A proper fraction is smaller than one, so we have to consider: 1/100, 2/100, 3/100, 4/100, ..., 99/100.
To be proper, it must be in lowest term, so we can remove all of them that have even numerators, leaving these 50 fractions: 1/100, 3/100, 5/100, 7/100, ..., 99/100
Now, let us remove all those whose numerators are a multiple of 5: 5/100, 15/100, 25/100, 35/100, 45/100, 55/100, 65/100 75/100, 85/100, and 95/100.
After removing these 10 fractions, there will remain 40 fractions.