Final answer:
If the Earth's radius decreased with no change in mass, a person's weight would increase. The gravitational force is inversely proportional to the square of the radius, so a smaller radius means a stronger gravitational pull and higher weight.
Step-by-step explanation:
If the radius of the Earth decreased with no change in mass, your weight would increase. This is because weight is determined by the gravitational force exerted on an object. According to Newton's law of universal gravitation, the force of gravity is directly proportional to the mass of the two objects and inversely proportional to the square of the distance between their centers.
In the case of Earth, if the mass remains the same but the radius decreases, the distance between the center of the Earth and a person standing on the surface would be smaller, leading to an increase in the gravitational force and thus an increase in weight.
The question on how a person's weight would change if Earth had its present size but only one-third its present mass can be answered similarly. The gravitational force, and hence your weight, depends directly on the mass. If Earth's mass were one-third of its current mass, the gravitational force at the surface would be reduced correspondingly, causing a person to weigh one-third as much.