Question

Verify the following:
(i) $\frac{3}{7}\times \left(\frac{5}{6}+\frac{12}{13}\right)=\left(\frac{3}{7}\times \frac{5}{6}\right)+\left(\frac{3}{7}\times \frac{12}{13}\right)$
(ii) $\frac{-15}{4}\times \left(\frac{3}{7}+\frac{-12}{5}\right)=\left(\frac{-15}{4}\times \frac{3}{7}\right)+\left(\frac{-15}{4}\times \frac{-12}{5}\right)$
(iii) $\left(\frac{-8}{3}+\frac{-13}{12}\right)\times \frac{5}{6}=\left(\frac{-8}{3}\times \frac{5}{6}\right)+\left(\frac{-13}{12}\times \frac{5}{6}\right)$
(iv) $\frac{-16}{7}\times \left(\frac{-8}{9}+\frac{-7}{6}\right)=\left(\frac{-16}{7}\times \frac{-8}{9}\right)+\left(\frac{-16}{7}\times \frac{-7}{6}\right)$

Answer 1

$$\begin{gathered} \text{(i)} \\ \text{LHS=}\frac{3}{7}\times \left(\frac{5}{6}+\frac{12}{13}\right) \\ =\frac{3}{7}\times \left(\frac{65+72}{78}\right) \\ =\frac{3}{7}\times \frac{137}{78} \\ =\frac{137}{182} \\ \text{RHS}=\left(\frac{3}{7}\times \frac{5}{6}\right)+\left(\frac{12}{13}\times \frac{3}{7}\right) \\ =\frac{3\times 5}{7\times 6}+\frac{12\times 3}{13\times 7} \\ =\frac{15}{42}+\frac{36}{91} \\ =\frac{195+216}{546} \\ =\frac{411}{546} \\ =\frac{137}{182} \end{gathered}$$

∴ ​$\frac{3}{7}\times (\frac{5}{6}+\frac{12}{13})=(\frac{3}{7}\times \frac{5}{6})+(\frac{3}{7}\times \frac{12}{13})$

$$\begin{gathered} \text{(ii)} \\ \mathrm{LHS}=\frac{-15}{4}\times (\frac{3}{7}+\frac{-12}{5}) \\ =\frac{-15}{4}\times \left(\frac{15-84}{35}\right) \\ =\frac{-15}{4}\times \frac{-69}{35} \\ =\frac{(-15)\times (-69)}{140} \\ =\frac{1035}{140} \\ =\frac{207}{28} \\ \text{RHS}=(\frac{-15}{4}\times \frac{3}{7})+(\frac{-15}{4}\times \frac{-12}{5}) \\ =\frac{(-15)\times 3}{4\times 7}+\frac{(-15)\times (-12)}{4\times 5} \\ =\frac{-45}{28}+\frac{180}{20} \\ =\frac{-225+1260}{140} \\ =\frac{1035}{140} \\ =\frac{207}{28} \\ \therefore \frac{-15}{4}\times (\frac{3}{7}+\frac{-12}{5})=(\frac{-15}{4}\times \frac{3}{7})+(\frac{-15}{4}\times \frac{-12}{5}) \end{gathered}$$

(iii)

$$\begin{gathered} (\frac{-8}{3}+\frac{-13}{12})\times \frac{5}{6}=(\frac{-8}{3}\times \frac{5}{6})+(\frac{-13}{12}\times \frac{5}{6}) \\ \text{LHS}=(\frac{-8}{3}+\frac{-13}{12})\times \frac{5}{6} \\ =\frac{-32-13}{12}\times \frac{5}{6} \\ =\frac{-45}{12}\times \frac{5}{6} \\ =\frac{-45\times 5}{12\times 6} \\ =\frac{-225}{72} \\ =\frac{-225÷9}{72÷9} \\ =-\frac{25}{8} \\ \text{RHS}=(\frac{-8}{3}\times \frac{5}{6})+(\frac{-13}{12}\times \frac{5}{6}) \\ =\frac{-8\times 5}{3\times 6}+\frac{(-13)\times 5}{12\times 6} \\ =\frac{-40}{18}+\frac{-65}{72} \\ =\frac{-160-65}{72} \\ =\frac{-225}{72} \\ =\frac{-225÷9}{72÷9} \\ =\frac{-25}{8} \\ \therefore (\frac{-8}{3}+\frac{-13}{12})\times \frac{5}{6}=(\frac{-8}{3}\times \frac{5}{6})+(\frac{-13}{12}\times \frac{5}{6}) \end{gathered}$$

(iv)

$$\begin{gathered} \frac{-16}{7}\times (\frac{-8}{9}+\frac{-7}{6})=(\frac{-16}{7}\times \frac{-8}{9})+(\frac{-16}{7}\times \frac{-7}{6}) \\ \text{LHS}=\frac{-16}{7}\times (\frac{-8}{9}+\frac{-7}{6}) \\ =\frac{-16}{7}\times \left(\frac{-16-21}{18}\right) \\ =\frac{-16}{7}\times \frac{-37}{18} \\ =\frac{592}{126} \\ =\frac{296}{63} \\ \text{RHS}=(\frac{-16}{7}\times \frac{-8}{9})+(\frac{-16}{7}\times \frac{-7}{6}) \\ =\frac{128}{63}+\frac{112}{42} \\ =\frac{256+336}{126} \\ =\frac{592}{126} \\ =\frac{296}{63} \\ \therefore \frac{-16}{7}\times (\frac{-8}{9}+\frac{-7}{6})=(\frac{-16}{7}\times \frac{-8}{9})+(\frac{-16}{7}\times \frac{-7}{6}) \end{gathered}$$