Invertendo, Alternendo, Componendo & Dividendo Property

Duration: 15 min

This video lesson is available to enrolled students.

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Inside: a video lesson and guided study material.

Module outline

  1. Child Development and Pedagogy: विकास की अवधारणा / Concept of Development, विकास के मनोवैज्ञानिक सिद्धांत / Psychological Theories of Development, व्यक्तिगत भिन्नताएँ, व्यक्तित्व एवं जेंडर / Individual Differences, Personality and Gender, बुद्धि एवं सृजनात्मकता / Intelligence and Creativity, अधिगम एवं इसके सिद्धांत / Learning and Its Theories, बालक कैसे सोचते एवं सीखते हैं / How Children Think and Learn, संवेग, अभिप्रेरणा, मानसिक स्वास्थ्य एवं निर्देशन / Emotion, Motivation, Mental Health and Guidance, समावेशी शिक्षा / Inclusive Education, शिक्षण-अधिगम प्रक्रिया / Teaching-Learning Process, शिक्षण विधियाँ एवं रणनीतियाँ / Teaching Methods and Strategies, आकलन एवं मूल्यांकन / Assessment and Evaluation, पाठ्यचर्या और विद्यालयी गतिविधियाँ / Curriculum and School Activities, पाठ योजना और शिक्षण कौशल / Lesson Planning and Teaching Skills, शिक्षण संसाधन और शैक्षिक प्रौद्योगिकी / Teaching Resources and Educational Technology, कक्षा प्रबंधन और सम्प्रेषण / Classroom Management and Communication, शिक्षक पहचान और व्यावसायिक विकास / Teacher Identity and Professional Development, शैक्षिक विचारक, उद्देश्य एवं राष्ट्रीय नीतियाँ / Educational Thinkers, Aims and National Policies, विद्यालय नेतृत्व और प्रशासन / School Leadership and Administration, जीवन कौशल, मूल्य, शांति एवं पर्यावरण शिक्षा / Life Skills, Values, Peace and Environmental Education, पर्यावरण अध्ययन का शिक्षणशास्त्र / Pedagogy of Environmental Studies, विज्ञान शिक्षण शास्त्र / Science Pedagogy, संविधान, आयोग, अधिनियम एवं योजनाएँ / Constitution, Commissions, Acts and Schemes in Education, कंप्यूटर हार्डवेयर एवं परिधीय उपकरण / Computer Hardware and Peripherals
  2. Language I — Hindi: भाषा, लिपि और वर्ण-विचार, संधि, शब्द-स्रोत और शब्द-रचना, समास, शब्द-भेद: विकारी और अविकारी, लिंग, वचन, कारक, काल और वाच्य, वाक्य-विचार और शुद्ध लेखन, शब्द-भंडार, रस, छंद, अलंकार और शब्द-शक्ति, हिंदी साहित्य, अपठित बोध, हिंदी शिक्षणशास्त्र
  3. Language II — English: Sounds, Spelling and Word Formation, Parts of Speech, Nouns and Pronouns, Articles, Determiners and Adjectives, Verbs and Modals, Tenses and Subject–Verb Agreement, Adverbs, Prepositions and Conjunctions, Sentence Structure, Voice and Narration, Vocabulary, Sentence Correction and Writing Skills, Poetry and English Literature, Reading Comprehension, Pedagogy of English Language
  4. Language II — Urdu: رسم الخط، املا اور تلفظ, ذخیرۂ الفاظ, قواعد, محاورے اور ضرب الامثال, شعر و سخن, اردو ادب کی تاریخ, تفہیمِ عبارت, اردو کی تدریس
  5. Language II — Sanskrit: वर्णविचारः एवं संज्ञाप्रकरणम्, सन्धिप्रकरणम्, शब्दरूपाणि, धातुरूपाणि, कारकाणि एवं विभक्तयः, उपसर्गाः एवं अव्ययानि, प्रत्ययप्रकरणम्, समासप्रकरणम्, वाच्यप्रकरणम्, संख्याज्ञानम्, शब्दज्ञानम् एवं अनुवादः, अपठितावबोधनम्, संस्कृतसाहित्यम् एवं काव्यशास्त्रम्, संस्कृतशिक्षणशास्त्रम्
  6. Mathematics: संख्या पद्धति एवं स्थानीय मान / Number System & Place Value, मूलभूत संक्रियाएँ, सरलीकरण एवं पैटर्न / Basic Operations, Simplification & Patterns, गुणनखंड, विभाज्यता, म.स.प. एवं ल.स.प. / Factors, Divisibility, HCF & LCM, भिन्न एवं दशमलव / Fractions & Decimals, वर्ग, घन, घातांक एवं करणी / Squares, Cubes, Exponents & Surds, मापन: लंबाई, भार, धारिता, समय एवं मुद्रा / Measurement: Length, Mass, Capacity, Time & Money, बीजगणित / Algebra, अनुपात, समानुपात, औसत एवं आयु / Ratio, Proportion, Average & Ages, प्रतिशत, लाभ-हानि एवं ब्याज / Percentage, Profit–Loss & Interest, समय-कार्य एवं चाल-दूरी / Time & Work and Speed–Distance, रेखाएँ, कोण, आकृतियाँ एवं निर्देशांक / Lines, Angles, Shapes & Coordinates, त्रिभुज, चतुर्भुज, वृत्त एवं त्रिकोणमिति / Triangles, Quadrilaterals, Circles & Trigonometry, क्षेत्रमिति: परिमाप, क्षेत्रफल एवं आयतन / Mensuration: Perimeter, Area & Volume, आँकड़ा प्रबंधन, सांख्यिकी एवं प्रायिकता / Data Handling, Statistics & Probability, गणित शिक्षणशास्त्र / Mathematics Pedagogy
  7. Science: प्रौद्योगिकी, मापन एवं गति / Technology, Measurement and Motion, बल, दाब, कार्य एवं ऊर्जा / Force, Pressure, Work and Energy, ऊष्मा एवं ध्वनि / Heat and Sound, प्रकाश / Light, विद्युत एवं चुंबकत्व / Electricity and Magnetism, ब्रह्मांड एवं प्राकृतिक परिघटनाएँ / Universe and Natural Phenomena, पदार्थ एवं उसकी संरचना / Matter and Its Structure, रासायनिक अभिक्रियाएँ, अम्ल, क्षार एवं लवण / Chemical Reactions, Acids, Bases and Salts, धातु, अधातु एवं दैनिक उपयोग की सामग्री / Metals, Non-metals and Materials of Daily Use, सजीव जगत: वर्गीकरण, कोशिका एवं ऊतक / Living World: Classification, Cell and Tissues, पादपों में पोषण, परिवहन एवं जनन / Plants: Nutrition, Transport and Reproduction, मानव शरीर एवं अंग तंत्र / Human Body and Organ Systems, भोजन, पोषण एवं स्वास्थ्य / Food, Nutrition and Health, पर्यावरण एवं पारिस्थितिकी / Environment and Ecology, पर्यावरण, पारिस्थितिकी और वन्यजीव / Environment, Ecology and Wildlife, प्राकृतिक संसाधन, प्रदूषण और जलवायु परिवर्तन / Natural Resources, Pollution and Climate Change
  8. Social Studies/Social Science: प्राचीन भारत का इतिहास / Ancient Indian History, मध्यकालीन भारत का इतिहास / Medieval Indian History, औपनिवेशिक भारत एवं समाज सुधार / Colonial India and Social Reform, भारतीय राष्ट्रीय आंदोलन एवं स्वतंत्र भारत / Indian National Movement and Independent India, विश्व इतिहास एवं अंतरराष्ट्रीय संबंध / World History and International Relations, पृथ्वी, ग्लोब एवं मानचित्र / Earth, Globe and Maps, भौतिक भूगोल / Physical Geography, प्राकृतिक वनस्पति, वन्य जीवन एवं विश्व के प्रदेश / Natural Vegetation, Wildlife and World Regions, भारत का भूगोल / Geography of India, संसाधन एवं विकास / Resources and Development, सामाजिक एवं राजनीतिक जीवन / Social and Political Life, भारतीय संविधान / Indian Constitution, सरकार: केंद्र, राज्य एवं स्थानीय / Government: Union, State and Local, अर्थशास्त्र एवं भारतीय अर्थव्यवस्था / Economics and Indian Economy, सामाजिक विज्ञान शिक्षणशास्त्र / Social Science Pedagogy, पृथ्वी, आकाश और आपदाएँ / Earth, Sky and Disasters, उत्तर प्रदेश सामान्य ज्ञान / Uttar Pradesh GK
  9. अतिरिक्त अभ्यास / Extra Practice (beyond syllabus): भारतीय अर्थव्यवस्था: आँकड़े एवं समसामयिकी / Indian Economy: Data & Current Affairs, भारतीय कला एवं संस्कृति / Indian Art & Culture, विज्ञान एवं प्रौद्योगिकी / Science & Technology, पर्यावरण एवं जलवायु समसामयिकी / Environment & Climate Affairs, रक्षा एवं सुरक्षा / Defence & Security, स्थिर सामान्य ज्ञान / Static General Knowledge, पुरस्कार एवं सम्मान / Awards & Honours, सरकारी योजनाएँ, शासन एवं आधारभूत संरचना / Government Schemes, Governance & Infrastructure, पासा एवं घन / Dice & Cubes
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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