Your intuition is right. Think of an average as a "tug of war" between two groups: if the groups were of the same size and strength, the average would be straight down the middle. If X is a greater group than Y, it "tugs" the average closer to X than to Y.
Not only is a weighted average always closer to the larger group, but the group's relative sizes can even be inferred from "how close" the average is to the larger group. The secret is that the ratio the distance is divided is equal to the ratio of the group's size, and vice versa.
For instance, the groups in your question have a distance of 3 years between them (between 26-29). The average is 27, which divides the distance of 3 at a ratio of 1:2 (one year from X=26, 2 years from Y=29).
If the distance division ratio is 1:2, then the group's sizes will also be 1:2 - just take care to reverse the ratio, as X is actually larger than y, so the ratio is 2:1.

Likewise, if the question had given you the group's size/weight ratio and asked for the average, you could find it using the same principle:
X average of 26
Y average of 29
X has twice as many members.
What is the average age of group Z?
Tell yourself the principle: if the X:Y ratio is 2:1, then the average is closer to x (the greatest group). How close? for a 2:1 ratio, the distance between 26 and 29 (a distance of 3) should also divide at 2:1, so the average is 27: it is 1 away from x, and 2 away from Y.

Great, thanks!