Since we "color" with "crayons," we are looking for an answer choice that includes a tool (e.g., crayon) and how it is used (e.g., coloring). The only answer choice that fits this criteria is "knife is to cutting" because we use knives to cut up food!

A “wrench” is a tool used by a “mechanic,” so to solve this analogy, you need to determine which of these answer choices is a tool often used by a farmer. The correct answer is a “plow”; a “farmer” uses a “plow” to break up soil and allow seeds to be planted. It might be reasonable to assume that a farmer uses a “hammer” as well, but it is not anywhere near as closely associated with the profession of farming as a “plow” is. Other professions, like carpenters, might use hammers, but only farmers are associated with plows. To provide further help, another name for the profession of farming is “agriculture.” 

Just as a pilot flies a plane, a captain steers a ship, so "captain" is the correct answer.

Since firefighters often use hoses while doing their job, we are looking for an answer in which the first word is an item that the occupation in the second word uses while working. The best answer choice is "easel is to artist," because artists often paint or sketch their creations on easels.

The analogy in question relates a profession to a tool associated with that profession. "Lawyer is to ball," "soldier is to comb," "dancer is to phone," and "chef is to drum" do not represent the given relationship. Artists often uses paintbrushes just as surgeons use scalpels, so "surgeon is to scalpel" is the correct answer.

A "carpenter" works with "wood," by definition, so the answer be a tool followed by its user. A mason similarly works with stone.

The key here is determining how the kind of person given relates to the given object. An "electrician" works with "light bulbs," or user to tool, so the answer must be a user and his tool. A "plumber" works with "toilets," so that must be the correct answer. None of the other answers describe a user-tool relationship. A "book" is a product made by an "author." A "cup" is a vessel for "water." "Nurses" are people, and thus cannot be a part of a user tool relationship.

Doctors work with patients just as teachers work with students, so "patients" is the correct answer.

A fan moves air by blowing it around, so you need to pick out an answer choice that describes what a hose moves. A hose moves water, so "water" is the correct answer. None of the other answer choices are things that are moved by hose, so none of them can be the correct answer.

Since veterinarians help animals when they are sick or hurt, we need to pick out a word that represents an occupation whose practitioners help out people when they are sick or hurt. While both "doctor" and "hospital" may look like correct answers, "hospital" can't be the correct answer because it is the name of the place where people go when they are sick or hurt, and not the job title of the people who help them. "Doctor" is the correct answer because just like veterinarians help animals when they are injured, doctors help people when they are injured.