# Using commutativity and associativity of addition of rational numbers,

Question:

Using commutativity and associativity of addition of rational numbers, express each of the following as a rational number:

(i) $\frac{2}{5}+\frac{7}{3}+\frac{-4}{5}+\frac{-1}{3}$

(ii) $\frac{3}{7}+\frac{-4}{9}+\frac{-11}{7}+\frac{7}{9}$

(iii) $\frac{2}{5}+\frac{8}{3}+\frac{-11}{15}+\frac{4}{5}+\frac{-2}{3}$

(iv) $\frac{4}{7}+0+\frac{-8}{9}+\frac{-13}{7}+\frac{17}{21}$

Solution:

(i) We have:

$\frac{2}{5}+\frac{7}{3}+\frac{-4}{5}+\frac{-1}{3}$

$=\left(\frac{2}{5}+\frac{-4}{5}\right)+\left(\frac{7}{3}+\frac{-1}{3}\right)$

$=\left(\frac{2-4}{5}\right)+\left(\frac{7-1}{3}\right)$

$=\frac{-2}{5}+\frac{6}{3}$

$=\frac{-6+30}{15}$

$=\frac{24}{15}$

$=\frac{8}{5}$

(ii) We have:

$\frac{3}{7}+\frac{-4}{9}+\frac{-11}{7}+\frac{7}{9}$

$=\left(\frac{3}{7}+\frac{-11}{7}\right)+\left(\frac{-4}{9}+\frac{7}{9}\right)$

$=\left(\frac{3-11}{7}\right)+\left(\frac{-4+7}{9}\right)$

$=\frac{-8}{7}+\frac{3}{9}$

$=\frac{-72+21}{63}$

$=\frac{-51}{63}$

$=\frac{-17}{21}$

(iii) We have:

$\frac{2}{5}+\frac{8}{3}+\frac{-11}{15}+\frac{4}{5}+\frac{-2}{3}$

$=\left(\frac{2}{5}+\frac{4}{5}\right)+\left(+\frac{8}{3}+\frac{-2}{3}\right)+\frac{-11}{15}$

$=\left(\frac{2+4}{5}\right)+\left(\frac{8-2}{3}\right)+\frac{-11}{15}$

$=\frac{6}{5}+\frac{6}{3}+\frac{-11}{15}$

$=\frac{18+30-11}{15}$

$=\frac{37}{15}$

(iv) We have;

$\frac{4}{7}+0+\frac{-8}{9}+\frac{-13}{7}+\frac{17}{21}$

$=\left(\frac{4}{7}+\frac{-13}{7}\right)+\left(\frac{-8}{9}\right)+\frac{17}{21}$

$=\left(\frac{4-13}{7}\right)+\left(\frac{-8}{9}\right)+\frac{17}{21}$

$=\frac{-9}{7}+\frac{-8}{9}+\frac{17}{21}$

$=\frac{-81-56+51}{63}$

$=\frac{-86}{63}$