change the recurring decimal 0.51 into a fraction

f = 0.5151515151.....

100f = 51.5151515151.....

 

Subtract the first from the second

99f = 51

 

Divide both sides by 99

f = 51/99

 

So 0.5151515151..... = 51/99

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Check:

$${\frac{{\mathtt{51}}}{{\mathtt{99}}}} = {\frac{{\mathtt{17}}}{{\mathtt{33}}}} = {\mathtt{0.515\: \!151\: \!515\: \!151\: \!515\: \!2}}$$

(The final 2 appears only because of rounding to a finite number of places)

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