ODDLY:- I looked at this one as soon as I got on the site and
  started off by writing down all of the ones up to 199. 111, 113,
  115, 117.......197, 199. I got the answer to all of them which
  was 3875. I then changed the first digit to 3 to make it 311,
  313... 399. Then did the same for all the rest. My final answer
  when I had added it all up was 69,375. <br />
  Nicely done Adam from Poltair Community School and Sports
  College, Cornwall! <br />
  A big thank you too to Michael for your elegant method of finding
  the sum of all 3 digit numbers each of whose digits is odd:
  <br />
  I think the answer is 69,375. <br />
  1. I estimated the total c 50,000. <br />
  2. I wrote out all the 3-digit numbers 100-199 which had odd
  digits only, and observed a pattern: value of 100 occurred 25
  times. <br />
  3. Decided that if all those numbers were written out, the values
  of 100, 300, 500, 700, and 900 would each occur 25 times; the
  values of 10, 30, 50, 70, and 90 would do the same; and the
  values of 1, 3, 5, 7, and 9 similarly. <br />
  4. Therefore, the sum can be simplified to:
  25(100+300+500+700+900) + 25(10+30+50+70+90) + 25(1+3+5+7+9) =
  69,375 5. <br />
  This is still very large! <br />
  We can simplify it further to: 25(111+333+555+777+999) =
  69,375<br />