Question

Using suitable properties of rational numbers, evaluate the following-
(a) 4/3 + 3/5 + -2/5 + -11/5
(b) -5/9 × 4/5 - 1/54 - 3/10 × -5/9

Answer 1

a) $\frac{4}{3}+\frac{3}{5}+\left(-\frac{2}{5}\right)+\left(\frac{-11}{5}\right)$
From associative and commutative law if follows that we can arrange the terms in suitable way.
Hence rearranging it gives
$$\begin{gathered} \frac{3}{5}-\frac{2}{5}-\frac{11}{5}+\frac{4}{3} \\ \Rightarrow \frac{{\displaystyle 3-2-11}}{5}+\frac{4}{3} \\ \Rightarrow \frac{{\displaystyle -10}}{5}+\frac{{\displaystyle 4}}{{\displaystyle 3}} \\ \Rightarrow \frac{{\displaystyle -30+20}}{15}=-\frac{{\displaystyle 10}}{15}=-\frac{{\displaystyle 2}}{3} \end{gathered}$$
b)
$\frac{-5}{9}\times \frac{{\displaystyle 4}}{5}-\frac{{\displaystyle 1}}{54}-\frac{3}{10}\times \frac{{\displaystyle -5}}{9}$
Using distributive law
a(b+c)= ab +ac
and commutative law a+b=b+a
we can rearrange it as follow
$$\begin{gathered} \frac{-5}{9}\times \frac{4}{5}-\frac{3}{10}\times -\frac{{\displaystyle 5}}{9}-\frac{{\displaystyle 1}}{54} \\ \frac{-5}{9}\left[\frac{4}{5}-\frac{3}{10}\right]-\frac{{\displaystyle 1}}{54} \\ \Rightarrow \frac{-5}{9}\left[\frac{8-3}{10}\right]-\frac{{\displaystyle 1}}{54} \\ \Rightarrow \frac{-5}{9}\times \frac{5}{10}-\frac{{\displaystyle 1}}{54} \\ \Rightarrow \frac{-25}{90}-\frac{{\displaystyle 1}}{54} \\ \Rightarrow -\frac{5}{18}-\frac{{\displaystyle 1}}{54} \\ \Rightarrow \frac{-15-1}{54} \\ =\frac{{\displaystyle -16}}{54}=\frac{{\displaystyle -8}}{27} \end{gathered}$$