3.58 Найдите неизвестный член пропорции: 1 1 a) 3:2 = 2:t; 7 3 б) 3:s = 4:1; 3 1 6 в) у: = 8:2; 6 3 1 г) 5: = z: 12. 7 17

Решение:

a) $$3\frac{1}{7} : 2\frac{1}{3} = 2\frac{1}{6} : t$$

$$\frac{22}{7} : \frac{7}{3} = \frac{13}{6} : t$$ $$\frac{22}{7} \cdot t = \frac{7}{3} \cdot \frac{13}{6}$$ $$\frac{22}{7} \cdot t = \frac{91}{18}$$ $$t = \frac{91}{18} : \frac{22}{7} = \frac{91}{18} \cdot \frac{7}{22} = \frac{637}{396} = 1\frac{241}{396}$$

Ответ: $$t = 1\frac{241}{396}$$

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

$$\frac{10}{3} : s = \frac{14}{3} : \frac{7}{6}$$ $$\frac{14}{3} \cdot s = \frac{10}{3} \cdot \frac{7}{6}$$ $$\frac{14}{3} \cdot s = \frac{70}{18}$$ $$s = \frac{70}{18} : \frac{14}{3} = \frac{70}{18} \cdot \frac{3}{14} = \frac{70 \cdot 3}{18 \cdot 14} = \frac{5 \cdot 14 \cdot 3}{6 \cdot 3 \cdot 14} = \frac{5}{6}$$

Ответ: $$s = \frac{5}{6}$$

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

$$y : \frac{1}{6} = \frac{26}{3} : \frac{13}{6}$$ $$\frac{26}{3} \cdot \frac{1}{6} = y \cdot \frac{13}{6}$$ $$\frac{26}{18} = y \cdot \frac{13}{6}$$ $$y = \frac{26}{18} : \frac{13}{6} = \frac{26}{18} \cdot \frac{6}{13} = \frac{2 \cdot 13 \cdot 6}{3 \cdot 6 \cdot 13} = \frac{2}{3}$$

Ответ: $$y = \frac{2}{3}$$

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

$$\frac{36}{7} : \frac{6}{7} = z : 12$$ $$\frac{36}{7} \cdot 12 = z \cdot \frac{6}{7}$$ $$\frac{432}{7} = z \cdot \frac{6}{7}$$ $$z = \frac{432}{7} : \frac{6}{7} = \frac{432}{7} \cdot \frac{7}{6} = \frac{432 \cdot 7}{7 \cdot 6} = \frac{432}{6} = 72$$

Ответ: $$z = 72$$