jmc

Consider the numbers $1,2,3,...,10$. Jo would add these integers up as is, while Kate would round the first four down to 0, decreasing her sum by $1+2+3+4=10$, and would round the last six up to 10, increasing her sum by $5+4+3+2+1+0=15$. Thus, her sum is $-10+15=5$ more than Jo's sum for the numbers $1,2,3,...,10$. This same logic applies to the numbers $11,12,13,...,20$ also, and in general it applies to every ten numbers greater than 20. Since there are ten sets of ten numbers from 1 to 100, Kate's sum is $10\cdot 5=\boxed{50}$ more than Jo's sum.