Verified answer

a=2,6666...

10a=26,6666

10a-a=24

a=24/9

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a=1,202020....

100a=120,202020...

100a-a=119

a=119/99

=======================

a=1,62222...

10a=16,2222

100a=162,2222..

100a-10a=146

a=146/90

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a=2,5101010

10a=25,1010101

1000a=2510,101

1000a-10a=2485

a=2485/990

x = 2,6666

10x = 26,6666

9x = 26,6666-2,6666=24

x = 24/9

x = 1,202020

100x = 120,20202

99x = 119

x = 119/99

x = 1,6222

10x = 16,222

100x = 162,2222

90x = 146

x = 146/90 = 73/45

x=2,5101010

10x = 25,101010

1000x =2510,10101

990x = 2485

x = 2485/990 = 497/198

2,6666... = 2 + 0,6666... = 2 + 6/9 = 2 + 2/3 = (6 + 2) / 3 = 8/3

1,202020... = 1 + 0,202020... = 1 + 20/99 = (99 + 20) / 99 = 119/99

1,6222... = 1,6 + 0,02222... = 16/10 + 2/90 = (144 + 2) / 90 = 146/90 = 73/45

2,5101010... = 2,5 + 0,0101010... = 25/10 + 10/990 = (2475 + 10) / 990 = 2485/990 = 497/198