Запишите периодические дроби в виде обыкновенных дробей:
а) 1,(0); 0,(3); 0,(7);
б) 0,1(2); 1,12(3); 7,5(4);
в) 0,(12); 1,0(12); 8,7(21);
г) 23,5(0); 23,5(1); 23,5(13); 23,5(127).
$1,(0) = 1,000... = \frac{1}{1}$
0,(3)
x = 0,(3)
10x = 3,(3)
10x − x = 3,(3) − 0,(3)
9x = 3
$x = \frac{3}{9} = \frac{1}{3}$
$0,(3) = \frac{1}{3}$
0,(7)
x = 0,(7)
10x = 7,(7)
10x − x = 7,(7) − 0,(7)
9x = 7
$x = \frac{7}{9}$
$0,(7) = \frac{7}{9}$
0,1(2)
x = 0,1(2)
10x = 1,(2)
100x = 12,(2)
100x − 10x = 12,(2) − 1,(2)
90x = 11
$x = \frac{11}{90}$
$0,1(2) = \frac{11}{90}$
1,12(3)
x = 1,12(3)
100x = 112,(3)
1000x = 1123,(3)
1000x − 100x = 1123,(3) − 112,(3)
900x = 1011
$x = \frac{1011}{900}$
$1,12(3) = \frac{1011}{900}$
7,5(4)
x = 7,5(4)
10x = 75,(4)
100x = 754,(4)
100x − 10x = 754,(4) − 75,(4)
90x = 679
$x = \frac{679}{90}$
$7,5(4) = \frac{679}{90}$
0,(12)
x = 0,(12)
100x = 12,(12)
100x − x = 12,(12) − 0,(12)
99x = 12
$x = \frac{12}{99} = \frac{4}{33}$
$0,(12) = \frac{4}{33}$
1,0(12)
x = 1,0(12)
10x = 10,(12)
1000x = 1012,(12)
1000x − 10x = 1012,(12) − 10,(12)
990x = 1002
$x = \frac{1002}{990} = \frac{334}{330} = \frac{167}{165}$
$0,(12) = \frac{167}{165}$
8,7(21)
x = 8,7(21)
10x = 87,(21)
1000x = 8721,(21)
1000x − 10x = 8721,(21) − 87,(21)
990x = 8634
$x = \frac{8634}{90} = \frac{4327}{495}$
$8,7(21) = \frac{4327}{495}$
$23,5(0) = 23,5000... = 23,5 = 23\frac{5}{10} = 23\frac{1}{2} = 23\frac{47}{2}$
23,5(1)
x = 23,5(1)
10x = 235,(1)
100x = 2351,(1)
100x − 10x = 2351,(1) − 235,(1)
90x = 2116
$x = \frac{2116}{90} = \frac{1058}{45}$
$23,5(1) = \frac{1058}{45}$
23,5(13)
10x = 235,(13)
1000x = 23513,(13)
1000x − 10x = 23513,(13) − 235,(13)
990x = 23278
$x = \frac{23278}{990} = \frac{11639}{495}$
$23,5(13) = \frac{11639}{495}$
23,5(127)
x = 23,5(127)
10x = 235,(127)
10000x = 235127,(127)
10000x − 10x = 234892
9990x = 234892
$x = \frac{234892}{9990} = \frac{117446}{4995}$
$23,5(127) = \frac{117446}{4995}$
Пожауйста, оцените решение
