51. 
0.232323….. = 0.23 = 23/99.
0.232323….. = 0.23 = 23/99.
52. 
0.125125…. =0.125 = 125/999
0.125125…. =0.125 = 125/999
53. 0.47 = 47/99.
54. 
0.36 = 36/39 =4/11.
0.36 = 36/39 =4/11.
55. 
1 + 0.2 = 1/0.2 = 10/2 = 5; 0.2 = 0.222 ….; (0.2)² = 0.04.
1 + 0.2 = 1/0.2 = 10/2 = 5; 0.2 = 0.222 ….; (0.2)² = 0.04.
0.04 < 0.2 < 0.22…. <5.
Since 0.04 is the least, so (0.2)²is the least.
56. 
6.46 = 6 + 0.46 = 6 + 46/99 = 594 +46/99 = 640 / 99.
6.46 = 6 + 0.46 = 6 + 46/99 = 594 +46/99 = 640 / 99.
57. 
0.57 = 57 – 5/90 =2/90 = 26/45.
0.57 = 57 – 5/90 =2/90 = 26/45.
58. 
0.84181 = 84181 – 841/99000 = 83340 / 99000 = 463/550
0.84181 = 84181 – 841/99000 = 83340 / 99000 = 463/550
Required difference = (550 – 463) = 87.
59. 
4.12 = 4 + 0.12 = 4 + 12 – 1 / 90 = 4 11/90
4.12 = 4 + 0.12 = 4 + 12 – 1 / 90 = 4 11/90
60. 2. + 0.136 = 2 + 136 – 1 /990 = 2 + 3/22 = 2 3/22
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