55. Найдите неизвестный член пропорции: a) \(3 : x = 6 : 1\frac{1}{3}\) b) \(4\frac{2}{5} : x = 8\frac{2}{5} : 2\frac{1}{2}\) c) \(7\frac{1}{2} : 2 = 3\frac{2}{3} : y\) d) \(6\frac{1}{2} : 3 = 3\frac{1}{4} : y\)

Решение:

1. а) \(3 : x = 6 : 1\frac{1}{3}\)
   \( x = \frac{3 \cdot 1\frac{1}{3}}{6} = \frac{3 \cdot \frac{4}{3}}{6} = \frac{4}{6} = \frac{2}{3} \)
2. б) \(4\frac{2}{5} : x = 8\frac{2}{5} : 2\frac{1}{2}\)
   \( x = \frac{4\frac{2}{5} \cdot 2\frac{1}{2}}{8\frac{2}{5}} = \frac{\frac{22}{5} \cdot \frac{5}{2}}{\frac{42}{5}} = \frac{11}{\frac{42}{5}} = 11 \cdot \frac{5}{42} = \frac{55}{42} = 1\frac{13}{42} \)
3. в) \(7\frac{1}{2} : 2 = 3\frac{2}{3} : y\)
   \( y = \frac{2 \cdot 3\frac{2}{3}}{7\frac{1}{2}} = \frac{2 \cdot \frac{11}{3}}{\frac{15}{2}} = \frac{\frac{22}{3}}{\frac{15}{2}} = \frac{22}{3} \cdot \frac{2}{15} = \frac{44}{45} \)
4. г) \(6\frac{1}{2} : 3 = 3\frac{1}{4} : y\)
   \( y = \frac{3 \cdot 3\frac{1}{4}}{6\frac{1}{2}} = \frac{3 \cdot \frac{13}{4}}{\frac{13}{2}} = \frac{\frac{39}{4}}{\frac{13}{2}} = \frac{39}{4} \cdot \frac{2}{13} = \frac{3 \cdot 2}{2} = 3 \)

Ответ: а) \(x = \frac{2}{3}\); б) \(x = 1\frac{13}{42}\); в) \(y = \frac{44}{45}\); г) \(y = 3\).