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Convert 8.690909 . . . into a single fraction using infinite geometric series.

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8.690909....
=8.6 + .09 + .0009 + .000009+...
= 86/10 + .09/(1-.01)
= 86/10 + .09/.99
= 86/10 + 9/99
= 86/10 + 1/11
= 478/55

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