$$2\frac{2}{3}:1\frac{1}{7}\cdot3\frac{3}{7}\cdot(-\frac{1}{2})$$
$-4$
$$\frac{\left(2\times 3+2\right)\times 7}{3\left(1\times 7+1\right)}\times \frac{3\times 7+3}{7}\left(-\frac{1}{2}\right)$$
$$\frac{\left(6+2\right)\times 7}{3\left(1\times 7+1\right)}\times \frac{3\times 7+3}{7}\left(-\frac{1}{2}\right)$$
$$\frac{8\times 7}{3\left(1\times 7+1\right)}\times \frac{3\times 7+3}{7}\left(-\frac{1}{2}\right)$$
$$\frac{56}{3\left(1\times 7+1\right)}\times \frac{3\times 7+3}{7}\left(-\frac{1}{2}\right)$$
$$\frac{56}{3\left(7+1\right)}\times \frac{3\times 7+3}{7}\left(-\frac{1}{2}\right)$$
$$\frac{56}{3\times 8}\times \frac{3\times 7+3}{7}\left(-\frac{1}{2}\right)$$
$$\frac{56}{24}\times \frac{3\times 7+3}{7}\left(-\frac{1}{2}\right)$$
$$\frac{7}{3}\times \frac{3\times 7+3}{7}\left(-\frac{1}{2}\right)$$
$$\frac{7}{3}\times \frac{21+3}{7}\left(-\frac{1}{2}\right)$$
$$\frac{7}{3}\times \frac{24}{7}\left(-\frac{1}{2}\right)$$
$$\frac{7\times 24}{3\times 7}\left(-\frac{1}{2}\right)$$
$$\frac{24}{3}\left(-\frac{1}{2}\right)$$
$$8\left(-\frac{1}{2}\right)$$
$$\frac{8\left(-1\right)}{2}$$
$$\frac{-8}{2}$$
$$-4$$
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