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

To do:

We have to simplify the given expressions.

Solution:

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

$=\frac{1 \times 1}{2 \times 2}+\frac{5 \times 7}{3 \times 2}-\frac{13 \times 1}{2 \times 3}$

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

$=\frac{3+70-26}{12}$           (LCM of $4,6=12$)

$=\frac{73-26}{12}$

$=\frac{47}{12}$

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

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

$=\frac{1 \times 1}{2 \times 7}-\frac{5 \times(-1)}{7 \times 3}+\frac{3 \times 9}{7 \times 2}$

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

$=\frac{3+10+81}{42}$             (LCM of $14,21=42$)

$=\frac{94}{42}$

$=\frac{47}{21}$

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

$=\frac{13 \times(-5)}{3 \times 2} \times \frac{7 \times 8}{3 \times 5}+\frac{3 \times 1}{5 \times 2}$

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

$=\frac{-13 \times 28}{3 \times 3}+\frac{3}{10}$

$=\frac{-364}{9}+\frac{3}{10}$

$=\frac{-3640+27}{90}$             (LCM of $9,10=90$)

$=\frac{-3613}{90}$

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

$=\frac{1 \times 5}{11 \times 2}-\frac{3 \times 1}{3}+\frac{1 \times 2}{13 \times 1}$

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

$=\frac{65-286+44}{286}$        (LCM of $22,13=286$)

$=\frac{109-286}{286}$

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