Easy way:

Choose the hundreds digit 9 ways, 1,2,3,4,5,6,7,8,9
Choose the tens digit 10 ways, 0,1,2,3,4,5,6,7,8,9
Choose the units digit 1 way, 5.

Answer 9*10*1 = 90 ways

OR:

An integer with a 5 in the units place is just a 2-digit number with
a 5 placed on the end.

So first we list all the 2-digit numbers:

10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 
70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 
80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 
90, 91, 92, 93, 94, 95, 96, 97, 98, 99,

There are obviously 90 of them.

Now put a 5 on the end of each of them:

105, 115, 125, 135, 145, 155, 165, 175, 185, 195, 
205, 215, 225, 235, 245, 255, 265, 275, 285, 295, 
305, 315, 325, 335, 345, 355, 365, 375, 385, 395, 
405, 415, 425, 435, 445, 455, 465, 475, 485, 495, 
505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 
605, 615, 625, 635, 645, 655, 665, 675, 685, 695, 
705, 715, 725, 735, 745, 755, 765, 775, 785, 795, 
805, 815, 825, 835, 845, 855, 865, 875, 885, 895, 
905, 915, 925, 935, 945, 955, 965, 975, 985, 995,

There are 90 of them too..

Explanation without writing them all out:

How many 1-digit numbers are there?

These 9: 1,2,3,4,5,6,7,8,9

How many 2-digit numbers?

There are 99 integers between 1 and 99 inclusive.
From this we subtract the 9 1 digit numbers.

So there are 99-9 or 90 2 digit numbers.

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Put a 5 on the end of each of the 90 2-digit numbers.

Answer: 90

Edwin