It is a hypothetical clock. Remember a day still has same number of minutes as before (i.e. 24*60 = 1440); the clock still has to travel 720 minutes for one turn, but now divided in 10 blocks (hours). Hence, each block = 720/10 = 72 minutes (previously it was 60 minutes). Also, each block is 360 (full circle) / 10 (no of new hours) = 36 degrees in angular distance. You can imagine how this clock will look like in your mind now.

As I said above, no of minutes do not change, hence, 3:45 on the old clock means at the time of the day when 225 (3*60 + 45) minutes have elapsed. Now you can imagine how these number of minutes must look like on the new clock (10 blocks). Because each block is now of 72 minutes, 225 = (72*3 + 9) i.e. 3 hour blocks and 9 minute blocks on the new clock. Inter block angular distance = 36 degrees (see above), hence angular distance of third block to the vertical = 36*3 = 108 degrees. We need to add the angular distance covered by the hour block to travel 9 minutes (remember each hour block has 72 minutes and 36 degrees), hence this angular distance = 9/72*36 = 4.5 degrees. Add both to get (108 + 4.5) = 112.5 degrees.

Now imagine only the minute hand on this new clock. The minute hand of the old clock travels 60 minutes in one round turn of 360 degrees. On the new clock, the minute hand has to cover 72 minutes (as number of theoretical hours has reduced). So 360/72 = 5 degrees per minutes. That gives, 9 minutes = 9*5 = 45 degrees. This is the angular distance that minutes hand now will have the vertical.

Difference between new minute and hour hand = 112.5 -45 = 67.5 degrees.

Thanks. Interesting. I read it as either 1) do the maths or 2) the angle of the hands doesn't change but the clock face does (to 20 hours) - as in "think outside the question".