$$5-3\frac{1}{2}+5\frac{1}{4}\div3+16\frac{1}{4}\times\frac{2}{65})$$
$\frac{15}{4}=3.75$
$$5-\frac{6+1}{2}+\frac{\frac{5\times 4+1}{4}}{3}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$5-\frac{7}{2}+\frac{\frac{5\times 4+1}{4}}{3}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{10}{2}-\frac{7}{2}+\frac{\frac{5\times 4+1}{4}}{3}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{10-7}{2}+\frac{\frac{5\times 4+1}{4}}{3}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{3}{2}+\frac{\frac{5\times 4+1}{4}}{3}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{3}{2}+\frac{5\times 4+1}{4\times 3}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{3}{2}+\frac{20+1}{4\times 3}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{3}{2}+\frac{21}{4\times 3}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{3}{2}+\frac{21}{12}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{3}{2}+\frac{7}{4}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{6}{4}+\frac{7}{4}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{6+7}{4}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{13}{4}+\frac{16\times 4+1}{4}\times \frac{2}{65}$$
$$\frac{13}{4}+\frac{64+1}{4}\times \frac{2}{65}$$
$$\frac{13}{4}+\frac{65}{4}\times \frac{2}{65}$$
$$\frac{13}{4}+\frac{65\times 2}{4\times 65}$$
$$\frac{13}{4}+\frac{2}{4}$$
$$\frac{13+2}{4}$$
$$\frac{15}{4}$$
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$\frac{3 \cdot 5}{2 ^ {2}} = 3\frac{3}{4} = 3.75$
