$$2 \frac{ 1 }{ 2 } \times 1 \frac{ 3 }{ 5 } -2 \frac{ 2 }{ 15 } \div 1 \frac{ 1 }{ 15 }$$
$2$
$$\frac{4+1}{2}\times \frac{1\times 5+3}{5}-\frac{\frac{2\times 15+2}{15}}{\frac{1\times 15+1}{15}}$$
$$\frac{5}{2}\times \frac{1\times 5+3}{5}-\frac{\frac{2\times 15+2}{15}}{\frac{1\times 15+1}{15}}$$
$$\frac{5}{2}\times \frac{5+3}{5}-\frac{\frac{2\times 15+2}{15}}{\frac{1\times 15+1}{15}}$$
$$\frac{5}{2}\times \frac{8}{5}-\frac{\frac{2\times 15+2}{15}}{\frac{1\times 15+1}{15}}$$
$$\frac{5\times 8}{2\times 5}-\frac{\frac{2\times 15+2}{15}}{\frac{1\times 15+1}{15}}$$
$$\frac{8}{2}-\frac{\frac{2\times 15+2}{15}}{\frac{1\times 15+1}{15}}$$
$$4-\frac{\frac{2\times 15+2}{15}}{\frac{1\times 15+1}{15}}$$
$$4-\frac{\left(2\times 15+2\right)\times 15}{15\left(1\times 15+1\right)}$$
$$4-\frac{2+2\times 15}{1+15}$$
$$4-\frac{2+30}{1+15}$$
$$4-\frac{32}{1+15}$$
$$4-\frac{32}{16}$$
$$4-2$$
$$2$$
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