Question

Using the rearrangement property find the sum:
(i) $\frac{4}{3}+\frac{3}{5}+\frac{-2}{3}+\frac{-11}{5}$
(ii) $\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}$
(iii) $\frac{-13}{20}+\frac{11}{14}+\frac{-5}{7}+\frac{7}{10}$
(iv) $\frac{-6}{7}+\frac{-5}{6}+\frac{-4}{9}+\frac{-15}{7}$

Answer 1

(i)
$\left(\frac{4}{3}+\frac{-2}{3}\right)+\left({\displaystyle \frac{3}{5}}+{\displaystyle \frac{-11}{5}}\right)$
= $\left(\frac{4-2}{3}\right)+\left(\frac{3-11}{5}\right)$
$$\begin{gathered} =\left({\displaystyle \frac{2}{3}}+{\displaystyle \frac{-8}{5}}\right) \\ =\left({\displaystyle \frac{10}{15}}+{\displaystyle \frac{-24}{15}}\right) \\ =\left({\displaystyle \frac{10-24}{15}}\right) \\ =\frac{-14}{15} \end{gathered}$$.


(ii)
$\left(\frac{-8}{3}+\frac{-11}{6}\right)+\left({\displaystyle \frac{-1}{4}}+{\displaystyle \frac{3}{8}}\right)$

=$\left({\displaystyle \frac{-16}{6}}+{\displaystyle \frac{-11}{6}}\right)+\left({\displaystyle \frac{-2}{8}}+{\displaystyle \frac{3}{8}}\right)$

=$\left(\frac{-16-11}{6}\right)+\left(\frac{-2+3}{8}\right)$

$$\begin{gathered} =\left(\frac{-27}{6}+\frac{1}{8}\right) \\ =\left({\displaystyle \frac{-108}{24}}+{\displaystyle \frac{3}{24}}\right) \\ =\frac{-105}{24} \end{gathered}$$
=$\frac{35}{8}$


(iii)
$\left(\frac{-13}{20}+\frac{7}{10}\right)+\left({\displaystyle \frac{11}{14}}+{\displaystyle \frac{-5}{7}}\right)$
=$\left({\displaystyle \frac{-13}{20}}+{\displaystyle \frac{14}{20}}\right)+\left({\displaystyle \frac{11}{14}}+{\displaystyle \frac{-10}{14}}\right)$
=$\left(\frac{-13+14}{20}\right)+\left(\frac{11-10}{14}\right)$
$$\begin{gathered} =\left(\frac{1}{20}+\frac{1}{14}\right) \\ =\left({\displaystyle \frac{7}{140}}+{\displaystyle \frac{10}{140}}\right) \\ =\left(\frac{7+10}{140}\right) \\ =\left({\displaystyle \frac{17}{140}}\right) \\ =\frac{17}{140} \end{gathered}$$.


(iv)
$\left(\frac{-6}{7}+\frac{-15}{7}\right)+\left({\displaystyle \frac{-5}{6}}+{\displaystyle \frac{-4}{9}}\right)$

=$\left(\frac{-6}{7}+\frac{-15}{7}\right)+\left({\displaystyle \frac{-15}{18}}+{\displaystyle \frac{-8}{18}}\right)$

=$\left(\frac{-6-15}{7}\right)+\left({\displaystyle \frac{-15-8}{18}}\right)$

$$\begin{gathered} =\left({\displaystyle \frac{-21}{7}}+{\displaystyle \frac{-23}{18}}\right) \\ =\left({\displaystyle \frac{-3}{1}}+{\displaystyle \frac{-23}{18}}\right) \\ =\left({\displaystyle \frac{-54}{18}}+{\displaystyle \frac{-23}{18}}\right) \\ =\left(\frac{-54-23}{18}\right) \\ =\frac{-77}{18} \end{gathered}$$