A grotto is a small cave, often an artificial one and a cavern is a large cave or part of a cave, so the comparison between the two terms is one of size. We thus need to pick out an answer choice that represents something that could be thought of as a very large ditch. Since a gorge is a narrow valley with a stream at the bottom, "canyon," which means a deep gorge, is the best answer, because just like a cavern can be thought of as a very large grotto, a canyon can be thought of as a large gorge.

A “metropolis” is defined as a large city, while a “village” is a small settlement in the countryside. A metropolis is a large-scale version of a village, and we need to pick out an answer choice that describes something of which the larger-scale version would be an ocean. While “puddle” and “storm” might each look like a potentially correct answer because like an ocean, they both involve water, a larger-scale storm would not be an ocean, so “storm” cannot be the correct answer. One could think of a large-scale puddle as an ocean however, as both are bodies of water, so “puddle” is the correct answer.

A semicolon is "half" of a colon, and a hemisphere is half of a sphere. 

A rivulet is a small river, so to solve this analogy you have to determine which of the answer choices is a larger version of a hamlet. A hamlet is a small town so the correct answer is “town.”

A pittance is a very small amount of money, so to solve this analogy you have to determine what a tidbit is a small amount of. A tidbit is used to describe a small piece of gossip or information.

The diameter of a circle is twice the size of the radius. A quadrant refers to a one-fourth section of something, so twice one-fourth is a half. 

A "stopwatch" is the device that is used to measure the time that is taken for a given event. For example, a coach or physical education instructor will use a stopwatch to clock the time taken to run a race. Thus, the analogy could be represented as, “Just as a stopwatch measures time, so a yardstick measures X.” Note that the options “yards” and “inches” are not as appropriate as “distance.” If the first half of the analogy had used (e.g.) “seconds” instead of “time,” those other options would be more acceptable. However, as things stand, the analogy indicates the general type of measure done by the device (time and distance respectively).

"Luminosity" is how bright something is, a measure of "brightness." So, to solve this analogy you need to determine what it is that "viscosity" measures. Well, "viscosity" is a measure of how thick something is, a measure of "thickness."

A "liter" is a measurement of liquid "capacity," a measurement of volume. So, to solve this analogy you need to determine which of these is measured by a "joule." Well, the correct answer is that a "joule" is used to measure "energy." Additionally, "luminosity" refers to brightness; "viscosity" refers to thickness. 

A "pedometer" is an object that measures how many steps you have "walked." So, to solve this analogy you need to determine what it is that an "odometer" measures. The correct answer is that an "odometer" measures how many miles you have "driven."