293. Проверьте, что следующие равенства верны: а) 0,222... = \(\frac{2}{9}\); б) 5,(6) = \(5\frac{2}{3}\); в) 0,818181... = \(\frac{9}{11}\); г) 0,(06) = \(\frac{2}{33}\); д) 0,4666... = \(\frac{7}{15}\); е) 2,8(12) = \(2\frac{134}{165}\).

Ответ: Все равенства верны.

Решение:

a) \(0,222... = \frac{2}{9}\) \(x = 0,222...\) \(10x = 2,222...\) \(10x - x = 2,222... - 0,222...\) \(9x = 2\) \(x = \frac{2}{9}\) Равенство верно. б) \(5,(6) = 5\frac{2}{3}\) \(5,(6) = 5 + 0,(6) = 5 + \frac{6}{9} = 5 + \frac{2}{3} = 5\frac{2}{3}\) Равенство верно. в) \(0,818181... = \frac{9}{11}\) \(x = 0,818181...\) \(100x = 81,818181...\) \(100x - x = 81,818181... - 0,818181...\) \(99x = 81\) \(x = \frac{81}{99} = \frac{9}{11}\) Равенство верно. г) \(0,(06) = \frac{2}{33}\) \(x = 0,(06)\) \(100x = 6,(06)\) \(100x - x = 6,(06) - 0,(06)\) \(99x = 6\) \(x = \frac{6}{99} = \frac{2}{33}\) Равенство верно. д) \(0,4666... = \frac{7}{15}\) \(x = 0,4666...\) \(10x = 4,666...\) \(100x = 46,666...\) \(100x - 10x = 46,666... - 4,666...\) \(90x = 42\) \(x = \frac{42}{90} = \frac{7}{15}\) Равенство верно. е) \(2,8(12) = 2\frac{134}{165}\) \(x = 2,8(12)\) \(10x = 28,(12)\) \(1000x = 2812,(12)\) \(1000x - 10x = 2812,(12) - 28,(12)\) \(990x = 2784\) \(x = \frac{2784}{990} = \frac{464}{165} = \frac{330 + 134}{165} = 2\frac{134}{165}\) Равенство верно.

Ответ: Все равенства верны.