To simplify the expression, let's first calculate the sum of the numbers from 1 to 101:

1 + 2 + 3 + ... + 99 + 100 + 101 = (101/2)(1 + 101) = 5151

Now, let's calculate the sum of the numbers from 1 to 100:

1 + 2 + 3 + ... + 98 + 99 + 100 = (100/2)(1 + 100) = 5050

Therefore, the expression becomes:

5151/101 - 5050/100 + 99 - 98 + ... + 3 - 2 + 1

Simplifying further:

51 - 50 + 99 - 98 + ... + 3 - 2 + 1

Now, let's group the terms:

(51 + 99 + ... + 3 + 1) - (50 + 98 + ... + 2)

The terms in each group form an arithmetic sequence with a common difference of 48 and a first term of 51 in the first group and 50 in the second group. We can use the formula for the sum of an arithmetic sequence to calculate each group's sum:

Sum of the first group = (n/2)(first term + last term) = (26/2)(51 + 1) = 676

Sum of the second group = (n/2)(first term + last term) = (25/2)(50 + 2) = 637.5

Therefore, the expression simplifies to:

676 - 637.5 = 38.5