Logical Reasoning

Logical Reasoning is a crucial section in the SBI PO Preliminary Examination, testing your ability to analyze information, identify patterns, and draw valid conclusions. It assesses your critical thinking skills and problem-solving aptitude. This section typically comprises various types of questions designed to evaluate your analytical and deductive reasoning capabilities.

Types of Logical Reasoning Questions

The questions in this section can be broadly categorized into the following types:

1. Puzzles

Puzzles are a common and often time-consuming part of Logical Reasoning. They present a set of conditions or clues, and you need to arrange or deduce the arrangement of people, objects, or situations based on these clues. Common puzzle types include:

• Seating Arrangement: Questions involving people sitting in a row, circle, or square, with clues about their positions relative to each other.
• Floor-Based Puzzles: People living on different floors of a building, with clues about their floor numbers and other details.
• Box-Based Puzzles: Items or people arranged in boxes, with clues about their order or contents.
• Blood Relations Puzzles: Family tree problems where you need to determine relationships between individuals based on given clues.
• Direction Sense Puzzles: Determining the final direction or distance of a person from their starting point based on a series of movements.
• Scheduling Puzzles: People or events scheduled on different days or times, with clues about their order.

2. Syllogisms

Syllogisms test your ability to understand the relationship between two or more statements (premises) and draw a logical conclusion. You need to determine if a given conclusion follows from the premises.

Key Concepts:

• Statements: The premises given in the question.
• Conclusions: The statements derived from the premises.
• Terms: Subject term, Predicate term, Middle term.
• Types of Statements: Universal Affirmative (All A are B), Universal Negative (No A are B), Particular Affirmative (Some A are B), Particular Negative (Some A are not B).

Venn Diagrams: A common method to solve syllogisms is by using Venn diagrams. You represent the statements graphically and then check if the conclusion holds true in all possible representations.

Example:

• Statements: All dogs are cats. All cats are black.
• Conclusion: All dogs are black.

In this case, the conclusion is valid. If all dogs are cats, and all cats are black, then logically, all dogs must be black.

3. Input-Output

These questions involve a given input (usually a sentence or a set of numbers) which is then transformed through a series of steps. You need to identify the logic behind each step and apply it to a new input to find the final output.

Steps to Solve:

1. Analyze the given input and its transformed output.
2. Identify the logic for each step. This might involve rearranging words, numbers based on their value, position, or other criteria.
3. Apply the identified logic to the new input, step-by-step.
4. Ensure consistency in applying the logic across all steps.

4. Coding-Decoding

In these questions, a certain code language is used, and you are given examples of words or sentences and their corresponding codes. You need to decipher the coding pattern and apply it to decode a new word or sentence.

Common Coding Patterns:

• Letter Shifting: Letters are shifted by a fixed number of positions in the alphabet (e.g., A becomes C, B becomes D).
• Letter Reversal: The order of letters in a word is reversed.
• Positional Coding: The position of letters or words in a sentence determines the code.
• Symbol/Number Coding: Letters or words are replaced by symbols or numbers based on a specific rule.

Example:

• If 'ROSE' is coded as 'SPTF', then 'TULIP' would be coded as 'UMJQP'. (Each letter is shifted one position forward in the alphabet).

5. Direction Sense

These questions test your understanding of directions (North, South, East, West) and distances. You are given a series of movements and need to determine the final position relative to the starting point, or the distance between two points.

Tips:

• Draw a diagram to visualize the movements.
• Keep track of the directions and distances.
• Remember the Pythagorean theorem if calculating diagonal distances: a² + b² = c².
• North is up, South is down, East is right, West is left.

6. Blood Relations

These problems involve understanding family relationships. You are given a set of statements about individuals and their relationships, and you need to deduce specific relationships or identify an individual based on the given information.

Tips:

• Draw a family tree to represent the relationships.
• Use symbols: Male (☐), Female (○).
• Relationships: Parent-child (vertical line), Spouse (horizontal line connecting two individuals).

7. Series Completion (Number, Alphabet, Alpha-numeric)

You are given a sequence of numbers, letters, or a combination, and you need to find the next term in the series based on the underlying pattern.

Common Patterns:

• Arithmetic progression (adding/subtracting a constant).
• Geometric progression (multiplying/dividing by a constant).
• Square/Cube numbers.
• Prime numbers.
• Alternating patterns.
• Difference between consecutive terms.

Inequalities

Inequalities, also known as 'Coded Inequalities' or 'Symbolic Inequalities', are a popular topic in the Reasoning Ability section. These questions test your ability to interpret coded symbols that represent standard mathematical inequality relations and deduce conclusions based on these relationships.

Understanding the Symbols

In these questions, standard symbols like '>', '<', '≥', '≤', and '=' are replaced by coded symbols. You are given the meaning of these coded symbols.

Commonly Used Coded Symbols and Their Meanings:

*Note: The symbols and their meanings can vary from question to question. Always refer to the given definitions.*

Types of Inequality Questions

There are two main types of inequality questions you'll encounter:

1. Direct Inequalities

In this type, the standard inequality symbols (>, <, ≥, ≤, =) are used directly. You are given a set of statements and need to determine which of the given conclusions logically follows from the statements.

Example:

• Statements: P > Q, Q ≤ R, R < S
• Conclusions:
• I. P > R
• II. Q < S

Analysis:

• To check P > R: We have P > Q and Q ≤ R. Since Q is less than or equal to R, we cannot definitively say P is greater than R. If Q=R, then P>Q=R. If QQ
• To check Q < S: We have Q ≤ R and R < S. Since Q is less than or equal to R, and R is less than S, it logically follows that Q must be less than S.

Answer: Only conclusion II follows.

2. Coded Inequalities

This is the more common type in competitive exams. Here, the symbols are coded, and you first need to decode them based on the given definitions.

Example:

• If 'P @ Q' means 'P > Q', 'P # Q' means 'P < Q', 'P $ Q' means 'P ≥ Q', 'P % Q' means 'P ≤ Q', and 'P & Q' means 'P = Q'.
• Statements: A $ B, B # C, C & D
• Conclusions:
• I. A @ C
• II. B # D

Decoding the Statements:

• A $ B means A ≥ B
• B # C means B < C
• C & D means C = D

So, the statements are: A ≥ B, B < C, C = D.

Analyzing the Conclusions:

• Conclusion I: A @ C means A > C. From A ≥ B and B < C, we cannot determine the relationship between A and C. It could be A > C, A < C, or A = C. Thus, A > C is not necessarily true.
• Conclusion II: B # D means B < D. We know B < C and C = D. Since C and D are equal, if B < C, then B must also be less than D.

Answer: Only conclusion II follows.

Key Principles for Solving Inequalities

To solve these problems effectively, remember the following rules regarding the flow of inequality:

• Consistent Direction: If all symbols between two variables point in the same direction (e.g., A > B > C), the conclusion follows the same direction (A > C).
• Mixed Directions: If symbols point in mixed directions (e.g., A > B < C), no definite conclusion can be drawn between the extreme variables (A and C).
• Equality: The equality symbol ('=') can be treated as either '>' or '<' when it is part of a '≥' or '≤' relation. For example, in A ≥ B, A can be greater than B or equal to B.
• The 'AND' Rule: For a conclusion to be valid, it must be true in ALL possible scenarios derived from the statements.
• The 'OR' Rule (for mutually exclusive conclusions): If two conclusions are mutually exclusive (e.g., A > B and A < B), and neither can be definitively proven, you check if one of them is possible. If one is possible and the other is not, the one that is possible is not necessarily true. However, if the relationship between the two variables is undecided (e.g., from A > B, B < C, we cannot determine A vs C), then 'A > C' or 'A < C' or 'A = C' are all possibilities. If the question asks if either 'A > C' or 'A < C' is true, and you know A cannot be equal to C, then one of them must be true.

Steps to Solve Coded Inequalities

1. Understand the Code: Carefully read and note down the meaning of each coded symbol provided in the question.
2. Decode the Statements: Rewrite the given statements using the standard inequality symbols.
3. Analyze the Relationships: For each conclusion, check the relationship between the variables involved by tracing the path through the statements.
4. Apply Logic: Use the rules of inequality (consistent direction, mixed directions, equality) to determine if the conclusion is definitely true, definitely false, or uncertain.
5. Check All Conclusions: Evaluate each conclusion independently. If the question asks for "which conclusion follows", select the one(s) that are necessarily true.

Data Sufficiency

Data Sufficiency (DS) questions are designed to test your ability to analyze the given information and determine if it is sufficient to answer a particular question, rather than finding the actual answer itself. You are given a question and two statements (Statement I and Statement II) containing some information. You need to decide which statement(s) provide enough information to arrive at a unique answer.

Understanding the Question Format

Each Data Sufficiency question is followed by five options:

• A) Statement I alone is sufficient, but Statement II alone is not sufficient.
• B) Statement II alone is sufficient, but Statement I alone is not sufficient.
• C) Either Statement I or Statement II is sufficient.
• D) Both Statement I and Statement II are sufficient, but neither statement alone is sufficient.
• E) Both Statement I and Statement II are not sufficient.

How to Approach Data Sufficiency Questions

The key is to determine sufficiency, not to find the answer. Follow these steps:

Step 1: Understand the Question

Read the main question carefully. What exactly are you being asked to find? Identify the variables and the relationship you need to establish.

Step 2: Analyze Statement I

Consider Statement I alone. Assume it is true. Can you answer the main question uniquely using only this information?

• If yes: The answer is likely A or C. Proceed to Step 3 to differentiate.
• If no: The answer is likely B, D, or E. Proceed to Step 3.

Step 3: Analyze Statement II

Consider Statement II alone. Assume it is true. Can you answer the main question uniquely using only this information?

• If yes: And Statement I was "no", the answer is B. If Statement I was also "yes", the answer is C.
• If no: And Statement I was "yes", the answer is A. If Statement I was also "no", proceed to Step 4.

Step 4: Analyze Both Statements Together

If neither Statement I nor Statement II alone is sufficient, consider both statements together. Assume both are true. Can you answer the main question uniquely now?

• If yes: And both statements alone were "no", the answer is D.
• If no: Even with both statements combined, you cannot answer the question. The answer is E.

Examples

Example 1: Arithmetic DS

Question: What is the value of x?

Statement I: x + y = 10

Statement II: 2x + 2y = 20

Analysis:

• Statement I alone: x + y = 10. We have one equation with two variables. We cannot find a unique value for x (x could be 5, 6, 4, etc.). Insufficient.
• Statement II alone: 2x + 2y = 20. If we divide by 2, we get x + y = 10. This is the same information as Statement I. Insufficient.
• Both statements together: Both statements provide the same equation (x + y = 10). Combining them doesn't give new information. We still cannot find a unique value for x. Insufficient.

Answer: E) Both Statement I and Statement II are not sufficient.

Example 2: Arithmetic DS

Question: What is the value of x?

Statement I: x + y = 10

Statement II: x - y = 2

Analysis:

• Statement I alone: x + y = 10. Cannot find a unique value for x. Insufficient.
• Statement II alone: x - y = 2. Cannot find a unique value for x. Insufficient.
• Both statements together: We have a system of two linear equations:
  x + y = 10
  x - y = 2
  Adding the two equations: (x + y) + (x - y) = 10 + 2 => 2x = 12 => x = 6.
  Since we can find a unique value for x (which is 6), the statements together are sufficient.

Answer: D) Both Statement I and Statement II are sufficient, but neither statement alone is sufficient.

Example 3: Geometry DS

Question: Is triangle ABC a right-angled triangle?

Statement I: The lengths of the sides of triangle ABC are 5, 12, and 13.

Statement II: The sum of the squares of two sides of triangle ABC is equal to the square of the third side.

Analysis:

• Statement I alone: Sides are 5, 12, 13. Check if the Pythagorean theorem holds: 5² + 12² = 25 + 144 = 169. And 13² = 169. Since 5² + 12² = 13², by the converse of the Pythagorean theorem, the triangle is right-angled. Sufficient.
• Statement II alone: Let the sides be a, b, c. The statement says a² + b² = c² (or some permutation). This is the definition/condition for a right-angled triangle based on the converse of the Pythagorean theorem. This statement *directly* tells us the triangle is right-angled. Sufficient.

Answer: C) Either Statement I or Statement II is sufficient.

Example 4: Word Problem DS

Question: What is the total number of students in Class X?

Statement I: The ratio of boys to girls in Class X is 3:2.

Statement II: The number of girls in Class X is 18.

Analysis:

• Statement I alone: Ratio of boys to girls is 3:2. This means for every 3 boys, there are 2 girls. Let boys = 3k and girls = 2k. We cannot find the total number of students (3k + 2k = 5k) without knowing the value of k. Insufficient.
• Statement II alone: Number of girls = 18. This only tells us the number of girls. We don't know the number of boys or the ratio. Insufficient.
• Both statements together: We know the ratio of boys to girls is 3:2, and the number of girls is 18. So, 2k = 18 => k = 9. The number of boys is 3k = 3 * 9 = 27. The total number of students is boys + girls = 27 + 18 = 45. We can find a unique total. Sufficient.

Answer: D) Both Statement I and Statement II are sufficient, but neither statement alone is sufficient.