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Write the repeating decimal as a fraction.\newline.61616161.61616161

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Full solution

Q. Write the repeating decimal as a fraction.\newline.61616161.61616161
  1. Denote decimal as xx: Let's denote the repeating decimal 0.61616161...0.61616161... as xx.
    x=0.61616161...x = 0.61616161...
    To isolate the repeating part, we multiply xx by 100100, since the repeating part is two digits long.
    100x=61.61616161...100x = 61.61616161...
    Now, we subtract the original xx from 100x100x to get rid of the repeating decimal part.
    100x−x=61.61616161...−0.61616161...100x - x = 61.61616161... - 0.61616161...
    0.61616161...0.61616161...00
    Now, we divide both sides by 0.61616161...0.61616161...11 to solve for xx.
    0.61616161...0.61616161...33
  2. Multiply by 100100: We check if the fraction can be simplified.\newlineBoth the numerator and the denominator have a common factor of 11, and there are no other common factors since 6161 is a prime number.\newlineTherefore, the fraction is already in its simplest form.

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