Group X has 5 men whose average age is 32 years and group Y has 5 men whose average age is 26 years. One man from X changes places with one man from Y, after which the two groups have exactly the same average age. By how many years is the man who left X older than the man who left Y?

X starts on $5\times 32=160$ years and Y on $5\times 26=130$ years. Call the man leaving X a and the man leaving Y $b$. After the swap X totals $160-a+b$ and Y totals $130-b+a$, and since both groups still hold 5 men, equal averages mean equal totals: $160-a+b=130-b+a$, so $2(a-b)=30$ and $a-b=15$. (Check with $a=40$ and $b=25$: both groups then total 145.) 6 is the gap between the two averages, which is only what one man must fix for five men each. 3 halves that gap. 30 is the gap between the two totals, one step short - the swap moves age in BOTH directions, so it closes twice its own size.