How much food?

Here are my thoughts about how much food to bring in a backpacking trip. Of course, your mileage may (and will) vary.

Calories needed = Base_Cal + Distance_Cal + ElevationGain_Cal
Base_Cal = weight*10.872 (for women the factor is 9.785)
Distance_Cal = 100 cal/mi
ElevationGain_Cal = 0.2 cal/ft

A 150lb male hiking 10mi with 2500 elevation gain would need: 150*10.872 + 100*10 + 2500*0.2 = 3130 calories

A 150lb male hiking 20mi with 4000 elevation gain would need: 150*10.872 + 200*10 + 4000*0.2 = 4430 calories

Now, food provides calories. In general, food has fat, protein, carbs, and non-caloric elements (such as water, packaging, etc.). Although this is a simplification, we can assume the following calorie densities:

Fat: 9 cal/gr = 255 cal/oz
Protein: 4 cal/gr = 113 cal/oz
Carbs: 4 cal/gr = 113 cal/oz
Non-caloric: 0 cal/gr = 0 cal/oz

So, in a 10mi/2500ft hike, a 150lb male would need 12oz (0.75lb) of pure fat (yuck!) or 27oz (1.69lb) of pure carbs. Of course, on top of this you need to add the "non caloric" elements that add a surprising amoung of weight. For example:

1.0oz slice of wheat bread provides only 0.60oz of caloric elements, so the calorie density of wheat bread is only 70cal/oz.
A double-portion package of Lasagna Mountain House has a calorie density of 127 cal/oz (without packaging). With the full packaging it goes down to only 98 cal/oz!

Given this, we can assume an average of 100cal/oz for most backpacking food. [Note: it's not impossible to achieve calorie densities of 150cal/oz by being very careful with packaging and the choice of food, for example, a 12oz ziplock bag full of M&Ms has a calorie density of 175 cal/oz)]

Therefore, for the 10mi/2500ft hike, the 150lb male hiker will need to carry 2lb of food; for the 20mi/4000ft hike, the same person will need 2.75lb of food (or 1.30lb and 1.85lb respectively if we assume a density of 150 cal/oz).

Can I carry less in my pack? (and more in my waist?)

For some of us, losing some weight in a hike is not a big deal, so a slightly calorie deficient diet is OK.
For example, a deficit of 3500 calories (1.5-2.2 pounds of food) will mean a lost of only 1 pound of body weight. Being happy on the trail... that's a different story :-)