company Z

Let's make some notations:
a = full time employees of company Z
b =part time employees of company Z

m = full time employees of division X
n =part time employees of division X

p = full time employees of division Y
q =part time employees of division Y

I'd say start with a bit of back solving. You're supposed to prove or disprove that:

m/n > (m + p)/(n + q)

mn + mq > mn + np

mq > np ----- divide by n

(m/n)*q > p ------ divide by q

m/n > p/q.

This means that what you're ultimately up against is proving that the ratio for division X is greater than the ratio for division Y.

1. tells you that (m + p)/(n + q) > p/q

mq + pq > np + pq

mq > np ---- divide by n

(m/n)*q > p ----- divide by q

m/n > p/q - what you were supposed to prove. So 1 is sufficient.

2. More than ½ of full-time employees of company Z are employees of div X translates to m > p. In the same time, more than ½ of part-time employees of company Z are employees of div Y means that q > n. This will obviously mean that m/n > p/q (since on the one side you're dividing the greater of m and p by the smaller of n and q). Again, this is sufficient.



There is an alternative way of looking at this: the ratio for the entire company will be something of a weighted average of the two division ratios. This means that it's somewhere between the two. However, you don't know if it's:
ratio division X < ratio total < ratio division Y

or

ratio division Y < ratio total < ratio division X.

Once you've established the relative position of any two elements (i.e. finding out that ratio X > ratio Y), you can pretty much answer the question (in my example, it's obviously the second case).