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Problem 19

Let F = .48181... be an infinite repeating decimal with the digits 8 and 1 repeating. When F is written as a fraction in lowest terms, by how much does the denominator exceed the numerator?

Solution

F = .4 + .081 + .00081 + ......
= (4/10) + .081(1 + .01 + .0001 + ....)
= .4 + .081 x (1 / (1-.01))
= (4/10) + (81/100)(1/.99)
= 53 / 110

Denominator = 110
Numerator = 53
Denominator - Numerator = 57