Invertendo, Alternendo, Componendo & Dividendo Property

Duration: 15 min

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Module outline

  1. About the course: About the Course
  2. Mathematical Aptitude / गणितीय अभिक्षमता: Number System / संख्या पद्धति, HCF and LCM / महत्तम समापवर्तक और लघुत्तम समापवर्त्य, Divisibility Rules / विभाज्यता के नियम, Percentage / प्रतिशत, Ratio and Proportion / अनुपात और समानुपात, Average / औसत, Profit, Loss and Discount / लाभ, हानि और बट्टा, Time and Work / समय और कार्य, Solid Mensuration / घन क्षेत्रमिति, Simple and Compound Interest / साधारण और चक्रवृद्धि ब्याज, Time, Speed and Distance / समय, चाल और दूरी, Age Problems / आयु संबंधी प्रश्न, Mixture and Alligation / मिश्रण और एलिगेशन, Algebra / बीजगणित, Simplification, Indices and Surds / सरलीकरण, घातांक और करणी, Pipe and Cistern / नल और टंकी, Trigonometry / त्रिकोणमिति, Statistics / सांख्यिकी, Data Interpretation / आँकड़ा निर्वचन, Arithmetic / अंकगणित, Sets, Relations, Functions and Discrete Mathematics / समुच्चय, संबंध, फलन और विविक्त गणित, Number Systems and Number Theory / संख्या पद्धति और संख्या सिद्धांत, Coordinate Geometry, Vectors and 3D Geometry / निर्देशांक ज्यामिति, सदिश और त्रिविमीय ज्यामिति, Polynomials, Equations and Complex Numbers / बहुपद, समीकरण और सम्मिश्र संख्याएँ
  3. Logical Reasoning / तार्किक तर्कशक्ति: Series / श्रेणी, Coding-Decoding / कूट लेखन एवं कूट वाचन, Analogy and Classification / सादृश्यता एवं वर्गीकरण, Blood Relations / रक्त संबंध, Direction and Distance / दिशा एवं दूरी, Order and Ranking / क्रम एवं रैंकिंग, Non-Verbal Reasoning / अशाब्दिक तर्क, Seating Arrangement and Puzzles / बैठक व्यवस्था एवं पहेलियाँ, Syllogisms / न्यायवाक्य, Venn Diagrams, Syllogism and Inequalities / वेन आरेख, न्याय वाक्य एवं असमानता, Mathematical Operations and Missing Numbers / गणितीय संक्रियाएँ एवं लुप्त संख्या, Alphabet and Number Test / वर्णमाला एवं संख्या परीक्षण, Statement-Based Reasoning / कथन आधारित तार्किक विवेचन, Analytical Reasoning (Counting Figures Reasoning) / विश्लेषणात्मक तर्क (आकृति गणना तर्क), Cubes, Dice and Figure Counting / घन, पासा एवं आकृतियों की गिनती, Calendar and Clock / कैलेंडर एवं घड़ी, Clock / घड़ी
  4. General Knowledge / सामान्य ज्ञान: Scientific Achievements and Research / वैज्ञानिक उपलब्धियाँ और अनुसंधान, Chemistry / रसायन विज्ञान, Biology / जीव विज्ञान, History / इतिहास, Geography / भूगोल, Polity / राजव्यवस्था, Indian Economy / भारतीय अर्थव्यवस्था, Current Affairs / समसामयिकी, Static General Knowledge / स्थैतिक सामान्य ज्ञान, State-Specific General Knowledge / राज्य विशेष सामान्य ज्ञान
  5. Art of Teaching / शिक्षण कला: Teaching-Learning Process and Classroom Management / शिक्षण-अधिगम प्रक्रिया एवं कक्षा प्रबंधन, Learners & Learning Process / अधिगमकर्ता और अधिगम प्रक्रिया, Factors Affecting Teaching / शिक्षण को प्रभावित करने वाले कारक, Teaching Methods and Approaches / शिक्षण विधियाँ एवं उपागम, Microteaching and Classroom Communication / सूक्ष्म शिक्षण एवं कक्षा संप्रेषण, Teaching Aids and Educational Technology / शिक्षण सहायक सामग्री एवं शैक्षिक तकनीकी, Assessment and Evaluation / आकलन एवं मूल्यांकन, Teacher: Qualities, Professional Ethics and Development / शिक्षक: गुण, व्यावसायिक नैतिकता एवं विकास, Educational Policies, Acts and Administration / शैक्षिक नीतियाँ, अधिनियम एवं प्रशासन, Curriculum, Arts and Co-curricular Areas / पाठ्यचर्या, कला एवं सह-शैक्षिक क्षेत्र, Guidance and Counselling / निर्देशन एवं परामर्श, Education of Children with Disabilities / दिव्यांग बच्चों की शिक्षा, Micro-Teaching & Teaching Skills / सूक्ष्म शिक्षण और शिक्षण कौशल, Lesson Planning / पाठ योजना
  6. Environmental Science / पर्यावरण विज्ञान: Environment / पर्यावरण
  7. Unit 1 | Digital Electronics: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  8. Unit 2 | COA: Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Instr Formats & Modes, Control Unit Design, Register Transfer and Microoperations, Programming the Basic Computer
  9. Unit 3 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  10. Unit 3 | Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  11. Unit 4 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  12. Unit 5 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  13. Unit 6 | Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Transaction Management, Concurrency Control
  14. Unit 7 | Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer
  15. Unit 8 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  16. Unit 9 | Object Oriented Programming: OOPS with C++
  17. Unit 10 | Web Development: HTML Basics, CSS, JavaScript-Basics, XML, Node.js, Git & Github, PHP
  18. Unit 11 | Theory Of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines
  19. Unit 12 | IoT: Emerging Tech Trends, Cloud Computing, Computer Networks, Internet of Things
  20. Unit 13 | Artificial Intelligence: Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Natural Language Processing, Artificial Neural Networks, Genetic Algorithms
  21. Unit 14 | E-Commerce: Electronic Data Interchange and E-Commerce, Role of Internet and Web Services, Security in Mobile and Wireless Computing, Physical Security and Cryptography Basics, Cyber Safety and Security
  22. Unit 15 | Multimedia: Multimedia
  23. Unit 16 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis, DSA using Python
  24. Full Length Mock Tests — General Paper:
  25. Full Length Mock Tests — Computer Science (Paper II):
  26. Full Length Mock Test – General + Computer Science:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lecture by Yash Jain Sir from Knowledge Gate Educator systematically teaches the core properties of proportion, focusing on Invertendo, Alternendo, Componendo, Dividendo, Convertendo, and the combined Componendo-Dividendo property. The presentation follows a consistent pedagogical pattern: each property is introduced with its formal algebraic rule using variables a, b, c, and d, followed by a concrete numerical example worked out step-by-step on the board. The instructor uses red ink for handwritten derivations, cross-multiplication checks, and visual arrows to highlight term transformations. The video progresses from simple inversion of ratios through term swapping, addition/subtraction operations on proportion terms, and culminates in the Componendo-Dividendo property, which is noted as frequently used in simplification problems. Each section includes verification steps where the instructor confirms equality through cross-multiplication or fraction simplification, reinforcing that these properties preserve proportional relationships. The lecture is structured for exam revision, emphasizing memorizable rules and their algebraic justifications.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title slide 'Properties of Proportion' presented by Yash Jain Sir from Knowledge Gate Educator, then transitions to the Invertendo Property. The on-screen rule states: 'For four numbers a, b, c, d if a : b = c : d, then b : a = d : c', meaning equal ratios have equal inverse ratios. The instructor writes the fraction form a/b = c/d on the right side, draws downward arrows to show inversion into b/a = d/c, and adds a cross-multiplication note 'ad = bc'. A concrete example is presented: 'Example: 6 : 10 = 9 : 15' leading to 'Therefore, 10 : 6 = 15 : 9'. The instructor underlines key terms like 'a, b, c, d' and 'inverse ratios' to emphasize the rule structure.

  2. 2:00 – 5:00 02:00-05:00

    The Invertendo Property example is verified through fraction work, showing '3/3 × 10/6 = 15/9 × 2/2' leading to '30/18 = 30/15'. The lecture then transitions to the Alternendo Property, whose rule reads: 'For four numbers a, b, c, d if a : b = c : d, then a : c = b : d'. The instructor writes 'a|b|c|d' to represent proportion terms and draws a red arrow indicating the swap of b and c. The example 'If 3 : 5 = 6 : 10 then 3 : 6 = 5 : 10' is worked with red handwritten arrows crossing fractions 3/5 and 6/10, circling '5 = 6' and '6 = 5', then writing 3/6 = 5/10. The view changes to the Componendo Property, stating 'if a : b = c : d then (a + b) : b :: (c + d) : d', with example '4 : 5 = 8 : 10' worked as (4+5)/5 = (8+10)/10, yielding 9/5 = 18/10 with a circled checkmark.

  3. 5:00 – 10:00 05:00-10:00

    The Dividendo Property slide states the rule 'For four numbers a, b, c, d if a : b :: c : d then (a - b) : b :: (c - d) : d', with terms underlined. A red handwritten fraction a/b = c/d with curved arrows is shown, then enclosed in a red box as (a-b)/b = (c-d)/d. The example '5 : 4 = 10 : 8' is worked as '(5 - 4) : 4 = (10 - 8) : 8 => 1 : 4 = 2 : 8', with red cross-out work marked by a check. The slide then retitles to 'Convertendo Property' with rule 'a : (a - b) :: c : (c - d)' and result line '5 : 1 = 10 : 2'. The worked example shows red handwritten fraction 5/4 = 10/8, then '5 : (5-4) = 10 : (10-8) => 5 : 1 = 10 : 2'. A proof slide derives (a-b)/b = (c-d)/d from a/b = c/d, with 'dividendo' written in red and underlined, and a boxed result a/(a-b) = c/(c-d).

  4. 10:00 – 14:37 10:00-14:37

    The Convertendo Property example is completed, then the lecture introduces the Componendo-Dividendo Property, noted on-screen as 'frequently used in simplification'. The rule states: 'For four numbers a, b, c, d If a : b :: c : d then (a + b) : (a - b) :: (c + d) : (c - d)'. The example '7 : 3 = 14 : 6' demonstrates that (7+3):(7-3) equals (14+6):(14-6), both simplifying to 5:2. The proof begins from a/b = c/d, adds 1 to both sides forming (a+b)/b = (c+d)/d, subtracts 1 forming (a-b)/b = (c-d)/d, then divides corresponding sides to conclude '(a + b) : (a - b) :: (c + d) : (c - d)'. The instructor highlights 'Comp' and 'Div' steps in red during the proof, writing out the division of corresponding sides explicitly on the right side.

The lecture builds a coherent progression through six proportion properties, each following the same teaching template: formal rule with variables, visual transformation using red ink arrows and boxes, concrete numerical example, and verification. Invertendo establishes that inverting both ratios preserves equality (a/b = c/d implies b/a = d/c). Alternendo shows that swapping the middle terms of a proportion yields another valid proportion (a:b = c:d implies a:c = b:d). Componendo and Dividendo introduce addition and subtraction of terms, producing (a+b):b = (c+d):d and (a-b):b = (c-d):d respectively. Convertendo combines subtraction with the original antecedent to give a:(a-b) = c:(c-d). The final Componendo-Dividendo property synthesizes both operations, deriving (a+b):(a-b) = (c+d):(c-d) by adding and subtracting 1 from a/b = c/d then dividing the results. The instructor consistently uses cross-multiplication (ad = bc) as a verification tool and emphasizes that these properties are algebraically justified transformations, not arbitrary rules. The note that Componendo-Dividendo is 'frequently used in simplification' signals its practical importance for exam problem-solving.

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