MHB Comparing Fractions with Large Numerators and Denominators

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The discussion centers on comparing two fractions, A and B, both with large numerators and denominators. Fraction A simplifies to 1 minus a small value related to the denominator, while fraction B simplifies similarly but with a different small value. The calculations reveal that fraction B is greater than fraction A. Participants express admiration for the mathematical elegance of the solution. The conclusion is that B is the larger fraction.
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Which of the fractions is greater?

$$A=\frac{\overbrace{333\cdots331}^{ \large{\text{2012 pieces}}}}{\underbrace{333\cdots334}_{ \large{\text{2012 pieces}}}}$$

or

$$B=\frac{\overbrace{222\cdots221}^{ \large{\text{2012 pieces}}}}{\underbrace{222\cdots223}_{ \large{\text{2012 pieces}}}}$$?
 
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A=1-\frac{1}{33...34/3}=1-\frac{1}{111..11 +\frac{1}{3}}

B=1-\frac{1}{22...23/2}=1-\frac{1}{111..11+\frac{1}{2}}

B is greater.
 
Very beautiful and impressive, M R!(Clapping)
 
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