Запишите периодическую дробь в виде обыкновенной дроби:1,(8) 0.1(2) 0.(59),0.(5) 1.12(3) 0.(12)

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1,(8) = 1 + 0.888... = 1 + 8/9 = 17/9

0.1(2) = 0.12222... = 0.1 + 0.022 + 0.0022... = 1/10 + 2/90 + 2/900 = 1/10 + 1/45 = 5/45 + 1/45 = 6/45 = 2/15

0.(59) = 0.595959... = 0.59 + 0.0000059... = 59/100 + 59/10^5 / (10^2 - 1) = 59/100 + 59/10^5 / 99 = 59/100 + 59/9900 = 59(1/100 + 1/9900) = 59(99 + 1)/9900 = 59 * 100 / 9900 = 5900/9900 = 590/990 = 59/99

0.(5) = 0.5555... = 5/9

1.12(3) = 1 + 0.123123... = 1 + 123/999 = 1 + 41/333 = 1 + 41/3 * 111 = 1 + 41/333 = 1 + 41/111 = 111/111 + 41/111 = 152/111

0.(12) = 0.121212... = 0.12 + 0.00001212... = 12/100 + 12/10^5 / (10^2 - 1) = 12/100 + 12/10^5 / 99 = 12/100 + 12/990 = 12(1/100 + 1/990) = 12(99 + 1)/990 = 12 * 100 / 990 = 1200/990 = 40/33

Таким образом:
1,(8) = 17/9
0.1(2) = 2/15
0.(59) = 59/99
0.(5) = 5/9
1.12(3) = 152/111
0.(12) = 40/33