Which Fraction Is Greater?

[anemone]

Which of the fractions is greater?

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

or

\(\displaystyle B=\frac{\overbrace{222\cdots221}^{ \large{\text{2012 pieces}}}}{\underbrace{222\cdots223}_{ \large{\text{2012 pieces}}}}\)?

 

[M R]

[tex]A=1-\frac{1}{33...34/3}=1-\frac{1}{111..11 +\frac{1}{3}}[/tex]

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

B is greater.

 

[anemone]

Very beautiful and impressive, M R!