Final answer:
The fat parachutist should deploy their backup parachute immediately to save the thin parachutist, as trying to fix the issue in mid-air or ignoring the signal could be fatal due to high terminal velocities.
Step-by-step explanation:
The best possible solution to save the life of the thin parachutist is for the fat parachutist to use a backup parachute immediately. When a main parachute fails, skydivers are trained to deploy their reserve parachute. This is essential to reducing terminal velocity and providing a safe landing speed. Waiting or trying to fix the issue in mid-air could be fatal due to the high speeds involved, which reach up to about 200 km/h in a spread-eagle position and 350 km/h in a headfirst position. Using a backup parachute will rapidly decelerate the parachutist, creating drag that counters the gravitational force, thereby significantly increasing the odds of survival.