Case 1. 

Let's count all the numbers in that range whose hundreds digits are 8.

These are the 100 numbers from 800 -- 899 inclusive.

Case 2. 

Let's count all the numbers in that range whose tens digits are 8.

There are 10 from 680-689 inclusive
There are 10 from 780-789 inclusive
There are 10 from 880-889 inclusive

That's 30 numbers with 8's for their tens digit.
 
Case 3.

Let's count all the numbers in that range whose ones digits are 8.

These are 608,618,628,...,898

If we take the 2-digit numbers from 60 -- 89, inclusive, and we
annex an 8 on the right end of them, that will be all the numbers
with an 8 for the units digit.

Since there are 30 2-digit numbers from 60 -- 89, inclusive that we
can put 8's on the end, that's 30 with an 8 for its units digit.

So that's 100+30+30 = 160 digits of 8 among the integers from 600 to 900.

[Notice that the reason there are 30 in cases 1 and 2 is because the 
numbers in case 3 are the ones in case 2 with the 2nd and 3rd digits 
switched.  So we could have done just one of them, and used that number 
for the other case.] 

Edwin