8 Найдите неизвестный член пропорции: a) 3\frac{1}{2}:2\frac{1}{7}=2\frac{1}{3}:t; б) 3\frac{1}{3}: s = 4\frac{2}{3}:1\frac{1}{6}; в) y:\frac{2}{3} = 8\frac{1}{6}:2\frac{1}{3}; г) 5\frac{1}{7}: \frac{6}{7} = z:\frac{12}{17}.

Решим пропорции.

a) $$3\frac{1}{2}:2\frac{1}{7}=2\frac{1}{3}:t$$

$$3\frac{1}{2} = \frac{7}{2}$$, $$2\frac{1}{7} = \frac{15}{7}$$, $$2\frac{1}{3}=\frac{7}{3}$$

$$\frac{7}{2}:\frac{15}{7}=\frac{7}{3}:t$$

Основное свойство пропорции: произведение крайних членов равно произведению средних членов.

$$\frac{7}{2} \cdot t = \frac{15}{7} \cdot \frac{7}{3}$$

$$\frac{7}{2} t = \frac{15}{3}$$

$$\frac{7}{2} t = 5$$

$$t=5:\frac{7}{2}$$, $$t=5 \cdot \frac{2}{7}$$, $$t=\frac{10}{7}$$, $$t=1\frac{3}{7}$$

б) $$3\frac{1}{3}: s = 4\frac{2}{3}:1\frac{1}{6}$$

$$3\frac{1}{3}=\frac{10}{3}$$, $$4\frac{2}{3}=\frac{14}{3}$$, $$1\frac{1}{6}=\frac{7}{6}$$

$$\frac{10}{3}:s=\frac{14}{3}:\frac{7}{6}$$

$$\frac{10}{3} \cdot \frac{7}{6} = s \cdot \frac{14}{3}$$

$$\frac{70}{18}=\frac{14}{3}s$$

$$\frac{14}{3}s=\frac{70}{18}$$

$$s=\frac{70}{18}:\frac{14}{3}$$, $$s=\frac{70}{18} \cdot \frac{3}{14}$$, $$s=\frac{5}{3}$$, $$s=1\frac{2}{3}$$

в) $$y:\frac{2}{3} = 8\frac{1}{6}:2\frac{1}{3}$$

$$8\frac{1}{6}=\frac{49}{6}$$, $$2\frac{1}{3}=\frac{7}{3}$$

$$y:\frac{2}{3}=\frac{49}{6}:\frac{7}{3}$$

$$y \cdot \frac{7}{3}=\frac{2}{3} \cdot \frac{49}{6}$$

$$\frac{7}{3}y=\frac{98}{18}$$

$$\frac{7}{3}y=\frac{49}{9}$$

$$y=\frac{49}{9}:\frac{7}{3}$$, $$y=\frac{49}{9} \cdot \frac{3}{7}$$, $$y=\frac{7}{3}$$, $$y=2\frac{1}{3}$$

г) $$5\frac{1}{7}: \frac{6}{7} = z:\frac{12}{17}$$

$$5\frac{1}{7}=\frac{36}{7}$$

$$\frac{36}{7}:\frac{6}{7}=z:\frac{12}{17}$$

$$\frac{36}{7} \cdot \frac{12}{17}=\frac{6}{7}z$$

$$\frac{6}{7}z=\frac{432}{119}$$

$$z=\frac{432}{119}:\frac{6}{7}$$, $$z=\frac{432}{119} \cdot \frac{7}{6}$$, $$z=\frac{36}{17}$$, $$z=2\frac{2}{17}$$

Ответ: a) $$t=1\frac{3}{7}$$, б) $$s=1\frac{2}{3}$$, в) $$y=2\frac{1}{3}$$, г) $$z=2\frac{2}{17}$$