Find the fraction that creates a repeating decimal that repeats certain digitsIs there any way to find the fraction x / y that, when converted to a decimal, repeats a series of digits z? For example: x / y = z . z z z z z z z z . . . or with actual numbers, x / y = 234.234234234... (z is 234) If this is impossible, is there a way that does the same but the value to the left of the decimal is not z?

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Find the fraction that creates a repeating decimal that repeats certain digitsIs there any way to find the fraction   x      /    y that, when converted to a decimal, repeats a series of digits   z? For example:       x        /        y    =  z  .  z  z  z  z  z  z  z  z  .  .  . or with actual numbers,   x      /    y  =  234.234234234... (z is 234) If this is impossible, is there a way that does the same but the value to the left of the decimal is not   z?

Marlee Norman · Accepted Answer

There&#039;s a nice fact (derived from the expression demonstrated by TonyK): The repeating decimal   0.  z  z  z  … can be represented by      z                  10                  l          (          z          )                    −      1      where   l  (  z  ) here denotes the number of digits of   z.Now if we want instead   z  .  z  z  z  … all we have to do is multiply the above expression by        10          l      (      z      )       (or as Joffan points out - simply add z), getting                    10                  l          (          z          )                    z                      10                  l          (          z          )                    −      1

Ezekiel Yoder · Accepted Answer

Here is a way that does not involve yet another formula for students (or you) to memorize. (I am not against formulas; I just realize that many people are.)The idea is to multiply the number by a power of ten that gets the same repeating pattern, then subtract. For your example,                    u                            =        234.234        …                            1000        u                            =        234234.234        …                            999        u                            =        234.234        …        −        234234.234        …                                          =        234000                            u                            =                  234000          999                                                  =                  26000          111

Find the fraction that creates a repeating decimal that repeats certain digits Is there any way to

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