3.45 Решите пропорцию: a) \(\frac{13,7}{4} = \frac{9t}{3,6}\); б) \(\frac{1}{3}a : 6 = 14 : 0,7\); в) \(\frac{1,5}{a + 0,03} = \frac{6,3}{0,21}\); г) \(\frac{4}{5} : 2,5 = 1\frac{1}{2} : (0,4 + b)\).

a) \(\frac{13,7}{4} = \frac{9t}{3,6}\)

По основному свойству пропорции:

\(4 \cdot 9t = 13,7 \cdot 3,6\)

\(36t = 49,32\)

\(t = \frac{49,32}{36}\)

\(t = 1,37\)

б) \(\frac{1}{3}a : 6 = 14 : 0,7\)

\(\frac{1}{3}a \cdot 0,7 = 6 \cdot 14\)

\(\frac{7}{30}a = 84\)

\(a = 84 : \frac{7}{30}\)

\(a = 84 \cdot \frac{30}{7}\)

\(a = 12 \cdot 30\)

\(a = 360\)

в) \(\frac{1,5}{a + 0,03} = \frac{6,3}{0,21}\)

По основному свойству пропорции:

\(6,3(a + 0,03) = 1,5 \cdot 0,21\)

\(6,3a + 0,189 = 0,315\)

\(6,3a = 0,315 - 0,189\)

\(6,3a = 0,126\)

\(a = \frac{0,126}{6,3}\)

\(a = 0,02\)

г) \(\frac{4}{5} : 2,5 = 1\frac{1}{2} : (0,4 + b)\)

\(\frac{4}{5} : \frac{5}{2} = \frac{3}{2} : (0,4 + b)\)

\(\frac{4}{5} \cdot (0,4 + b) = \frac{5}{2} \cdot \frac{3}{2}\)

\(\frac{16}{50} + \frac{4}{5}b = \frac{15}{4}\)

\(\frac{4}{5}b = \frac{15}{4} - \frac{16}{50}\)

\(\frac{4}{5}b = \frac{15 \cdot 25 - 8 \cdot 8}{100}\)

\(\frac{4}{5}b = \frac{375 - 64}{100}\)

\(\frac{4}{5}b = \frac{311}{100}\)

\(b = \frac{311}{100} : \frac{4}{5}\)

\(b = \frac{311}{100} \cdot \frac{5}{4}\)

\(b = \frac{311}{20 \cdot 4}\)

\(b = \frac{311}{80}\)

\(b = 3\frac{71}{80}\)

Ответ: a) 1,37; б) 360; в) 0,02; г) \(3\frac{71}{80}\)