Quantiative-Aptitude
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) 🟪
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:
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:
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:
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
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.
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.
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:
09 Percentage
This guide summarizes the key concepts from your Percentage PDF, making the topic simple with step-by-step explanations, formulas, conversion tables, solved examples, cheatsheets, practice questions (with answers), and visual aids. 1. What is a Percentage? Definition: A percentage is a fraction with a denominator of 100, denoted by the symbol %. Example: $ 10% = \frac{10}{100} = 0.1 $1. 2. Key Formulas and Conversions A. Basic Percentage Formula To find $ y% $ of $ x $: $$ y\% \text{ of } x = x \times \frac{y}{100} $$Example: $ 25% $ of $ 200 = 200 \times \frac{25}{100} = 50 $12.
10 Profit and Loss
This guide breaks down the Profit and Loss chapter from your PDF for SSC and other competitive exams, using simple language, step-by-step formulas, practical examples, a cheatsheet, solved practice questions, and visual aids. 1. Key Concepts and Definitions Term Meaning Cost Price (CP) The price at which an item is purchased (includes all expenses: purchase, transport, etc.) Selling Price (SP) The price at which the item is sold (money received from the buyer) Marked Price (MP) The price written/tagged on the item (before discount) Profit (Gain) When SP > CP; $ Profit = SP - CP $ Loss When SP < CP; $ Loss = CP - SP $ Discount Reduction from the marked price; $ Discount = MP - SP $ 2. Essential Formulas Cheatsheet Quantity Formula Profit $ SP - CP $ Loss $ CP - SP $ Profit % (on CP) $ \frac{Profit}{CP} \times 100 $ Loss % (on CP) $ \frac{Loss}{CP} \times 100 $ SP (given Profit %) $ SP = \left(1 + \frac{Profit %}{100}\right) \times CP $ SP (given Loss %) $ SP = \left(1 - \frac{Loss %}{100}\right) \times CP $ CP (given Profit %) $ CP = \frac{100}{100 + Profit %} \times SP $ CP (given Loss %) $ CP = \frac{100}{100 - Loss %} \times SP $ Discount % $ \frac{Discount}{MP} \times 100 $ Markup % $ \frac{MP - CP}{CP} \times 100 $ Equivalent Discount $ X + Y - \frac{XY}{100} $ (for successive discounts of X% and Y%) 3. Visuals & Graphics A. Profit and Loss Flowchart +-----------------+ | Cost Price | +-----------------+ | +------------------+------------------+ | | +-----------+ +-----------+ | Selling | | Marked | | Price | | Price | +-----------+ +-----------+ | | | | +-----------+ +-----------+ | Profit or | | Discount | | Loss | +-----------+ +-----------+ B. Example Bar Chart CP: |========| (Rs. 100) SP: |==========| (Rs. 120) Profit: |==| (Rs. 20) 4. Solved Examples Example 1: Profit Calculation A shopkeeper buys a pen for Rs. 8 and sells it for Rs. 10.
11 Ratio and Proportion
This guide breaks down the Ratio and Proportion PDF into simple explanations, key formulas, solved examples, a cheatsheet, practice questions (with answers), and visual aids for better understanding. 1. Ratio: Definition and Basics Ratio compares two quantities by division. Written as $ a : b $, which means $ \frac{a}{b} $. Antecedent: First term (a), Consequent: Second term (b). Example: Ratio 5:9 means antecedent = 5, consequent = 9.
12 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) Identity Formula 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 = ? $
13 Speed, Time, and Distance
This summary provides a comprehensive, easy-to-understand guide to the Speed, Time, and Distance PDF, including definitions, key formulas, solved examples, a cheatsheet, practice questions with answers, and visual aids. 1. Core Concepts and Definitions Term Definition Units Speed Rate at which distance is covered. m/s, km/hr Time Duration taken to cover the distance. seconds, minutes, hr Distance Length of the path traveled. meters, kilometers Key Relationship:
14 Time and Work
This guide simplifies the Time and Work topic from your PDF, including core concepts, key formulas, solved examples, a cheatsheet, practice questions (with answers), and visual aids for clarity. 1. Core Concepts Work: Any task to be completed (e.g., building a wall, filling a tank). Time: Duration taken to finish the work. Efficiency: Amount of work done per unit time (higher efficiency = less time). 2. Key Formulas (Cheatsheet) Formula/Rule Use Case If A completes work in n days, A’s 1 day work = 1/n Basic calculation Work = Time × Rate of Work Find total work done Rate of Work = 1 / Time Work done per unit time Time = 1 / Rate of Work Find time from rate Total Work = Efficiency × Number of Days Work done by a person/group If A is x times as efficient as B, time ratio = 1:x Comparing times for same work If A & B together finish in T days: 1/T = 1/A + 1/B Combined work If A, B, C together: 1/T = 1/A + 1/B + 1/C Three people together If x men do W1 work in D1 days, x men do W2 in D2 days: Work/men/days relationship M₁D₁/W₁ = M₂D₂/W₂ If A takes x days more than (A+B) and B takes y days more (A+B) finish in √(xy) days 3. Visual: Work-Time-Efficiency Relationship Work / \ Time Efficiency \ / (Inverse) More efficiency → less time for same work. More people → less time for same work. 4. Solved Examples Example 1: Two People Working Together Q: A does a work in 10 days, B in 15 days. How long together?
15 Trigonometry
This guide summarizes the Trigonometry PDF for competitive exams, making the topic simple with clear explanations, formulas, solved examples, cheatsheets, practice questions (with answers), and visual aids. 1. What is Trigonometry? Trigonometry is the study of relationships between the sides and angles of triangles, especially right triangles. “Tri” = three, “gono” = sides, “metry” = measurement. 2. The Six Trigonometric Ratios In a right triangle:
16 Qunatiative Aptitude
SSC CHSL Exam Study Material for Exam Syllabus