Invertendo, Alternendo, Componendo & Dividendo Property

Duration: 15 min

This video lesson is available to enrolled students.

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Module outline

  1. Paper - 1 | Unit - 1 | Teaching Aptitude: शिक्षण की अवधारणा एवं उद्देश्य / Concept and Objectives of Teaching, शिक्षण के स्तर / Levels of Teaching, शिक्षार्थी की विशेषताएँ / Learner's Characteristics, अधिगम: प्रकृति एवं सिद्धांत / Learning: Nature and Theories, संज्ञान, स्मृति एवं अधिसंज्ञान / Cognition, Memory and Metacognition, शिक्षण को प्रभावित करने वाले कारक / Factors Affecting Teaching, प्रभावी शिक्षण एवं शिक्षक / Effective Teaching and the Teacher, अनुदेशनात्मक उद्देश्य एवं पाठ नियोजन / Instructional Objectives and Lesson Planning, शिक्षण विधियाँ / Methods of Teaching, ऑफ़लाइन एवं ऑनलाइन शिक्षण विधियाँ / Offline and Online Teaching Methods, शिक्षण सहायक प्रणाली / Teaching Support System, मूल्यांकन प्रणाली / Evaluation Systems, Nature, Objectives & Characteristics of Teaching, Learners & Learning Process, Teaching-Learning Aids & ICT Integration
  2. Paper - 1 | Unit - 2 | Research Aptitude: अनुसंधान: अर्थ, लक्षण व वैज्ञानिक विधि / Research: Meaning, Features and Scientific Method, अनुसंधान के प्रतिमान एवं उपागम / Research Paradigms and Approaches, अनुसंधान के प्रकार / Types of Research, चर एवं मापन / Variables and Measurement, अनुसंधान की विधियाँ / Methods of Research, अनुसंधान के चरण / Steps of Research, प्रतिचयन / Sampling, आँकड़ा संग्रह के उपकरण एवं विधियाँ / Tools and Methods of Data Collection, आँकड़ा विश्लेषण एवं शोध सांख्यिकी / Data Analysis and Statistics in Research, शोध-प्रबंध एवं लेख लेखन / Thesis and Article Writing, अनुसंधान में ICT का अनुप्रयोग / Application of ICT in Research, शोध नैतिकता / Research Ethics, Introduction to Research, Validity & Reliability, Research Paradigms & Types of Research, Research Ethics, Writing & Publication
  3. Paper - 1 | Unit - 3 | Comprehension: प्रत्यक्ष बोध / Literal Comprehension, गद्यांश में शब्दावली और भाषा / Vocabulary and Language in Context, निष्कर्षात्मक बोध / Inferential Comprehension, आलोचनात्मक बोध / Critical Comprehension, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions)
  4. Paper - 1 | Unit - 4 | Communication: संचार: अर्थ, प्रकृति एवं प्रक्रिया / Communication: Meaning, Nature and Process, संचार के प्रकार एवं स्तर / Types and Levels of Communication, मौखिक एवं अमौखिक संचार / Verbal and Non-verbal Communication, संकेत विज्ञान एवं अर्थ / Semiotics and Meaning, संचार के मॉडल एवं सिद्धांत / Models and Theories of Communication, प्रभावी संचार / Effective Communication, संचार की बाधाएँ / Barriers to Communication, कक्षा संचार / Classroom Communication, जनसंचार माध्यम: प्रकार एवं विकास / Mass Media: Types and Evolution, जनसंचार एवं समाज / Mass Media and Society, डिजिटल एवं नव माध्यम / Digital and New Media, Communication Basics, Language and Semiotics, Mass Communication, Mass Media, Journalism, General Knowledge and General Studies related to Communication
  5. Paper - 1 | Unit - 5 | Mathematical Reasoning and Aptitude: संख्या पद्धति एवं सरलीकरण / Number System and Simplification, बीजगणित एवं आयु / Algebra and Ages, प्रतिशतता / Percentage, अनुपात, समानुपात, साझेदारी एवं मिश्रण / Ratio, Proportion, Partnership and Mixtures, औसत / Averages, लाभ, हानि एवं बट्टा / Profit, Loss and Discount, साधारण एवं चक्रवृद्धि ब्याज / Simple and Compound Interest, समय एवं कार्य / Time and Work, चाल, समय एवं दूरी / Speed, Time and Distance, क्षेत्रमिति एवं ज्यामिति / Mensuration and Geometry, क्रमचय, संचय एवं प्रायिकता / Permutation, Combination and Probability, श्रेणी / Series, कूटलेखन-कूटवाचन / Coding-Decoding, संबंध, दिशा एवं क्रम / Relationships, Directions and Ranking, व्यवस्था, पहेलियाँ, घड़ी एवं कैलेंडर / Arrangements, Puzzles, Clock and Calendar, Series (Number and Letter Series) (Numerical Relations and Reasoning), Ratio and Proportion (Ratios), Powers and Exponents (Surds and Indices), Directions (Direction Test), Analytical Reasoning (Counting Figures Reasoning), Verbal Analogy (Word Based Analogy), Divisibility Rules, Miscellaneous, Probability
  6. Paper - 1 | Unit - 6 | Logical Reasoning: युक्ति की संरचना / Structure of Arguments, पद, परिभाषा एवं भाषा के उपयोग / Terms, Definitions and Uses of Language, निरुपाधिक प्रतिज्ञप्तियाँ / Categorical Propositions, निरुपाधिक न्यायवाक्य / Categorical Syllogism, वेन आरेख / Venn Diagrams, निगमन, आगमन एवं सादृश्यता / Deduction, Induction and Analogy, तर्कदोष / Fallacies, भारतीय तर्कशास्त्र: प्रमाण / Indian Logic: Pramanas, अनुमान एवं व्याप्ति / Anumana and Vyapti, हेत्वाभास / Hetvabhasa, Syllogisms, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Venn Diagram, School Of Thoughts, Western Logic
  7. Paper - 1 | Unit - 7 | Data Interpretation: आँकड़े एवं आलेखीय निरूपण / Data and Graphical Representation, आँकड़ा निर्वचन: तालिका / Data Interpretation: Tables
  8. Paper - 1 | Unit - 8 | Information and Communication Technology: कंप्यूटर की मूल बातें और हार्डवेयर / Computer Fundamentals and Hardware, मेमोरी, भंडारण और डेटा निरूपण / Memory, Storage and Data Representation, कंप्यूटर सॉफ्टवेयर / Computer Software, ऑफिस अनुप्रयोग / Office Applications, मल्टीमीडिया और फ़ाइल प्रारूप / Multimedia and File Formats, कंप्यूटर नेटवर्क / Computer Networks, इंटरनेट और वर्ल्ड वाइड वेब / Internet and World Wide Web, ई-मेल और ऑनलाइन संचार / Email and Online Communication, साइबर सुरक्षा / Cyber Security, डिजिटल नैतिकता, बौद्धिक संपदा और समाज / Digital Ethics, IPR and Society, क्लाउड और वितरित कंप्यूटिंग / Cloud and Distributed Computing, उभरती प्रौद्योगिकियाँ / Emerging Technologies, शिक्षा और शासन में ICT / 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, Artificial Intelligence, 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
  9. Paper - 1 | Unit - 9 | People Development and Environment: पारितंत्र और जैव विविधता / Ecosystems and Biodiversity, मानव और पर्यावरण की अंतःक्रिया / Human and Environment Interaction, विकास और पर्यावरण / Development and Environment, प्राकृतिक संसाधन: वन, जल और मृदा / Natural Resources – Forests, Water and Soil, ऊर्जा संसाधन / Energy Resources, वायु प्रदूषण / Air Pollution, जल प्रदूषण और उपचार / Water Pollution and Treatment, ध्वनि प्रदूषण / Noise Pollution, अपशिष्ट प्रबंधन / Waste Management, पर्यावरणीय स्वास्थ्य और विषविज्ञान / Environmental Health and Toxicology, जलवायु परिवर्तन और वैश्विक पर्यावरणीय समस्याएँ / Climate Change and Global Environmental Issues, प्राकृतिक प्रकोप और आपदा प्रबंधन / Natural Hazards and Disaster Management, भारतीय पर्यावरण कानून और नीतियाँ / Indian Environmental Laws and Policies, अंतर्राष्ट्रीय पर्यावरण समझौते / International Environmental Agreements, Ecosystem, Biomes & Environmental Issues, Noise Pollution and Waste Management, Natural Resources, Energy & Disaster Management
  10. Paper - 1 | Unit - 10 | Higher Education System: प्राचीन भारत में शिक्षा / Education in Ancient India, प्राचीन भारत के उच्च शिक्षा केंद्र / Ancient Centres of Higher Learning, औपनिवेशिक काल की शिक्षा नीतियाँ और आयोग / Colonial Education Policies and Commissions, औपनिवेशिक कॉलेज, विश्वविद्यालय व सुधारक / Colonial Colleges, Universities and Reformers, स्वतंत्र भारत में शिक्षा आयोग और नीतियाँ / Education Commissions and Policies after Independence, राष्ट्रीय शिक्षा नीति 2020 / National Education Policy 2020, उच्च शिक्षा संस्थानों के प्रकार / Types of Higher Education Institutions, अधिगम कार्यक्रम और अकादमिक क्रेडिट / Learning Programmes and Academic Credits, व्यावसायिक, तकनीकी और कौशल आधारित शिक्षा / Professional, Technical and Skill-Based Education, मूल्य शिक्षा और पर्यावरण शिक्षा / Value Education and Environmental Education, शासन, प्रशासन और वित्तपोषण / Governance, Administration and Financing, उच्च शिक्षा का नियमन, प्रत्यायन और रैंकिंग / Regulation, Accreditation and Ranking, Jainism & Buddhism, Pre-Independence Commissions, Post-Independence Policy, Higher Education Structure & Accreditation, Universities & Learning Programmes, Types of Education & NEP 2020
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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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