Given that x=111110/111111, y=222221/222223, z=333331/333334, which is the biggest and which is the smallest?

We know that if p> q, then:

p/(p+1) - q/(q+1)

= (p-q)/((p+1)(q+1)), and so:

p/(p+1)> q/(q+1), whenever both p and q are positive with p> q .....(*)

Now,

x= 111110/111111

y = 222221/222223 = (111110+ 1/2)/(111111+ 1/2)

z = 333331/333334 = (11110 + 1/3) /(111111+ 1/3)

Since 11110 < 11110 + 1/3< 11110 + 1/2, it now follows from (*) that:

x< z< y

Consequently, it follows that x is the smallest.

*Edited on ***July 26, 2007, 9:54 am**