Find the mixed number or rational fraction in lowest terms represented by the repeating decimal. 1.363636?

2 Answers

Oddman Profile
Oddman answered
Count the number of repeating digits. When they start at the decimal point, as these do, you can find the fraction by putting those digits over an equal number of 9s. There are two (2) repeating digits, so a suitable denominator is 99 (two 9s). Your number is
  1 36/99 = 1 4/11
When the repeating digits don't start at the decimal point, you can do something similar. Suppose the number were 1.8363636...
Set x = 1.8363636...
Now, compute (10^2)*x = 183.6363636...    (the exponent 2 is the number of repeating digits)
Take the difference
  (100x) - (x) = 183.6363636 - 1.8363636 = 183.6 - 1.8 = 181.8    (the repeating digits cancel in the subtraction)
Now, we have
  99x = 181.8
  x = 181.8/99 = 1818/990 = 101/55 = 1 46/55

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