Easy Floors.

0-99 will give floor value 0 for both

101-199 will give floor value 1 for both

202-299 will give floor value 2 for both

....... & so on till 9999-9999 which gives floor value 99 for both

After this no number gives same floor value for both (x/101) & (x/100)

So number of values for x:

100+99+98+......1=5050

No need for complicated algebra or inequalities, just observe and the answer will come! The possible values of $x$ are as follows:

$0<x<100$ 100 possible values

$100<x<200$ 99 values

$201<x<300$ 98 values

$302<x<400$ 97 values

$403<x<500$ 96 values....

Clearly the possibilities are in arithmetic series and that's no surprise; the lower extreme of the inequalities above needs to be $1$ less than the multiples of $101$ .

Summing it up: $100+99+98+97+....+2+1=5050$