This clause:

gives us both the units and the hundreds digits.  The smallest the hundreds
digit can be is 1 because a 3-digit number can't begin with 0.  And the
largest it can be in order to be able to add 8 to it and still get a
one-digit number is also 1.  So the hundreds digit can only be 1, and the
hundreds digit is 8 more than 1 or 9.  

Then since

the tens digit is half of 1+9, or 10, and half of 10 is 5, so the
number has hundreds digit 1, tens digit 5, and units digit 9. 

Answer = 159

Edwin