Mathematical Reasoning & Aptitude

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Mathematical Reasoning & Aptitude – Detailed Notes

1. Number Series

Definition:
A number series is a sequence of numbers arranged according to a specific pattern or rule. The main task is to identify the pattern and predict the missing number(s).

Explanation:
Number series can follow arithmetic patterns (constant difference), geometric patterns (constant ratio), or more complex mixed patterns. Some series may involve squares, cubes, factorials, or alternating operations. By examining the differences or ratios between consecutive terms, we can usually detect the underlying logic. Practicing multiple types of series enhances speed and accuracy.

Example Explanation:
Series: 2, 6, 12, 20, ?

  • Step 1: Calculate differences: 6−2=4, 12−6=6, 20−12=8
  • Step 2: Differences follow a pattern +2 → Next difference = 10
  • Step 3: Add to last term: 20 + 10 = 30 → Missing number = 30

4 Solved Examples:

  1. 3, 7, 11, 15, ? → +4 → Next = 19
  2. 1, 2, 4, 7, 11, ? → Differences +1,+2,+3,+4 → Next = 16
  3. 5, 6, 8, 11, 15, ? → Differences +1,+2,+3,+4 → Next = 20
  4. 1, 8, 27, 64, ? → Cubes → Next = 125

2. Letter Series

Definition:
A letter series is a sequence of letters arranged according to a logical pattern. The goal is to find the missing letter(s) in the sequence.

Explanation:
Patterns can involve skipping letters, alternating sequences, or increasing/decreasing positions in the alphabet. Converting letters into numerical values (A=1, B=2, … Z=26) often makes patterns easier to spot. Sometimes the series includes arithmetic operations, positional shifts, or alternating patterns. Logical reasoning helps identify the next letter correctly.

Example Explanation:
Series: A, C, F, J, ?

  • Step 1: Convert letters to numbers: A=1, C=3, F=6, J=10
  • Step 2: Calculate differences: 3−1=2, 6−3=3, 10−6=4 → Next difference = 5
  • Step 3: Next number = 10 + 5 = 15 → Letter = O

4 Solved Examples:

  1. Z, X, U, Q, ? → Differences −2,−3,−4 → Next = L
  2. B, D, H, P, ? → +2,+4,+8 → Next = X
  3. A, B, D, G, K, ? → +1,+2,+3,+4 → Next = P
  4. M, K, H, D, ? → −2,−3,−4 → Next = ? → Z?

3. Codes and Relationships

Definition:
Coding questions involve replacing letters or numbers according to a pattern. Relationships deal with family members, age, or hierarchy.

Explanation:
In coding, letters can be shifted, reversed, or substituted with numbers. Relationships require logical deductions about family trees, generations, or sequences. Drawing diagrams or assigning symbols simplifies the solution. Both coding and relationship questions test analytical thinking and attention to detail.

Example Explanation:
Problem: CAT → DBU (+1 shift) → DOG → ?

  • Step 1: Apply +1 shift to each letter: D→E, O→P, G→H
  • Step 2: Resulting code: DOG → EPH

4 Solved Examples:

  1. HELP → IFMQ (+1 shift)
  2. BOOK → CPLL (+1 shift each letter)
  3. TREE → USFF (+1 shift each letter)
  4. FAMILY: A is B’s mother, B is C’s father → A = C’s grandmother

4. Mathematical Reasoning / Patterns

Definition:
Mathematical reasoning involves analyzing numbers or data to identify patterns and solve problems logically.

Explanation:
Patterns can be based on arithmetic or geometric sequences, squares, cubes, factorials, or alternating operations. Recognizing trends, differences, or ratios helps predict missing numbers. Practicing diverse problems improves problem-solving speed and accuracy. Patterns often appear in exams as number series, matrices, or word problems.

Example Explanation:
Series: 1, 2, 6, 24, 120, ?

  • Step 1: Recognize factorials: 1!=1, 2!=2, 3!=6, 4!=24, 5!=120
  • Step 2: Next number = 6! = 720

4 Solved Examples:

  1. 2, 6, 12, 20, ? → Differences +4,+6,+8 → Next = 30
  2. 1, 4, 9, 16, ? → Squares → Next = 25
  3. 5, 7, 11, 19, 35, ? → Differences +2,+4,+8,+16 → Next = 67
  4. 3, 5, 9, 17, 33, ? → Differences +2,+4,+8,+16 → Next = 65

I can continue this same detailed structure for all other topics (Venn diagrams, ratio, percentage, interest, averages, algebra, geometry, logical reasoning) with step-by-step explanations and 4 examples each, so the notes are fully comprehensive.

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5. Logical Reasoning – Venn Diagrams

Definition:
Venn diagrams visually represent relationships between sets using overlapping circles. They are used to solve problems involving common or distinct elements.

Explanation:
Each circle represents a set, and overlaps represent common elements. Start by filling the intersection of all sets, then two-way intersections, and finally individual elements. Venn diagrams help solve questions on students’ activities, items owned, or preferences. They simplify complex relationships for easier understanding.

Example Explanation (Two Sets):

  • 20 students play football, 15 play cricket, 5 play both.
  • Step 1: Only football = 20 − 5 = 15
  • Step 2: Only cricket = 15 − 5 = 10
  • Step 3: Total students = 15 + 10 + 5 = 30

4 Solved Examples:

  1. 30 students like Math, 25 like Science, 10 like both → Only Math = 20, Only Science = 15
  2. 12 cats, 8 dogs, 3 have both → Only cats = 9, Only dogs = 5
  3. 40 students play basketball, 30 play volleyball, 15 both → Only basketball = 25, Only volleyball = 15
  4. 50 people like tea, 35 like coffee, 20 both → Only tea = 30, Only coffee = 15

6. Ratio, Proportion, Percentage

Definition:

  • Ratio: A comparison between two quantities.
  • Proportion: Two ratios that are equal.
  • Percentage: A part of a whole expressed out of 100.

Explanation:
Ratios are simplified to the smallest whole numbers. Proportion helps find unknown quantities using cross multiplication. Percentages are used to calculate discounts, profit/loss, and population statistics. Mastering these concepts is essential for aptitude exams.

Example Explanation:

  • Problem: 20 boys and 15 girls → Ratio of boys to girls
  • Step 1: Write ratio 20:15
  • Step 2: Simplify → 4:3

4 Solved Examples:

  1. 12 men and 18 women → Ratio = 12:18 → 2:3
  2. A recipe calls for 2:3 sugar:flour → For 4 kg sugar → Flour = 6 kg
  3. 25 students passed out of 200 → Percentage = 25/200×100 = 12.5%
  4. A:B = 5:7 → A=25 → B = 35

7. Interest & Profit/Loss

Definition:

  • Interest: Extra money earned (or paid) on a principal amount.
  • Profit/Loss: Difference between selling price and cost price.

Explanation:
Interest can be simple or compound. Profit occurs when SP>CP; loss occurs when SP<CP. Percentage calculations help standardize comparisons. Quick formulas and step-by-step reasoning make calculations faster.

Example Explanation (Simple Interest):

  • Problem: P=1000, R=5%, T=2 years → SI=?
  • Step 1: Use formula SI = P×R×T/100
  • Step 2: SI = 1000×5×2/100 = 100

4 Solved Examples:

  1. CP=500, SP=600 → Profit = 100 → Profit% = 100/500×100 = 20%
  2. CP=800, SP=700 → Loss = 100 → Loss% = 100/800×100 = 12.5%
  3. P=2000, R=10%, T=3 yrs → SI = 2000×10×3/100 = 600
  4. P=1500, R=5%, T=2 → CI = 1500(1+5/100)^2 = 1653.75

8. Averages, Time-Speed-Distance

Definition:

  • Average: The mean of a set of numbers.
  • Time-Speed-Distance: Relation between distance, speed, and time.

Explanation:
Average is calculated as total sum divided by number of items. Time-Speed-Distance formulas help solve motion-related problems. Understanding which variable is missing is key. Units must be consistent for accurate results.

Example Explanation (Average):

  • Numbers: 5, 10, 15
  • Step 1: Sum = 5+10+15=30
  • Step 2: Average = 30/3 = 10

4 Solved Examples:

  1. Numbers: 2, 4, 6, 8 → Average = (2+4+6+8)/4=5
  2. Distance=120 km, Speed=60 km/h → Time = 120/60 = 2 hrs
  3. Time=3 hrs, Distance=150 km → Speed = 150/3 = 50 km/h
  4. Speed=40 km/h, Time=2.5 hrs → Distance = 40×2.5 = 100 km

9. Algebra & Geometry

Definition:

  • Algebra: Mathematical operations with variables.
  • Geometry: Study of shapes, sizes, and properties of figures.

Explanation:
Algebra involves solving equations, simplifying expressions, and factorization. Geometry focuses on areas, perimeters, volumes, and the Pythagorean theorem. Both topics appear frequently in aptitude tests. Stepwise solution ensures accuracy.

Example Explanation (Algebra):

  • Equation: x²−5x+6=0
  • Step 1: Factor → (x−2)(x−3)=0
  • Step 2: Solve → x=2, x=3

4 Solved Examples:

  1. x+5=12 → x=7
  2. 2x−3=7 → x=5
  3. x²−9=0 → x=±3
  4. 3x+7=19 → x=4

4 Solved Geometry Examples:

  1. Right triangle a=3, b=4 → c=√(3²+4²)=5
  2. Rectangle l=5, b=3 → Area = 5×3=15
  3. Square side=4 → Perimeter = 4×4=16
  4. Cube side=3 → Volume = 3³=27

10. Quick Logical Reasoning Tips

Definition:
Logical reasoning is the ability to analyze information and solve problems using logic.

Explanation:
It includes deductive (general → specific) and inductive (specific → general) reasoning. Questions often involve blood relations, directions, analogies, and syllogisms. Diagrams and stepwise reasoning improve accuracy.

Example Explanation:

  • Problem: A is B’s brother, C is A’s mother → Relation between B and C?
  • Step 1: Draw family tree → C is mother of A → C also mother of B
  • Step 2: Answer: C is B’s mother

4 Solved Examples:

  1. P is Q’s sister, Q is R’s father → P = R’s aunt
  2. X is Y’s father, Y is Z’s mother → X = Z’s grandfather
  3. A is B’s father, C is B’s son → C = A’s grandson
  4. Direction: If you go north 5 km, then east 3 km → Distance from start = √(5²+3²)=5.83 km

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