Does a headwind affect the climb gradient? When studying for my German PPL exam, I came across this question:



For those that do not speak German I'll try to translate the question, but bear with me as it is kinda picky what I'm trying to get across here:


  During flight on a straight track with constant speed
  
  
  headwind will increase the gradient of climb  
  headwind will decrease the gradient of climb  
  headwind will increase the distance needed to descent 50m  
  headwind does not affect the gradient of climb at all
  


As you can see answer 4 is supposed to be the correct one. I disagree, so I started searching for a reason, yet I couldn't find anything useful. Let me explain my train of thoughts:

I didn't really know the "gradient of climb", I always used the two terms "angle of climb" and "rate of climb". Thus I'm trying to figure out which one the gradient of climb is. Common sense makes me think it is the same as the angle of climb (gradient does just sound like an angle). This opinion is supported by SIDs, which have a PDG (Procedure Design Gradient) that is given in percent, just like the gradient of climb.

Assuming the gradient of climb is indeed the same as the angle of climb, the only way for the "correct" answer to be confirmed is by taking the distance traveled during the climb relative to the air, not to the ground (angle of climb is defined as the height gained divided by the horizontal distance traveled in a certain amount of time). This would mean that it is used with IAS, rather than GS, which in my opinion absolutely makes no sense ("Hey look, that mountain is 10NM away, we are climbing with 600 ft/min. and going 100 KIAS and need 3500ft more to go over the top, we're all good!" - Well, add a tailwind of 25kts, the pilot would still think he's good to go but it'd result in CFIT...).

The question I have is: How is answer 4 the correct one?
The gradient of climb is the ratio of the increase of altitude to horizontal distance through the air, not over the ground. 
The definition used by the UK CAA in CAP 698 is:


  Climb Gradient  
  
  The ratio, in the same units of measurement, expressed
  as a percentage, as obtained from the formula:
  - $$\text{Gradient} = \frac{\text{Change in Height}}{\text{Horizontal Distance}} \times 100 \% $$


If you have a headwind or a tailwind it makes no difference to your climb gradient because your airplane is moving relative to the air mass. 

Climb gradient is not the same as rate of climb, although they are related. Rate of climb is altitude over a period of time, gradient is climb over distance travelled. There is another question which covers the difference in more detail.