$$2\frac{3}{4}\div\frac{7}{8}\times(\frac{1}{3}+\frac{1}{4})+\frac{5}{7}\div\frac{3}{4}\times\frac{3}{7}$$
$\frac{659}{294}\approx 2.241496599$
$$\frac{\left(2\times 4+3\right)\times 8}{4\times 7}\left(\frac{1}{3}+\frac{1}{4}\right)+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{2\left(3+2\times 4\right)}{7}\left(\frac{1}{3}+\frac{1}{4}\right)+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{2\left(3+8\right)}{7}\left(\frac{1}{3}+\frac{1}{4}\right)+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{2\times 11}{7}\left(\frac{1}{3}+\frac{1}{4}\right)+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{22}{7}\left(\frac{1}{3}+\frac{1}{4}\right)+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{22}{7}\left(\frac{4}{12}+\frac{3}{12}\right)+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{22}{7}\times \frac{4+3}{12}+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{22}{7}\times \frac{7}{12}+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{22\times 7}{7\times 12}+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{22}{12}+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{11}{6}+\frac{\frac{5}{7}}{\frac{3}{4}}\times \frac{3}{7}$$
$$\frac{11}{6}+\frac{5}{7}\times \frac{4}{3}\times \frac{3}{7}$$
$$\frac{11}{6}+\frac{5\times 4}{7\times 3}\times \frac{3}{7}$$
$$\frac{11}{6}+\frac{20}{21}\times \frac{3}{7}$$
$$\frac{11}{6}+\frac{20\times 3}{21\times 7}$$
$$\frac{11}{6}+\frac{60}{147}$$
$$\frac{11}{6}+\frac{20}{49}$$
$$\frac{539}{294}+\frac{120}{294}$$
$$\frac{539+120}{294}$$
$$\frac{659}{294}$$
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$\frac{659}{2 \cdot 3 \cdot 7 ^ {2}} = 2\frac{71}{294} = 2.241496598639456$
