Question

Zamień ułamek okresowy na zwykły:
a) 0,(6) =
b) 0,(54) =
c) 4,(3) =
d) 0,5(3) =
e) 20,2(7) =
f) 0,1(5) =
g) 0,2(2) =
h) 0,9(4) =
i) 0,(72) =

Answer 1

A) 0,(6) = 0,6666....
x = 0,666.....
10x = 6,666....
10x - x = 6,666....-0,666
9x = 6  |:9
x = 6/9 = 2/3
0,(6) = 2/3

b) 0,(54) = 0,545454...
x = 0,545454....
100x = 54,545454....
100x - x = 54,545454...-0,545454...
99x = 54  |:99
x = 54/99 = 6/11
0,(54) = 6/11

c)
4,(3) = 4,3333....
x = 4,33333.....
10x = 43,33333....
10x - x = 43,3333.... - 4,33333....
9x = 39  |:9
x = 39/9 = 4 3/9 = 4 1/3
4,(3) = 4 1/3

d) 0,5(3) = 0,533333....
x = 0,53333....
100x = 53,3333....
100x - x  = 53,33333...-0,5333....
99x = 52,8   |:99
x = 52,8 /99 = 528/990 = 264/495
0,5(3) = 264/495

e) 20,2(7) = 20,2 + 0,0(7) = 20 1/5 + 0,07777....
x = 0,07777...
10x = 0,7777....
10x-x = 0,7777... - 0,07777...
9x = 0,7  |:9
x =  0,7/9 = 7/90
20,2(7) = 20 1/5  + 7/90 = 20 18/90  + 7/90 = 20 25/90

f) 0,1(5) = 0,155555....
x = 0,15555....
10x = 1,555555.....
10x - x = 1,55555.... - 0,15555....
9x = 1,4  |:9
x = 1,4/9 = 14/90 = 7/45
0,1(5) = 7/45

g) 0,2(2) = 0,(2) =  0,222222....
x = 0,222222....
10x = 2,22222...
10x - x = 2,22222...-0,22222...
9x = 2  |:9
x = 2/9
0,2(2) = 2/9

h) 0,9(4) = 0,944444....
x = 0,944444.....
10x = 9,44444.....
10x - x = 9,44444.... - 0,944444....
9x = 8,5  |:9
x = 8,5/9 = 85/90 = 17/18
0,9(4) = 17/18

i) 0,(72) = 0,727272.....
x = 0,727272.....
100x = 72,727272....
100x - x = 72,727272...-0,727272....
99x = 72  |:99
x = 72/99 = 24/33 = 8/11
0,(72) = 8/11