Verify the following:

Solution:

(i) $\mathrm{LHS}=\frac{3}{7} \times\left(\frac{5}{6}+\frac{12}{13}\right)$

$=\frac{3}{7} \times\left(\frac{65+72}{78}\right)$

$=\frac{3}{7} \times \frac{137}{78}$

$=\frac{137}{182}$

RHS $=\left(\frac{3}{7} \times \frac{5}{6}\right)+\left(\frac{12}{13} \times \frac{3}{7}\right)$

$=\frac{3 \times 5}{7 \times 6}+\frac{12 \times 3}{13 \times 7}$

$=\frac{15}{42}+\frac{36}{91}$

$=\frac{195+216}{546}$

$=\frac{411}{546}$

$=\frac{137}{182}$

$\therefore \frac{3}{7} \times\left(\frac{5}{6}+\frac{12}{13}\right)=\left(\frac{3}{7} \times \frac{5}{6}\right)+\left(\frac{3}{7} \times \frac{12}{13}\right)$

(ii) $\mathrm{LHS}=\frac{-15}{4} \times\left(\frac{3}{7}+\frac{-12}{5}\right)$

$=\frac{-15}{4} \times\left(\frac{15-84}{35}\right)$

$=\frac{-15}{4} \times \frac{-69}{35}$

$=\frac{(-15) \times(-69)}{140}$

$=\frac{1035}{140}$

$=\frac{207}{28}$

$\mathrm{RHS}=\left(\frac{-15}{4} \times \frac{3}{7}\right)+\left(\frac{-15}{4} \times \frac{-12}{5}\right)$

$=\frac{(-15) \times 3}{4 \times 7}+\frac{(-15) \times(-12)}{4 \times 5}$

$=\frac{-45}{28}+\frac{180}{20}$

$=\frac{-225+1260}{140}$

$=\frac{1035}{140}$

$=\frac{207}{28}$

$\therefore \frac{-15}{4} \times\left(\frac{3}{7}+\frac{-12}{5}\right)=\left(\frac{-15}{4} \times \frac{3}{7}\right)+\left(\frac{-15}{4} \times \frac{-12}{5}\right)$

(iii) $\left(\frac{-8}{3}+\frac{-13}{12}\right) \times \frac{5}{6}=\left(\frac{-8}{3} \times \frac{5}{6}\right)+\left(\frac{-13}{12} \times \frac{5}{6}\right)$

$\mathrm{LHS}=\left(\frac{-8}{3}+\frac{-13}{12}\right) \times \frac{5}{6}$

$=\frac{-32-13}{12} \times \frac{5}{6}$

$=\frac{-45}{12} \times \frac{5}{6}$

$=\frac{-45 \times 5}{12 \times 6}$

$=\frac{-225}{72}$

$=\frac{-225 \div 9}{72 \div 9}$

$=-\frac{25}{8}$

$\mathrm{RHS}=\left(\frac{-8}{3} \times \frac{5}{6}\right)+\left(\frac{-13}{12} \times \frac{5}{6}\right)$

$=\frac{-8 \times 5}{3 \times 6}+\frac{(-13) \times 5}{12 \times 6}$

$=\frac{-40}{18}+\frac{-65}{72}$

$=\frac{-160-65}{72}$

$=\frac{-225}{72}$

$=\frac{-225 \div 9}{72 \div 9}$

$=\frac{-25}{8}$

$\therefore\left(\frac{-8}{3}+\frac{-13}{12}\right) \times \frac{5}{6}=\left(\frac{-8}{3} \times \frac{5}{6}\right)+\left(\frac{-13}{12} \times \frac{5}{6}\right)$

(iv) $\frac{-16}{7} \times\left(\frac{-8}{9}+\frac{-7}{6}\right)=\left(\frac{-16}{7} \times \frac{-8}{9}\right)+\left(\frac{-16}{7} \times \frac{-7}{6}\right)$

$\mathrm{LHS}=\frac{-16}{7} \times\left(\frac{-8}{9}+\frac{-7}{6}\right)$

$=\frac{-16}{7} \times\left(\frac{-16-21}{18}\right)$

$=\frac{-16}{7} \times \frac{-37}{18}$

$=\frac{592}{126}$

$=\frac{296}{63}$

$\mathrm{RHS}=\left(\frac{-16}{7} \times \frac{-8}{9}\right)+\left(\frac{-16}{7} \times \frac{-7}{6}\right)$

$=\frac{128}{63}+\frac{112}{42}$

$=\frac{256+336}{126}$

$=\frac{592}{126}$

$=\frac{296}{63}$

$\therefore \frac{-16}{7} \times\left(\frac{-8}{9}+\frac{-7}{6}\right)=\left(\frac{-16}{7} \times \frac{-8}{9}\right)+\left(\frac{-16}{7} \times \frac{-7}{6}\right)$