Notes
SSC CHSL Exam
Qunatiative Aptitude
Simplification

Simplification

This summary explains the Simplification chapter from your PDF for SSC and other competitive exams. You’ll find key concepts, formulas, solved examples, a cheatsheet, practice questions, and visual aids for better understanding1.

1. Number Basics

  • Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
  • Types of Numbers:
    • Natural numbers: 1, 2, 3, …
    • Whole numbers: 0, 1, 2, 3, …
    • Integers: …, -3, -2, -1, 0, 1, 2, 3, …
    • Real numbers: All rational and irrational numbers (e.g., 2.8, -10, 3.13)
    • Even numbers: 2, 4, 6, …
    • Odd numbers: 1, 3, 5, …
    • Prime numbers: 2, 3, 5, 7, 11, …

2. Algebraic Identities (Cheatsheet)

IdentityFormula
Square of sum$ (a+b)^2 = a^2 + 2ab + b^2 $
Square of difference$ (a-b)^2 = a^2 - 2ab + b^2 $
Difference of squares$ a^2 - b^2 = (a+b)(a-b) $
Cube of sum$ (a+b)^3 = a^3 + b^3 + 3ab(a+b) $
Cube of difference$ (a-b)^3 = a^3 - b^3 - 3ab(a-b) $
Sum of cubes$ a^3 + b^3 = (a+b)(a^2 - ab + b^2) $
Difference of cubes$ a^3 - b^3 = (a-b)(a^2 + ab + b^2) $

3. Addition & Subtraction Tricks

Example 1

Q: $ 8 + 88 + 888 + 8888 + 88888 = ? $

Trick: Factor out 8: $ 8 \times (1 + 11 + 111 + 1111 + 11111) = 8 \times 12345 = 98760 $

Example 2

Q: $ 0.8 + 0.88 + 0.888 + 0.8888 + 0.88888 = ? $

Trick: $ 8 \times (0.1 + 0.11 + 0.111 + 0.1111 + 0.11111) = 8 \times 0.54321 = 4.34568 $

Example 3

Q: $ 8.8 + 8.88 + 8.888 + 8.8888 + 8.88888 = ? $

Trick: Add the whole numbers: $ 8 \times 5 = 40 $ Add decimals from previous: $ 4.34568 $ Total: $ 44.34568 $

4. Multiplication Shortcuts

Type 1: Unit Digits Add to 10, Other Digits Same

Q: $ 43 \times 47 $

  • $ 4 \times (4+1) = 20 $
  • $ 3 \times 7 = 21 $
  • Answer: 2021

Type 2: Tens Digits Add to 10, Units Same

Q: $ 46 \times 66 $

  • $ (4 \times 6) + 6 = 30 $
  • $ 6 \times 6 = 36 $
  • Answer: 3036

Type 3: Both Units 5, Numbers Differ by 10

Q: $ 75 \times 65 $

  • $ 6 \times (7+1) = 48 $
  • Write 75
  • Answer: 4875

5. Squares and Cubes

  • Squares of 1–50: Memorize for quick calculation.
  • Example: $ 56^2 $
    • Split as 5 and 6.
    • $ (5/6)^2 = 5^2 / 2 \times 5 \times 6 / 6^2 = 25 / 60 / 36 $
    • Combine: $ 3136 $

6. Fractions

  • Proper Fraction: Numerator < Denominator (e.g., $ \frac{3}{5} $)
  • Improper Fraction: Numerator > Denominator (e.g., $ \frac{7}{5} $)
  • Mixed Fraction: Whole number + proper fraction (e.g., $ 8 \frac{7}{2} $)

7. VBODMAS Rule

Order of Operations:

  • Vinculum (bar)
  • Brackets: (), {}, []
  • Orders (powers, roots)
  • Division
  • Multiplication
  • Addition
  • Subtraction

Table:

BODMAS
BracketsOrdersDivideMultiplyAddSubtract

8. VBODMAS Examples

Example 1

$ 35 \div 7 \times 5 = ? $

  • Divide first: $ 35 \div 7 = 5 $
  • Multiply: $ 5 \times 5 = 25 $

Example 2

$ 35 \div 5 $ of $ 7 = ? $

  • ‘Of’ first: $ 5 \times 7 = 35 $
  • Then divide: $ 35 \div 35 = 1 $

Example 3

$ 48 \div 12 $ of $ 2 + [3 + 17 \times 2] $

  • Solve inside brackets: $ 17 \times 2 = 34 $, $ 3 + 34 = 37 $
  • ‘Of’: $ 12 \times 2 = 24 $
  • Divide: $ 48 \div 24 = 2 $
  • Add: $ 2 + 37 = 39 $

Example 4

$ 2 \div 2 \div 2 \div 2 \div 2 \div 2 = ? $

  • Left to right: $ 2 \div 2 = 1 $, $ 1 \div 2 = 0.5 $, $ 0.5 \div 2 = 0.25 $, etc.
  • Final answer: $ 1/16 $

9. Practice Questions

  1. $ 7 + 77 + 777 + 7777 = ? $
  2. $ 0.9 + 0.99 + 0.999 + 0.9999 = ? $
  3. $ 53 \times 57 $ (use special multiplication)
  4. $ 85 \times 75 $
  5. $ (a+b)^2 $ when $ a = 12, b = 8 $
  6. $ (a-b)^3 $ when $ a = 10, b = 4 $
  7. $ 25 + 75 \div 5 \times 2 $
  8. $ 16 \div 2 $ of $ 4 + 2 $
  9. $ 3.5 + 4.5 + 5.5 + 6.5 $
  10. $ 2 \div 2 \div 2 $

10. Visuals & Graphics

A. VBODMAS Pyramid 

    Vinculum
     Brackets
      Orders
      Divide
      Multiply
      Add
      Subtract

B. Multiplication Pattern

For numbers like 43 × 47:

|--4--|3| × |--4--|7|
Same tens, units add to 10

11. Quick Reference Table

TopicKey Formula/Rule
AdditionCombine like terms, factor when possible
MultiplicationUse patterns for quick calculation
Squares$ (a+b)^2 = a^2 + 2ab + b^2 $
Cubes$ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $
FractionsProper, Improper, Mixed
VBODMASOrder: V > B > O > D > M > A > S

12. Tips & Tricks

  • Memorize squares and cubes up to at least 25 for speed.
  • Use multiplication patterns for numbers close to each other.
  • Always follow VBODMAS for order of operations.
  • For decimals, align digits for addition/subtraction.

With these explanations, formulas, examples, cheatsheet, practice questions, and visuals, you’ll be ready to master Simplification for any exam!

⁂

  1. Simplification.pdf ↩︎

Ratio and ProportionSpeed, Time, and Distance

Related Chapters 📂

  • Algebraic Equations 📚✨

    01 Algebraic Equations 📚✨

    Algebraic equations are polynomial equations used in quantitative aptitude exams. They can be of degree 1 (linear), degree 2 (quadratic), or degree 3 (cubic)1. Linear Equations (Degree 1) ➗ Form: $ ax + c = 0 $ Examples: $ 2x + 3y = 4 $ $ x + y + z = 10 $ Solving Linear Equations: Use substitution or elimination to find variable values. Example: $ 2x + 3y = 13 $ …(1) $ 3x + 2y = 12 $ …(2) Multiply and subtract to eliminate a variable: $ 3 \times (2) - 2 \times (1) $ → $ 5x = 10 $ → $ x = 2 $ Substitute $ x $ into (1): $ y = 3 $ Result: $ x < y $ ✅ Another Example: $ 4x + 5y = 14 $ …(1) $ 2x + 3y = 5 $ …(2) Multiply (2) by 2: $ 4x + 6y = 10 $ Subtract (1): $ y = -4 $, then $ x = 1 $ Result: $ x > y $ 👍 Quadratic Equations (Degree 2) 🟪

  • Average

    02 Average

    1. What is an Average? An average (also called arithmetic mean) is a value that represents the central or typical value in a set of numbers. It is calculated by dividing the sum of all values by the number of values. Formula: $$ \text{Average} = \frac{\text{Sum of Terms}}{\text{Number of Terms}} $$Example: If you have the numbers 10, 20, and 30: $$ \text{Average} = \frac{10 + 20 + 30}{3} = \frac{60}{3} = 20 $$ 2. Why Use Averages? Averages help us:

  • Data Interpretation (DI)

    03 Data Interpretation (DI)

    Data Interpretation is about understanding and analyzing data given in tables, charts, or graphs to answer questions. Here’s a simple summary with emojis for each type! Key Tips for DI Success 📝 Read Carefully 👀: Always read the full question and data set. Analyze Data 🧐: Don’t get overwhelmed—break it down. Check Units ⚖️: Watch for different units (e.g., km/h vs. seconds). Approximate When Needed 🔍: If options are far apart, round numbers for speed. Use Last Digit Tricks 🔢: Sometimes, just the last digit helps pick the answer. Practice Mental Math 🧠: The faster you calculate, the more time you save! Types of Data Interpretation 📈 Tabular DI 📋 Pie Chart 🥧 Bar Graph 📊 Line Graph 📉 Caselet DI 📄 Tabular DI 📋 Data is shown in a table (rows & columns). Example:

  • Geometery

    04 Geometery

    1. Fundamental Concepts of Geometry 1.1 Points, Lines, Segments, and Rays Point: An exact location in space with no size. Example: “.” (a dot on paper) Line Segment: The shortest path joining two points, with definite length. Example: Segment $AB$ (see diagram below) Ray: A line segment extended infinitely in one direction. Example: Ray $AB$ starts at $A$ and passes through $B$, continuing forever. Visual Aid: 1.2 Types of Lines Type Description Example/Diagram Intersecting Lines Meet at a point (point of intersection) “X” shape Concurrent Lines Three or more lines intersect at the same point Star-like pattern Parallel Lines Never meet, always same distance apart " Transversal A line that cuts two or more lines at distinct points See below Visual Aid: 1.3 Angles and Their Types Angle Type Measure (Degrees) Example/Diagram Right $90^\circ$ Corner of a square Acute $< 90^\circ$ $45^\circ, 60^\circ$ Obtuse $90^\circ < x < 180^\circ$ $120^\circ$ Straight $180^\circ$ A straight line Reflex $180^\circ < x < 360^\circ$ $270^\circ$ Example: An angle of $60^\circ$ is acute; $135^\circ$ is obtuse. Visual Aid: 1.4 Angle Relationships Relationship Description Example Complementary Sum to $90^\circ$ $30^\circ$ and $60^\circ$ Supplementary Sum to $180^\circ$ $130^\circ$ and $50^\circ$ Vertically Opposite Angles Opposite angles formed by intersecting lines (equal) See diagram Angle Bisector Divides an angle into two equal parts See diagram Visual Aid: Practice Questions: Fundamental Concepts Name the type of angle for $110^\circ$. If two angles are complementary and one is $35^\circ$, what is the other? Draw two parallel lines and a transversal. Mark and name a pair of corresponding angles. Answers:

  • Interest

    05 Interest

    This summary breaks down the key concepts from the provided PDF on Interest for government exam preparation, with clear explanations, formulas, examples, cheat sheets, and practice questions. Key Definitions Interest: The extra amount paid for borrowing money or received for lending money. Principal (P): The original amount borrowed or lent. Amount (A): The total sum after adding interest to the principal ($A = P + \text{Interest}$). Rate of Interest (r): The percentage charged or earned on the principal per year. Time (t): The period for which money is borrowed or deposited (usually in years). Types of Interest Simple Interest (SI) Definition: Interest calculated only on the principal for every year. Formula: $$ SI = \frac{P \times r \times t}{100} $$Where: - $P$ = Principal - $r$ = Rate per annum - $t$ = Time in years

  • Mesuration

    06 Mesuration

    Mensuration is the mathematics of measuring geometric figures-calculating their area, perimeter, and volume. This summary breaks down the key concepts, formulas, solved examples, and practice questions from the provided PDF, making it easy to understand and apply for exams. 1. What is Mensuration? Mensuration is the branch of mathematics that deals with the measurement of 2D (plane) and 3D (solid) shapes, including their area, perimeter, surface area, and volume12. 2. Types of Shapes 2D Shapes (Plane) 3D Shapes (Solid) Triangle, Square, Cube, Cuboid, Cylinder, Rectangle, Circle, Sphere, Cone, Pyramid, Parallelogram, Rhombus Prism, Hemisphere 2D shapes: Only length and breadth; measure area & perimeter. 3D shapes: Length, breadth, height/depth; measure volume & surface area12. 3. Key Formulas Cheat Sheet Triangles Perimeter: $ a + b + c $ Area (General): $ \frac{1}{2} \times base \times height $ Area (Heron’s Formula): $ \sqrt{s(s-a)(s-b)(s-c)} $, where $ s = \frac{a+b+c}{2} $ Area (Equilateral): $ \frac{\sqrt{3}}{4}a^2 $ Height (Equilateral): $ \frac{\sqrt{3}}{2}a $3 Quadrilaterals Square Area: $ a^2 $ Perimeter: $ 4a $ Diagonal: $ a\sqrt{2} $ Rectangle Area: $ l \times b $ Perimeter: $ 2(l + b) $ Diagonal: $ \sqrt{l^2 + b^2} $3 Parallelogram Area: $ base \times height $ Perimeter: $ 2(a + b) $ Rhombus Area: $ \frac{d_1 \times d_2}{2} $ Perimeter: $ 4a $3 Trapezium Area: $ \frac{1}{2} \times (sum of parallel sides) \times height $ Perimeter: Sum of all sides Circle Area: $ \pi r^2 $ Circumference: $ 2\pi r $ Diameter: $ 2r $3 3D Shapes Cube Volume: $ a^3 $ Surface Area: $ 6a^2 $ Diagonal: $ a\sqrt{3} $ Cuboid Volume: $ l \times b \times h $ Surface Area: $ 2(lb + bh + hl) $ Cylinder Curved Surface Area: $ 2\pi rh $ Total Surface Area: $ 2\pi r(r + h) $ Volume: $ \pi r^2 h $ Cone Curved Surface Area: $ \pi r l $ (l = slant height) Total Surface Area: $ \pi r(r + l) $ Volume: $ \frac{1}{3}\pi r^2 h $ Sphere Surface Area: $ 4\pi r^2 $ Volume: $ \frac{4}{3}\pi r^3 $32 4. Solved Examples Example 1: Triangle Area (Heron’s Formula) Q: Sides are 12m, 13m, 11m. Find area and height with respect to side 12m.

  • Mixture and Alligation

    07 Mixture and Alligation

    This summary simplifies the key points of the Mixture and Alligation PDF, including core concepts, formulas, worked examples, a cheatsheet, and all practice questions with clear answers. Core Concepts Alligation: A rule to find the ratio in which two or more ingredients at given prices must be mixed to obtain a mixture of a desired price. Mean Price: The cost price per unit of the mixture.

  • Number System

    08 Number System

    This guide breaks down the essential points from your Number System PDF, making it easy to understand for competitive exams. It includes definitions, formulas, worked examples, cheatsheets, visuals, and practice questions with solutions. 1. What is a Number System? A number system is a method of representing numbers using a set of symbols and rules. It’s used for counting, measuring, and labeling. The most common systems are: