Question

Simplify:
(i) $\left(\frac{13}{9}\times \frac{-15}{2}\right)+\left(\frac{7}{3}\times \frac{8}{5}\right)+\left(\frac{3}{5}\times \frac{1}{2}\right)$
(ii) $\left(\frac{3}{11}\times \frac{5}{6}\right)-\left(\frac{9}{12}\times \frac{4}{3}\right)+\left(\frac{5}{13}\times \frac{6}{15}\right)$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)(\frac{13}{9}\times \frac{-15}{2})+(\frac{7}{3}\times \frac{8}{5})+(\frac{3}{5}\times \frac{1}{2}) \\ =(\frac{13}{3\times 3}\times \frac{-3\times 5}{2})+\left(\frac{56}{15}\right)+\left(\frac{3}{10}\right)=\frac{-65}{6}+\frac{56}{15}+\frac{3}{10} \\ =\frac{-65\times 5}{6\times 5}+\frac{56\times 2}{15\times 2}+\frac{3\times 3}{10\times 3}=\frac{-325}{30}+\frac{112}{30}+\frac{9}{30}=\frac{-325+112+9}{30}=\frac{-204}{30}=\frac{-34}{5} \\ \left(\mathrm{ii}\right)(\frac{3}{11}\times \frac{5}{6})-(\frac{9}{12}\times \frac{4}{3})+(\frac{5}{13}\times \frac{6}{15}) \\ =(\frac{3}{11}\times \frac{5}{2\times 3})-\left(\frac{3\times 3}{4\times 3}\times \frac{4}{3}\right)+\left(\frac{5}{13}\times \frac{3\times 2}{5\times 3}\right)=\frac{5}{22}-\frac{1}{1}+\frac{2}{13} \\ =\frac{5\times 13}{22\times 13}-\frac{1\times 286}{1\times 286}+\frac{2\times 22}{13\times 22}=\frac{65}{286}-\frac{286}{286}+\frac{44}{286}=\frac{65-286+44}{286}=\frac{-177}{286} \end{gathered}$$