# Analytical Reasoning

Analytical reasoning, also known as logical reasoning, is a problem-solving method that focuses on identifying patterns and using logic to fill in missing pieces. This form of reasoning is slightly more detached from inferences and opinions, and places great emphasis on factual evidence.

Deductive Reasoning

Deductive reasoning, sometimes called top-down logic, is the formation of a specific conclusion from a general premise or, in some cases, multiple hypotheses. Deductive reasoning is best for situations with multiple variables that must be accounted for and handled.

Inductive Reasoning

Inductive reasoning, also known as bottom-up logic, is the direct opposite of deductive reasoning as it forms plausible conclusions from a specific observation. Inductive reasoning works best when using context and practical intelligence to resolve an issue.

Numerical Reasoning

Numerical reasoning is the ability to apply logic to problems that include data, charts and graphs. Someone who has strong numerical reasoning skills is proficient in basic mathematical functions, statistical interpretation, and algorithms to name a few.

Abstract Reasoning

Abstract reasoning is the ability to identify patterns, extract main ideas, and analyze information. Abstract reasoning is an important factor of problem solving and decision making. This form of reasoning is complex and includes various aspects.

Analytical reasoning test can be difficult to study for because it examines your raw critical thinking skills. If you are looking to optimize and improve your score, the best way to prepare is running through sample questions or completing practice tests. By doing this, you can become familiar with the idea of the content and the learn how to pace yourself under the time constraint.

## Directions:

Each set of questions in this section is based on a scenario with a set of conditions. The questions are to be answered on the basis of what can be logically inferred from the scenario and conditions. For each question, choose the response that most accurately and completely answers the question.

### Passage for Question 1

A university library budget committee must reduce exactly five of eight areas of expenditure—G, L, M, N, P, R, S, and W—in accordance with the following conditions:

1. If both G and S are reduced, W is also reduced.
2. If N is reduced, neither R nor S is reduced.
3. If P is reduced, L is not reduced.
4. Of the three areas L, M, and R, exactly two are reduced.

#### Question 1

If both M and R are reduced, which one of the following is a pair of areas neither of which could be reduced?

1. G, L
2. G, N
3. L, N
4. L, P
5. P, S

Solution : L, N

This question concerns a committee’s decision about which five of eight areas of expenditure to reduce. The question requires you to suppose that M and R are among the areas that are to be reduced, and then to determine which pair of areas could not also be among the five areas that are reduced.

The fourth condition given in the passage on which this question is based requires that exactly two of M, R, and L are reduced. Since the question asks us to suppose that both M and R are reduced, we know that L must not be reduced:

• Reduced: M, R
• Not reduced: L

The second condition requires that if N is reduced, neither R nor S is reduced. So N and R cannot both be reduced. Here, since R is reduced, we know that N cannot be. Thus, adding this to what we’ve determined so far, we know that L and N are a pair of areas that cannot both be reduced if both M and R are reduced:

• Reduced: M, R
• Not reduced: L, N

Therefore the correct answer is L,N.

### Passage for Questions 2 and 3

Seven piano students—T, U, V, W, X, Y, and Z—are to give a recital, and their instructor is deciding the order in which they will perform. Each student will play exactly one piece, a piano solo. In deciding the order of performance, the instructor must observe the following restrictions:

1. X cannot play first or second.
2. W cannot play until X has played.
3. Neither T nor Y can play seventh.
4. Either Y or Z must play immediately after W plays.
5. V must play either immediately after or immediately before U plays.

#### Question 2

If V plays first, which one of the following must be true?

1. T plays sixth.
2. X plays third.
3. Z plays seventh.
4. T plays immediately after Y.
5. W plays immediately after X.

Solution :

This question deals with an ordering relationship defined by a set of conditions concerning when seven piano students will perform. One way to approach this problem is to write down the seven recital slots in order from left to right, as illustrated below. Student V is shown in the first slot, as specified by the supposition that “V plays first”:

 1 2 3 4 5 6 7 V

We can immediately fill in one of the empty slots in the recital schedule. The condition that “V must play either immediately after or immediately before U plays” tells us that U must occupy the second slot in the schedule. This is shown below:

 1 2 3 4 5 6 7 V U

Since the question asks us what must be true, we can eliminate incorrect responses by showing that they could be false. Response (A) is incorrect because the statement that “T plays sixth” is not necessarily true—we can place T in one of the slots other than sixth and still meet all the conditions of the problem. One such recital schedule, with T playing third, is shown below:

 1 2 3 4 5 6 7 V U T X W Y Z

This schedule can be derived as follows:

1. With V, U, and T in the first three positions, there are four positions left for W, X, Y, and Z.
2. W must come after X—because of the condition that “W cannot play until X has played”—so if X is fourth and W is fifth, this condition will be met.
3. This leaves two possible slots for Y and Z. Y cannot play seventh because of the condition that “Neither T nor Y can play seventh.” Suppose, then, that Y is sixth and Z is seventh.

A check will verify that this schedule meets the conditions of the problem, including the one that “Either Y or Z must play immediately after W plays.”

The schedule shown above also demonstrates that response (B) is incorrect. In it, X plays fourth, so it is not correct that the statement, “X plays third,” must be true.

Response (C), “Z plays seventh,” is the credited response. We can show Z must be seventh by demonstrating that:

• all the conditions can be met with Z in the seventh slot, and
• some of the conditions would be violated with Z in any slot other than seventh.

To demonstrate that Z can play seventh, you can refer to the schedule that was developed for the discussion of response (A), above. In it, Z plays seventh, and the supposition given in the question and all the conditions in the passage are met.

To demonstrate that Z cannot play in a slot other than seventh, we can attempt to find another student to play seventh. We already know that neither U nor V can play seventh. Hence, there are four remaining players: T, W, X, and Y. However, a review of the conditions shows that none of those players can play seventh:

• The third condition states that “Neither T nor Y can play seventh.”
• W can’t play seventh, because there must be a slot following W’s in order to meet the condition, “Either Y or Z must play immediately after W plays.” If W plays seventh, then there is no such slot left for Y or Z.
• For a similar reason X can’t play seventh, because there must be a slot following X’s in order to meet the condition, “W cannot play until X has played.”

Since Z can play seventh and no other player can, then the statement that Z must play seventh is correct and (C) is the credited response.

Response (D) is incorrect because it is not necessarily true that “T plays immediately after Y.” In our discussion of response (A), we developed a schedule in which T plays third and Y plays sixth, yet all conditions are satisfied.

Response (E) is incorrect because, as shown in the schedule below, it is not necessarily true that “W plays immediately after X.” This schedule is obtained by simply reversing the order of players W and Y in the schedule we developed in the analysis of response (A).

A review will show that all of the suppositions given in the question and all the conditions in the passage are met by this schedule:

 1 2 3 4 5 6 7 V U T X W Y Z

This was a difficult question, based on the number of test takers who answered it correctly when it appeared on the LSAT. The most commonly selected incorrect answer choices were (B) and (E). In answering this question, it is important to derive information not explicitly mentioned in the passage, such as that W cannot perform seventh.

#### Question 3

If U plays third, what is the latest position in which Y can play?

1. first
2. second
3. fifth
4. sixth
5. seventh

Solution :

This question involves the same original conditions as the previous problem, but it begins with an additional supposition: “U plays third.” You must determine what effect this supposition would have on the possible positions in which Y can appear in the recital schedule.

The correct response is (D): Y can play as late as sixth. The schedule below shows a recital order that meets all the conditions and has Y performing in the sixth position:

 1 2 3 4 5 6 7 T V U X W Y Z

One strategy for arriving at this solution is to work backward to see which position is the latest in which we can place Y and at the same time produce a recital schedule that meets all the conditions.

Using that approach, we immediately see that Y cannot play as late as seventh, because of the condition that “Neither T nor Y can play seventh.” Backing up and placing Y sixth, we can begin to fill in the schedule, as follows:

 1 2 3 4 5 6 7 U Y

This schedule has five empty slots, into which we must fit players T, V, W, X, and Z. The following is a series of reasoning steps that can be used:

1. From our analysis of the previous question, we know that players T, W, and X cannot play seventh, but that Z can, so we can tentatively place Z in the seventh slot.
2. We also know that “Either Y or Z must play immediately after W plays.” If we place W in the fifth slot, this condition will be met.
3. By placing V in the second slot, we can meet the condition that “V must play either immediately after or immediately before U plays.”
4. We must place the remaining two players, T and X, in the two remaining slots, the first and the fourth. Because the first condition states that “X cannot play first ...,” we will place X in the fourth slot and T in the first. These positions will meet the conditions that apply to T and X: T will avoid playing seventh and X will play before W.
5. Since Y can play as late as sixth, response (D) is the correct solution.

### Passage for Question 4

A charitable foundation awards grants in exactly four areas—medical services, theater arts, wildlife preservation, and youth services—each grant being in one of these areas. One or more grants are awarded in each of the four quarters of a calendar year. Additionally, over the course of a calendar year, the following must obtain:

1. Grants are awarded in all four areas.
2. No more than six grants are awarded.
3. No grants in the same area are awarded in the same quarter or in consecutive quarters.
4. Exactly two medical services grants are awarded.
5. A wildlife preservation grant is awarded in the second quarter.

#### Question 4

If a wildlife preservation grant and a youth services grant are awarded in the same quarter of a particular calendar year, then any of the following could be true that year EXCEPT:

1. A medical services grant is awarded in the second quarter.
2. A theater arts grant is awarded in the first quarter.
3. A theater arts grant is awarded in the second quarter.
4. A wildlife preservation grant is awarded in the fourth quarter.
5. A youth services grant is awarded in the third quarter.

Solution :

This question deals with the awarding of grants during the quarters of a calendar year. One way to approach this problem is to set up a simple table with columns representing the four quarters. Since the fifth condition in the passage states that “[a] wildlife preservation grant is awarded in the second quarter,” we know that all possible solutions for any question based on the passage must include a wildlife preservation grant awarded in the second quarter, which we can represent like this:

 1 2 3 4 W

The particular question here begins with the added supposition that “a wildlife preservation grant and a youth services grant are awarded in the same quarter of a particular calendar year.” One possible way this could be satisfied is to have a youth services grant awarded in the second quarter in addition to the wildlife grant awarded in that quarter:

 1 2 3 4 W Y

Another possibility would be to have a wildlife preservation grant and a youth services grant both being awarded in some quarter other than the second quarter. Given the condition that “[n]o grants in the same area are awarded in the same quarter or in consecutive quarters,” the only quarter in which a wildlife preservation grant could be awarded in addition to the second quarter is the fourth quarter. So the only alternative way to satisfy the added supposition is if both a wildlife preservation grant and a youth services grant are awarded in the fourth quarter:

 1 2 3 4 W W Y

So far, then, we’ve determined that for this question there must be a youth services grant awarded in the second quarter or the fourth quarter.

Each of the incorrect answer choices for this question is a statement that could be true. The question asks you to identify the exception; that is, you need to find the statement that cannot be true. The correct answer choice is (E), which states: “A youth services grant is awarded in the third quarter.” This could not be true without violating the third condition, which specifies that “[n]o grants in the same area are awarded in the same quarter or in consecutive quarters.” We saw above that a youth services grant must either be awarded in the second quarter or the fourth quarter. On either possibility, awarding a youth services grant in the third quarter would result in two consecutive quarters where the youth services grant is awarded:

 1 2 3 4 W Y Y

or:

 1 2 3 4 W Y W Y

In both cases, two youth services grants would be awarded in consecutive quarters, in violation of the third condition.

To see that each of the other answer choices could be true, it will suffice to construct a possible outcome for each one that is consistent with the supposition given in the question and the conditions in the passage. Consider the following possible outcome:

 1 2 3 4 T M W Y T M

A quick check of the conditions shows that this satisfies all of the conditions for the problem:

• A wildlife preservation grant and a youth services grant are awarded in the same quarter of a particular calendar year.
• Grants are awarded in all four areas. (The table includes at least one of each of the four letters—M, T, W, and Y.)
• No more than six grants are awarded. (The table contains exactly six entries.)
• No grants in the same area are awarded in the same quarter or in consecutive quarters. (In the table above, only T and M are repeated, and neither repetition appears in the same or consecutive columns.)
• Exactly two medical services grants are awarded. (The table contains exactly two M’s, in columns 2 and 4.)
• A wildlife preservation grant is awarded in the second quarter.

Notice that in this possible outcome, a medical services grant is awarded in the second quarter (answer choice (A)) and a theater arts grant is awarded in the first quarter (answer choice (B)). So answer choices (A) and (B) are both incorrect.

Now consider the following possible outcome:

 1 2 3 4 M T W M W Y

A check of the conditions shows that this satisfies the supposition and all of the conditions. In this outcome, a theater arts grant is awarded in the second quarter (answer choice (C)) and a wildlife preservation grant is awarded in the fourth quarter (answer choice (D)). So answer choices (C) and (D) are also incorrect.