Question

Verify each of the following:
(i) $\left(\frac{5}{7}\times \frac{12}{13}\right)\times \frac{7}{18}=\frac{5}{7}\times \left(\frac{12}{13}\times \frac{7}{8}\right)$
(ii) $\frac{-13}{24}\times \left(\frac{-12}{5}\times \frac{35}{36}\right)=\left(\frac{-13}{24}\times \frac{-12}{5}\right)\times \frac{35}{36}$
(iii) $\left(\frac{-9}{5}\times \frac{-10}{3}\right)\times \frac{21}{-4}=\frac{-9}{5}\times \left(\frac{-10}{3}\times \frac{21}{-4}\right)$

Answer 1

(i)

$\left(\frac{5}{7}\times \frac{12}{13}\right)\times \frac{7}{18}=\frac{5}{7}\times \left(\frac{12}{13}\times \frac{7}{18}\right)$

$$\begin{gathered} \mathrm{LHS}=\left(\frac{5}{7}\times \frac{12}{13}\right)\times \frac{7}{18} \\ =\frac{5\times 12}{7\times 13}\times \frac{7}{18} \\ =\frac{60}{91}\times \frac{7}{18} \\ =\frac{420}{1638} \\ =\frac{10}{39} \end{gathered}$$


$$\begin{gathered} \mathrm{RHS}=\frac{5}{7}\times \left(\frac{12}{13}\times \frac{7}{18}\right) \\ =\frac{5}{7}\times \frac{12\times 7}{13\times 18} \\ =\frac{5}{7}\times \frac{84}{234} \\ =\frac{420}{1638} \\ =\frac{10}{39} \end{gathered}$$

∴ ​$\left(\frac{5}{7}\times \frac{12}{13}\right)\times \frac{7}{18}=\frac{5}{7}\times \left(\frac{12}{13}\times \frac{7}{18}\right)$


(ii)

$\frac{-13}{24}\times \left(\frac{-12}{5}\times \frac{35}{36}\right)=\left(\frac{-13}{24}\times \frac{-12}{5}\right)\times \frac{35}{36}$

$$\begin{gathered} \mathrm{LHS}=\frac{-13}{24}\times \left(\frac{-12}{5}\times \frac{35}{36}\right) \\ =\frac{-13}{24}\times \frac{(-12)\times 35}{5\times 36} \\ =\frac{-13}{24}\times \frac{-420}{180} \\ =\frac{5460}{4320} \\ =\frac{91}{72} \end{gathered}$$


$$\begin{gathered} \mathrm{RHS}=\left(\frac{-13}{24}\times \frac{-12}{5}\right)\times \frac{35}{36} \\ =\frac{(-13)\times (-12)}{24\times 5}\times \frac{35}{36} \\ =\frac{156}{120}\times \frac{35}{36} \\ =\frac{156\times 35}{120\times 36} \\ =\frac{5460}{4320} \\ =\frac{91}{72} \end{gathered}$$

∴ ​$\frac{-13}{24}\times \left(\frac{-12}{5}\times \frac{35}{36}\right)=\left(\frac{-13}{24}\times \frac{-12}{5}\right)\times \frac{35}{36}$


(iii)

$\left(\frac{-9}{5}\times \frac{-10}{3}\right)\times \frac{21}{-4}=\frac{-9}{5}\times \left(\frac{-10}{3}\times \frac{21}{-4}\right)$

$$\begin{gathered} \mathrm{LHS}=\left(\frac{-9}{5}\times \frac{-10}{3}\right)\times \frac{21}{-4} \\ =\frac{(-9)\times (-10)}{5\times 3}\times \frac{21}{-4} \\ =\frac{90}{15}\times \frac{21}{-4} \\ =\frac{90\times 21}{15\times (-4)} \\ =-\frac{1890}{60} \\ =-\frac{63}{2} \end{gathered}$$


$$\begin{gathered} \mathrm{RHS}=\frac{-9}{5}\times \left(\frac{-10}{3}\times \frac{21}{-4}\right) \\ =\frac{-9}{5}\times \frac{(-10)\times 21}{3\times (-4)} \\ =\frac{-9}{5}\times \frac{210}{12} \\ =\frac{(-9)\times 210}{5\times 12} \\ =-\frac{1890}{60} \\ =\frac{-63}{2} \end{gathered}$$

∴ $(\frac{-9}{5}\times \frac{-10}{3})\times \frac{21}{-4}=\frac{-9}{5}\times (\frac{-10}{3}\times \frac{21}{-4})$