509. Ададҳоро ба намуди касри одӣ нависед: a) 0, (8); б) 0, (3); в) 0, (26); г) 2, (71); ғ) 0,2(3); д) 0,82 (45); e) 0. (5); ë)1, (72); ж) 0,4 (6); з) 0,01 (12); и) 0,1 (3); к) 2, (1); л) 0,21 (22); м) 0,13 (11); н) 0,2 (52).

Решение:

Давай представим каждое число в виде обыкновенной дроби. Начнем с первого примера, а затем перейдем к остальным.

a) 0, (8):

Пусть x = 0,(8) = 0,888...

Тогда 10x = 8,888...

Вычтем x из 10x: 10x - x = 8,888... - 0,888...

9x = 8

x = \(\frac{8}{9}\)

б) 0, (3):

Аналогично, x = 0,(3) = 0,333...

10x = 3,333...

10x - x = 3,333... - 0,333...

9x = 3

x = \(\frac{3}{9} = \frac{1}{3}\)

в) 0, (26):

x = 0,(26) = 0,262626...

100x = 26,262626...

100x - x = 26,262626... - 0,262626...

99x = 26

x = \(\frac{26}{99}\)

г) 2, (71):

x = 2,(71) = 2,717171...

x - 2 = 0,(71)

Пусть y = 0,(71) = 0,717171...

100y = 71,717171...

100y - y = 71,717171... - 0,717171...

99y = 71

y = \(\frac{71}{99}\)

x = 2 + \(\frac{71}{99} = \frac{2 \cdot 99 + 71}{99} = \frac{198 + 71}{99} = \frac{269}{99}\)

г) 0,2(3):

x = 0,2(3) = 0,2333...

10x = 2,333...

100x = 23,333...

100x - 10x = 23,333... - 2,333...

90x = 21

x = \(\frac{21}{90} = \frac{7}{30}\)

д) 0,82 (45):

x = 0,82(45) = 0,82454545...

100x = 82,454545...

10000x = 8245,454545...

10000x - 100x = 8245,454545... - 82,454545...

9900x = 8163

x = \(\frac{8163}{9900} = \frac{2721}{3300} = \frac{907}{1100}\)

e) 0. (5):

x = 0,(5) = 0,555...

10x = 5,555...

10x - x = 5,555... - 0,555...

9x = 5

x = \(\frac{5}{9}\)

ë) 1, (72):

x = 1,(72) = 1,727272...

x - 1 = 0,(72)

Пусть y = 0,(72) = 0,727272...

100y = 72,727272...

100y - y = 72,727272... - 0,727272...

99y = 72

y = \(\frac{72}{99} = \frac{8}{11}\)

x = 1 + \(\frac{8}{11} = \frac{11 + 8}{11} = \frac{19}{11}\)

ж) 0,4 (6):

x = 0,4(6) = 0,4666...

10x = 4,666...

100x = 46,666...

100x - 10x = 46,666... - 4,666...

90x = 42

x = \(\frac{42}{90} = \frac{7}{15}\)

з) 0,01 (12):

x = 0,01(12) = 0,01121212...

100x = 1,121212...

10000x = 112,121212...

10000x - 100x = 112,121212... - 1,121212...

9900x = 111

x = \(\frac{111}{9900} = \frac{37}{3300}\)

и) 0,1 (3):

x = 0,1(3) = 0,1333...

10x = 1,333...

100x = 13,333...

100x - 10x = 13,333... - 1,333...

90x = 12

x = \(\frac{12}{90} = \frac{2}{15}\)

к) 2, (1):

x = 2,(1) = 2,111...

x - 2 = 0,(1)

Пусть y = 0,(1) = 0,111...

10y = 1,111...

10y - y = 1,111... - 0,111...

9y = 1

y = \(\frac{1}{9}\)

x = 2 + \(\frac{1}{9} = \frac{18 + 1}{9} = \frac{19}{9}\)

л) 0,21 (22):

x = 0,21(22) = 0,212222...

100x = 21,2222...

10000x = 2122,2222...

10000x - 100x = 2122,2222... - 21,2222...

9900x = 2101

x = \(\frac{2101}{9900}\)

м) 0,13 (11):

x = 0,13(11) = 0,131111...

100x = 13,1111...

1000x = 131,1111...

1000x - 100x = 131,1111... - 13,1111...

900x = 118

x = \(\frac{118}{900} = \frac{59}{450}\)

н) 0,2 (52):

x = 0,2(52) = 0,2525252...

10x = 2,525252...

1000x = 252,525252...

1000x - 10x = 252,525252... - 2,525252...

990x = 250

x = \(\frac{250}{990} = \frac{25}{99}\)

Ответ: a) \(\frac{8}{9}\); б) \(\frac{1}{3}\); в) \(\frac{26}{99}\); г) \(\frac{269}{99}\); ғ) \(\frac{7}{30}\); д) \(\frac{907}{1100}\); e) \(\frac{5}{9}\); ë) \(\frac{19}{11}\); ж) \(\frac{7}{15}\); з) \(\frac{37}{3300}\); и) \(\frac{2}{15}\); к) \(\frac{19}{9}\); л) \(\frac{2101}{9900}\); м) \(\frac{59}{450}\); н) \(\frac{25}{99}\)

Отлично! Ты справился с этим заданием. Продолжай в том же духе!