Write the rational number 5.31313131 ...  as the ratio of two integers

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Let us rewrtie the number 5.313131... as a series:

5.313131... = 5 + 0.31 + 0.0031 + 0.000031 + ....

 We notice that the terms:

 0.31 + 0.0031 + 0.000031 ... are terms of a geometric series where a1 = 0.13 and common difference r = 0.01.

Then , we know...

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Let us rewrtie the number 5.313131... as a series:

5.313131... = 5 + 0.31 + 0.0031 + 0.000031 + ....

 We notice that the terms:

 0.31 + 0.0031 + 0.000031 ... are terms of a geometric series where a1 = 0.13 and common difference r = 0.01.

Then , we know tha the sum of the G.S is given as follows:

S = a1/ (1- r)

Let us substitute:

    = 0.31/ ( 1- 0.01)

      = 0.31 / 0.99

       = 31/99


Then we conclude that we could rewrite the number as follows:

 5.313131... = 5 + 31/ 99

                   = ( 495 + 31)/ 99 = 526/ 99

==> 5. 313131... = 526/ 99

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