Simplify:

Solution:

$(\mathrm{i})\left(\frac{3}{2} \times \frac{1}{6}\right)+\left(\frac{5}{3} \times \frac{7}{2}\right)-\left(\frac{13}{8} \times \frac{4}{3}\right)$

$=\frac{1}{4}+\frac{35}{6}-\frac{13}{6}$

$=\frac{1 \times 3+35 \times 2-13 \times 2}{12}$

$=\frac{3+70-26}{12}$

$=\frac{47}{12}$

(ii) $\left(\frac{1}{4} \times \frac{2}{7}\right)-\left(\frac{5}{14} \times \frac{-2}{3}\right)+\left(\frac{3}{7} \times \frac{9}{2}\right)$

$=\frac{1}{14}-\frac{-5}{21}+\frac{27}{14}$

$=\frac{1 \times 3+5 \times 2+27 \times 3}{42}$

$=\frac{94}{42}=\frac{47}{21}$

$(\mathrm{iii})\left(\frac{13}{9} \times \frac{-15}{2}\right)+\left(\frac{7}{3} \times \frac{8}{5}\right)+\left(\frac{3}{5} \times \frac{1}{2}\right)$

$=\frac{-13 \times 5}{6}+\frac{7 \times 8}{15}+\frac{3}{10}$

$=\frac{-65}{6}+\frac{56}{15}+\frac{3}{10}$

$=\frac{-65 \times 5+56 \times 2+3 \times 3}{30}$

$=\frac{-204}{30}=\frac{-34}{5}$

$($ iv $)\left(\frac{3}{11} \times \frac{5}{6}\right)-\left(\frac{9}{12} \times \frac{4}{3}\right)+\left(\frac{5}{13} \times \frac{6}{15}\right)$

$=\frac{5}{22}-1+\frac{2}{13}$

$=\frac{5 \times 13-286+2 \times 22}{286}$

$=\frac{65-286+44}{286}$

$=\frac{-177}{286}$