The recurring decimal representation 1.272727... is equivalent to which one of the following?

Correct Option: (c) 127/99

To convert the recurring decimal 1.272727... into a fraction, let x = 1.272727... (Equation 1). Since two digits are repeating, multiply both sides by 10^2 = 100. We get 100x = 127.272727... (Equation 2). Subtracting Equation 1 from Equation 2 gives: 100x - x = 127.272727... - 1.272727..., which simplifies to 99x = 126. Wait, let us re-evaluate carefully: 1.272727... = 1 + 0.272727... = 1 + 27/99. Simplifying 27/99 by dividing numerator and denominator by 9 gives 3/11. Thus, 1 + 3/11 = (11 + 3)/11 = 14/11? Let us check 127/99: 127/99 = 99/99 + 28/99, wait. Let x = 1.272727... then 100x = 127.2727... so 99x = 126. Thus x = 126/99 = 14/11. Wait, option (c) is 127/99, but let's check: 1.272727... as an improper fraction is 127 - 1 / 99 = 126 / 99 = 14 / 11, which matches option (b). Let us verify options: (c) 127/99 = 1.2828... Wait, let us check the question options properly. The options given are (a) 13/11, (b) 14/11, (c) 127/99, (d) 137/99. If x = 1.2727..., then 127/99 = 1.2828..., whereas 14/11 = 1.2727... Let us check option (b).