The least fraction to be subtracted from the expression $$\frac{{3\frac{1}{4} - \frac{4}{5}{\text{ of }}\frac{5}{6}}}{{4\frac{1}{3} \div \frac{1}{5} - \left( {\frac{3}{{10}} + 21\frac{1}{5}} \right)}}$$     to make it an integer?

$$\eqalign{ & {\text{According to question,}} \cr & \frac{{3\frac{1}{4} - \frac{4}{5}{\text{ of }}\frac{5}{6}}}{{4\frac{1}{3} \div \frac{1}{5} - \left( {\frac{3}{{10}} + 21\frac{1}{5}} \right)}}{\text{ }} \cr & \Rightarrow \frac{{\frac{{13}}{4} - \frac{4}{5} \times \frac{5}{6}}}{{\frac{{13}}{3} \times 5 - \left( {\frac{3}{{10}} + \frac{{106}}{5}} \right)}}{\text{ }} \cr & \Rightarrow \frac{{\frac{{13}}{4} - \frac{2}{3}}}{{\frac{{65}}{3} - \frac{3}{{10}} - \frac{{106}}{5}}} \cr & \Rightarrow \frac{{\frac{{39 - 8}}{{12}}}}{{\frac{{650 - 9 - 636}}{{30}}}} \cr & \Rightarrow \frac{{31}}{{12}} \times \frac{{30}}{5} \cr & \Rightarrow \frac{{31}}{2} \cr & \Rightarrow 15\frac{1}{2} \cr} $$
∴ Least fraction number should be subtracted is:
$${\text{15}}\frac{1}{2} - 15 = \frac{1}{2}$$     (least fraction number)