Analytical Reasoning

Structured reasoning with conditions and constraints

69 practice questionsPage updated 25 Sept 2026Reviewed by AISEA Editorial DeskPractice questions

Analytical reasoning solves a problem by translating its conditions into a structured model and finding the possibilities that satisfy all conditions together. In aptitude exams, the model may be an order, family tree, floor plan, color assignment, coordinate diagram, equation system or data-sufficiency test. The best approach is to record every constraint, apply the strongest ones first, keep valid cases separate and verify the final answer against the original wording.

Key formulas and rules

7 rules

Constraint Satisfaction System: $\text{Solution Set} = \bigcap_{i=1}^m \{x : C_i(x) = \text{True}\}$ , satisfying all conditions simultaneously

Cartesian Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Linear Equation System: $A\mathbf{x} = \mathbf{b}$ has a unique solution $\iff \det(A) \neq 0$

Contrapositive Logical Equivalence: $(P \implies Q) \iff (\neg Q \implies \neg P)$ ; note that $(Q \implies P)$ and $(\neg P \implies \neg Q)$ are formal fallacies

Pigeonhole Principle: If $n$ items are put into $k$ containers, then at least one container must contain $\lceil n / k \rceil$ items

Data Sufficiency Determination: Set of conditions $S$ determines variable $v \iff |\operatorname{Domain}(v \mid S)| = 1$

Solve Applications with Systems of EquationsOpenStax · Checked 2 Sep 2026Supports: translating word conditions into variables and systems, solving them and checking results

Logical ArgumentsOpenStax · Checked 2 Sep 2026Supports: conditional arguments, valid conclusions and the contrapositive

Solving Problems to Satisfy Multiple Constraints SimultaneouslyOpenStax · Checked 2 Sep 2026Supports: translating multiple constraints, solving jointly and checking non-negative feasible solutions

Analytical Reasoning

Key formulas and rules

Key concepts

Translate words into a model

Separate hard constraints from consequences

Use tables for order and assignment problems

Apply the strongest constraint first

Use casework without losing cases

Represent family relations as a graph

Convert directions into coordinates

Turn averages into equations

Test data sufficiency for uniqueness

Read conclusions and conditionals literally

Verify the answer against every clue

Worked examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

Key formulas and rules

Key concepts

Translate words into a model

Separate hard constraints from consequences

Use tables for order and assignment problems

Apply the strongest constraint first

Use casework without losing cases

Represent family relations as a graph

Convert directions into coordinates

Turn averages into equations

Test data sufficiency for uniqueness

Read conclusions and conditionals literally

Verify the answer against every clue

Worked examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Sources and review notes

Ready to test your understanding?

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