How much food for each person is need and fuel per various engine to go to Mars on the current 9-month trip there then 3 months on Mars and the 9 months back? Can time be saved on the trip to and from Mars if the food for the crew is replaced with fuel for the engine?
Fuel requirements will probably dominate food and other consumables by an order of magnitude or more, so you can't save mass by shortening the trip. The exact tradeoff depends on the assumptions you make.
I'm just going to consider the outbound leg of the flight. NASA's trajectory browser offers me two missions to Mars in the 2025-2035 timeframe:
- Optimized for ∆v (minimum fuel): 4250m/s from LEO to Mars orbit, 304 days flight time
- Optimized for duration: 6330m/s from LEO to Mars orbit, 128 days flight time
Assuming an open-loop environmental system, with no oxygen or water reclamation, we budget 9.15kg per crewperson per day for food, water, oxygen, hygienic wipes, clothing, etc.
For the 304-day flight and a 6-person crew, this comes to 16.7 tons. For the 128-day flight, 7 tons. So the short flight saves almost ten tons of supplies.
Assuming the spacecraft masses 100 tons, and uses a hydrogen-oxygen rocket at a specific impulse of ~450 (e.g. RL10) to do the trans-Mars injection maneuver, the long flight will require 161.9 tons of propellant; the short flight will require 319.5 tons. So the increased fuel cost is about 16 times the mass savings in consumables.
If you assume that water and oxygen are reclaimed, the mass requirement drops to about 3.45 kg/person-day, and the savings for the short flight are only 3.6 tons; the fuel cost is 40 times that.