Invertendo, Alternendo, Componendo & Dividendo Property

Duration: 15 min

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Module outline

  1. Paper - 1 | Unit - 1 | Teaching Aptitude: Concept and Objectives of Teaching / शिक्षण की अवधारणा एवं उद्देश्य, Levels of Teaching / शिक्षण के स्तर, Learner's Characteristics / शिक्षार्थी की विशेषताएँ, Learning: Nature and Theories / अधिगम: प्रकृति एवं सिद्धांत, Cognition, Memory and Metacognition / संज्ञान, स्मृति एवं अधिसंज्ञान, Factors Affecting Teaching / शिक्षण को प्रभावित करने वाले कारक, Effective Teaching and the Teacher / प्रभावी शिक्षण एवं शिक्षक, Instructional Objectives and Lesson Planning / अनुदेशनात्मक उद्देश्य एवं पाठ नियोजन, Methods of Teaching / शिक्षण विधियाँ, Offline and Online Teaching Methods / ऑफ़लाइन एवं ऑनलाइन शिक्षण विधियाँ, Teaching Support System / शिक्षण सहायक प्रणाली, Evaluation Systems / मूल्यांकन प्रणाली, Sāṃkhya / सांख्य, Individual Behaviour / व्यक्तिगत व्यवहार, Human Resource Management / मानव संसाधन प्रबंधन, Domains of Child Development / बाल्यावस्था में विकास के क्षेत्र, Biological Basis of Emotion / संवेग का जैविक आधार, Upanishads and Bhagavad Gita / उपनिषद और भगवद्गीता, Central and Peripheral Nervous Systems / केन्द्रीय और परिधीय तंत्रिका तंत्र
  2. Paper - 1 | Unit - 2 | Research Aptitude: Research: Meaning, Features and Scientific Method / अनुसंधान: अर्थ, लक्षण व वैज्ञानिक विधि, Research Paradigms and Approaches / अनुसंधान के प्रतिमान एवं उपागम, Types of Research / अनुसंधान के प्रकार, Variables and Measurement / चर एवं मापन, Methods of Research / अनुसंधान की विधियाँ, Steps of Research / अनुसंधान के चरण, Sampling / प्रतिचयन, Tools and Methods of Data Collection / आँकड़ा संग्रह के उपकरण एवं विधियाँ, Data Analysis and Statistics in Research / आँकड़ा विश्लेषण एवं शोध सांख्यिकी, Thesis and Article Writing / शोध-प्रबंध एवं लेख लेखन, Application of ICT in Research / अनुसंधान में ICT का अनुप्रयोग, Research Ethics / शोध नैतिकता
  3. Paper - 1 | Unit - 3 | Comprehension: Literal Comprehension, Vocabulary and Language in Context, Inferential Comprehension, Critical Comprehension, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Restoration and Eighteenth-Century Drama, Modern British Fiction, William Shakespeare, Mid-Twentieth-Century and Contemporary British Fiction, American Drama
  4. Paper - 1 | Unit - 4 | Communication: Communication: Meaning, Nature and Process / संचार: अर्थ, प्रकृति एवं प्रक्रिया, Types and Levels of Communication / संचार के प्रकार एवं स्तर, Verbal and Non-verbal Communication / मौखिक एवं अमौखिक संचार, Semiotics and Meaning / संकेत विज्ञान एवं अर्थ, Models and Theories of Communication / संचार के मॉडल एवं सिद्धांत, Effective Communication / प्रभावी संचार, Barriers to Communication / संचार की बाधाएँ, Classroom Communication / कक्षा संचार, Mass Media: Types and Evolution / जनसंचार माध्यम: प्रकार एवं विकास, Mass Media and Society / जनसंचार एवं समाज, Digital and New Media / डिजिटल एवं नव माध्यम, Morphology and Syntax, Public Health Nutrition / जन स्वास्थ्य पोषण, Newtonian Motion and Stability, GK- Current Affairs / GK- समसामयिकी, Indian Economy / भारतीय अर्थव्यवस्था, Art and Architecture / कला और स्थापत्य, India after Independence / स्वतंत्रता के बाद भारत, General Awareness (Legacy Paper 1) / सामान्य जागरूकता (पुराना पेपर 1)
  5. Paper - 1 | Unit - 5 | Mathematical Reasoning and Aptitude: Number System and Simplification / संख्या पद्धति एवं सरलीकरण, Algebra and Ages / बीजगणित एवं आयु, Percentage / प्रतिशतता, Ratio, Proportion, Partnership and Mixtures / अनुपात, समानुपात, साझेदारी एवं मिश्रण, Averages / औसत, Profit, Loss and Discount / लाभ, हानि एवं बट्टा, Simple and Compound Interest / साधारण एवं चक्रवृद्धि ब्याज, Time and Work / समय एवं कार्य, Speed, Time and Distance / चाल, समय एवं दूरी, Mensuration and Geometry / क्षेत्रमिति एवं ज्यामिति, Permutation, Combination and Probability / क्रमचय, संचय एवं प्रायिकता, Series / श्रेणी, Coding-Decoding / कूटलेखन-कूटवाचन, Relationships, Directions and Ranking / संबंध, दिशा एवं क्रम, Arrangements, Puzzles, Clock and Calendar / व्यवस्था, पहेलियाँ, घड़ी एवं कैलेंडर, Mathematical Foundations, जीवनी, Deconstruction, Gender and Queer Theory, Modern British and Irish Poetry, Central Force and Scattering, Chemical thermodynamics, Microscopic techniques, Philosophy of the Constitution / संविधान का दर्शन, Waves, Optics and Charged-Particle Motion, Membrane structure and function, Analysis, Markov Chains, Univariate and Joint Distributions, Indian Polity and Governance (Legacy Unit 10) / भारतीय राजव्यवस्था और शासन (पुराना यूनिट 10)
  6. Paper - 1 | Unit - 6 | Logical Reasoning: Structure of Arguments / युक्ति की संरचना, Terms, Definitions and Uses of Language / पद, परिभाषा एवं भाषा के उपयोग, Categorical Propositions / निरुपाधिक प्रतिज्ञप्तियाँ, Categorical Syllogism / निरुपाधिक न्यायवाक्य, Venn Diagrams / वेन आरेख, Deduction, Induction and Analogy / निगमन, आगमन एवं सादृश्यता, Fallacies / तर्कदोष, Indian Logic: Pramanas / भारतीय तर्कशास्त्र: प्रमाण, Anumana and Vyapti / अनुमान एवं व्याप्ति, Hetvabhasa / हेत्वाभास, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Aristotle / अरस्तू, Western Schools of Philosophy / पाश्चात्य दर्शन की विचारधाराएँ, Kant / कांट, Sadhana Paada, Vaiśeṣika / वैशेषिक
  7. Paper - 1 | Unit - 7 | Data Interpretation: Data and Graphical Representation / आँकड़े एवं आलेखीय निरूपण, Data Interpretation: Tables / आँकड़ा निर्वचन: तालिका
  8. Paper - 1 | Unit - 8 | Information and Communication Technology: Computer Fundamentals and Hardware, Memory, Storage and Data Representation, Computer Software, Office Applications, Multimedia and File Formats, Computer Networks, Internet and World Wide Web, Email and Online Communication, Cyber Security, Digital Ethics, IPR and Society, Cloud and Distributed Computing, Emerging Technologies, ICT in Education and Governance, Cyber Threats and Malware Attacks, Role of Internet and Web Services, Electronic Data Interchange and E-Commerce, Internet Fundamentals, Web Design & Development, Web Publishing & Hosting, Emerging Technologies, Terms & Abbreviations, Basics of COA, Introduction to OS, Introduction to CN, Data Communication, Net Layer: IP Addressing, Network Security, Number System, Number Representation, Society, Law & Ethics, Keyboard Shortcuts, Website, Browser & Services, Protection of Geographical Indications / भौगोलिक संकेतों का संरक्षण
  9. Paper - 1 | Unit - 9 | People Development and Environment: Ecosystems and Biodiversity / पारितंत्र और जैव विविधता, Human and Environment Interaction / मानव और पर्यावरण की अंतःक्रिया, Development and Environment / विकास और पर्यावरण, Natural Resources – Forests, Water and Soil / प्राकृतिक संसाधन: वन, जल और मृदा, Energy Resources / ऊर्जा संसाधन, Air Pollution / वायु प्रदूषण, Water Pollution and Treatment / जल प्रदूषण और उपचार, Noise Pollution / ध्वनि प्रदूषण, Waste Management / अपशिष्ट प्रबंधन, Environmental Health and Toxicology / पर्यावरणीय स्वास्थ्य और विषविज्ञान, Climate Change and Global Environmental Issues / जलवायु परिवर्तन और वैश्विक पर्यावरणीय समस्याएँ, Natural Hazards and Disaster Management / प्राकृतिक प्रकोप और आपदा प्रबंधन, Indian Environmental Laws and Policies / भारतीय पर्यावरण कानून और नीतियाँ, International Environmental Agreements / अंतर्राष्ट्रीय पर्यावरण समझौते, Environment / पर्यावरण, Geography / भूगोल, Factory and Industry Systems / कारखाना और उद्योग व्यवस्थाएँ, Earth Materials, Surface Features and Processes, Communicable Diseases and Immunization / संक्रामक रोग और प्रतिरक्षण
  10. Paper - 1 | Unit - 10 | Higher Education System: Education in Ancient India / प्राचीन भारत में शिक्षा, Ancient Centres of Higher Learning / प्राचीन भारत के उच्च शिक्षा केंद्र, Colonial Education Policies and Commissions / औपनिवेशिक काल की शिक्षा नीतियाँ और आयोग, Colonial Colleges, Universities and Reformers / औपनिवेशिक कॉलेज, विश्वविद्यालय व सुधारक, Education Commissions and Policies after Independence / स्वतंत्र भारत में शिक्षा आयोग और नीतियाँ, National Education Policy 2020 / राष्ट्रीय शिक्षा नीति 2020, Types of Higher Education Institutions / उच्च शिक्षा संस्थानों के प्रकार, Learning Programmes and Academic Credits / अधिगम कार्यक्रम और अकादमिक क्रेडिट, Professional, Technical and Skill-Based Education / व्यावसायिक, तकनीकी और कौशल आधारित शिक्षा, Value Education and Environmental Education / मूल्य शिक्षा और पर्यावरण शिक्षा, Governance, Administration and Financing / शासन, प्रशासन और वित्तपोषण, Regulation, Accreditation and Ranking / उच्च शिक्षा का नियमन, प्रत्यायन और रैंकिंग
  11. Full Notes: Paper 1 Full Notes
  12. Full Mock Tests:
  13. Previous Year Papers:
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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