}{15}and\frac{-7}{10}$

Answer 1

Solving $\frac{-2}{5}+(\frac{4}{15}+\frac{-7}{10})$

$\frac{-2}{5}+(\frac{4}{15}+\frac{-7}{10})=-\frac{2}{5}+(\frac{4\times 2}{15\times 2}+\frac{7\times 3}{10\times 3})$

(LCM of 15 and 10 is 30)

$=\frac{-2}{5}+(\frac{8}{30}-\frac{21}{30})$

$=\frac{-2}{5}+\frac{8-21}{30}$

$=\frac{-2}{5}+\frac{-13}{30}$

$=\frac{2\times 6}{5\times 6}-\frac{8-21}{30}$

(LCM of 5 and 30 is 30)

$=-\frac{12}{30}+\frac{-13}{30}$

$\frac{-12-13}{30}$

$=-\frac{25}{30}$ (equation 1)

Solving $(\frac{-2}{5}+\frac{4}{15})+\frac{-7}{10}$

$(\frac{-2}{5}+\frac{4}{15})+\frac{-7}{10}=(\frac{-2\times 3}{5\times 3}+\frac{4}{15})-\frac{-7}{10}$

(LCM of 5 and 15 is 15)

$=(-\frac{6}{15}+\frac{4}{15})-\frac{7}{10}$

$=\frac{-6+4}{15}-\frac{7}{10}$

$=-\frac{2}{15}-\frac{7}{10}$

$=\frac{2\times 2}{15\times 2}-\frac{7\times 3}{10\times 3}$ (LCM of 15 and 10 is 30)

$=-\frac{4}{30}-\frac{21}{30}$

$=\frac{-4-21}{30}$

$=-\frac{25}{30}$ (equation 2)

From (1) and (2), $\frac{-2}{5}+(\frac{4}{15}+\frac{-7}{10})=(\frac{-2}{5}+\frac{4}{15})+\frac{-7}{10}$

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