I don't have a convincing explaination. But since nobody has replied yet, I will give it a shot. Someone with a better insight, please provide a better explaination for sufficiency of condition i).
Let Xf = number of full time employees in division X
Let Xp = number of part time employees in division X
Let Yf = number of full time employees in division Y
Let Yp = number of part time employees in division Y
Let Zf = number of full time employees in Company Z
Let Zp = number of part time employees in Company Z
Therefore, Xf+Yf=Zf and Zp+Yp=Zp
Consider ii) since that's easy to evaluate first.
ii) More than half the full-time employees of Z are in Division X and more than half the part-time employees of Z are in Division Y
Lets say 60% of full-time employees are in division X and 60% of part-time employees are in division Y.
Therefore, Xf = 0.6Zf => Yf = 0.4Zf
and Yp= 0.6Zp => Xp = 0.4Zp
Therefore the ratio of full-time to part-time employees for division X compared to company Z is
Xf/Xp = 0.6Zf/0.4Zp = 1.5(Zf/Zp).
Thus ii) is sufficient condition.
At this point answer is narrowed down to B or D.
Now consider i)
i) The ratio of the number of full-time to part-time employees is less for division Y than that of company Z
Yf/Yp < Zf/Zp,
Yf/Yp < (Xf+Yf)/(Xp+Yp)
For the above to be true the ratio Xf/Xp must be greater than Zf/Zp
Consider a few examples involving both proper and improper fractions,
Yf/Yp=1/2 & Zf/Zp=2/3 => Xf/Xp=1/1
Yf/Yp=3/2 & Zf/Zp=2/1 => Xf/Xp=3/1
Thus i) is sufficient condition. Hence answer is D.
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