Addendo Property & Equivalent Ratio Property

Duration: 16 min

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Inside: a video lesson and guided study material.

Module outline

  1. About the Course: About the Course
  2. Section 1 | Language (Hindi): भाषा, लिपि और वर्ण, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, संज्ञा और सर्वनाम, पद-भेद और रूप-विकार, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, संधि, समास, विलोम शब्द, पर्यायवाची शब्द, सामान्य अशुद्धियाँ, अनेक शब्दों के लिए एक शब्द, शब्द-भंडार, अपठित बोध, Multi-Topic Content (Cross-Concept Material), वर्णमाला
  3. Section 1 | Language (उर्दू भाषा — 22 प्रश्न (निश्चित)): رسم الخط، حروفِ تہجی اور املا / Script, Alphabet and Spelling, اسم، ضمیر اور صفت / Nouns, Pronouns and Adjectives, الفاظ، محاورے اور ضرب الامثال / Vocabulary, Idioms and Proverbs, تفہیمِ عبارت، انشا اور ترجمہ / Comprehension, Writing and Translation, Comprehension
  4. Section 1 | Language (বাংলা / बांग्ला भाषा — 22 प्रश्न (निश्चित)): শব্দার্থ, বাগধারা ও ভাষাশুদ্ধি / Meaning, Idioms and Correct Usage, বাক্য ও বাচ্য / Sentences and Voice, শব্দগঠন ও শব্দভান্ডার / Word Formation and Vocabulary, অনুবাদ, বোধপরীক্ষণ ও রচনা / Translation, Comprehension and Composition
  5. Section 1 | Language (English): Vocabulary, Noun, Subject Verb Agreement (Verb Noun Agreement), Adjectives, Verbs and Tenses, Pronoun, Verb, Adverb, Prepositions and Conjunctions, Parts of Speech, Determiner, Conjunction, Modals, Transformation of Sentences, Direct and Indirect Speech, Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Sentence Structure and Common Errors, Reading Comprehension, Cloze Test (Cloze Passage), Verbal Analogy (Word Based Analogy), Facts, Inference and Judgements
  6. Section 2 | Primary Maths & Reasoning: अनुपात और समानुपात / Ratio and Proportion, Divisibility Rules, मिश्रण और एलिगेशन / Mixture and Alligation, Permutation and Combination, संख्या पद्धति / Number System, महत्तम समापवर्तक और लघुत्तम समापवर्त्य / HCF and LCM, Sequence and Series (Series), समय, चाल और दूरी / Time, Speed and Distance, श्रेणी / Series, कूट लेखन एवं कूट वाचन / Coding-Decoding, अशाब्दिक तर्क / Non-Verbal Reasoning, प्रतिशत / Percentage, औसत / Average, लाभ, हानि और बट्टा / Profit, Loss and Discount, साधारण और चक्रवृद्धि ब्याज / Simple and Compound Interest, समय और कार्य / Time and Work, घन क्षेत्रमिति / Solid Mensuration, Age Problems, Arithmetic, सरलीकरण, घातांक और करणी / Simplification, Indices and Surds, Algebra, Probability, Trigonometry, Syllogisms, बैठक व्यवस्था एवं पहेलियाँ / Seating Arrangement and Puzzles, रक्त संबंध / Blood Relations, दिशा एवं दूरी / Direction and Distance, Analytical Reasoning (Counting Figures Reasoning), गणितीय संक्रियाएँ एवं लुप्त संख्या / Mathematical Operations and Missing Numbers, कैलेंडर एवं घड़ी / Calendar and Clock, Clock, घन, पासा एवं आकृतियों की गिनती / Cubes, Dice and Figure Counting, वेन आरेख, न्याय वाक्य एवं असमानता / Venn Diagrams, Syllogism and Inequalities, Data Arrangement, Inequality, सादृश्यता एवं वर्गीकरण / Analogy and Classification, क्रम एवं रैंकिंग / Order and Ranking, समुच्चय, संबंध, फलन और विविक्त गणित / Sets, Relations, Functions and Discrete Mathematics, संख्या पद्धति और संख्या सिद्धांत / Number Systems and Number Theory, निर्देशांक ज्यामिति, सदिश और त्रिविमीय ज्यामिति / Coordinate Geometry, Vectors and 3D Geometry, बहुपद, समीकरण और सम्मिश्र संख्याएँ / Polynomials, Equations and Complex Numbers
  7. Section 2 | General Awareness (GK + Current Affairs): Polity, वैज्ञानिक उपलब्धियाँ और अनुसंधान / Scientific Achievements and Research, Biology, राज्य विशेष सामान्य ज्ञान / State-Specific General Knowledge, Chemistry, History, Geography, Indian Economy, Environment, समसामयिकी / Current Affairs, Static General Knowledge
  8. Section 3 | Computer System and Org. (CSO): Basics of COA, Floating Point Rep, Cache Memory Organization, Instr Formats & Modes, Programming the Basic Computer, Input Output Organisation, Register Transfer and Microoperations, Control Unit Design
  9. Section 3 | Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, SQL, File Organization & Indexing, Relational Algebra, Relational Calculus, Transaction Management, Concurrency Control
  10. Section 3 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Process Synchronization
  11. Section 3 | Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics
  12. Section 3 | Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Minimum Spanning Trees, Shortest Path Algos
  13. Section 3 | Theory Of Computation & Compiler Design: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization
  14. Section 3 | Emerging Trends (AI, Societal Ethics & MS Office): कंप्यूटर के मूल सिद्धांत / Computer Fundamentals, Emerging Tech Trends, Society, Law & Ethics, MS Office Applications, व्यापार, विपणन एवं उपभोक्ता संरक्षण / Trade, Marketing and Consumer Protection, साइबर सुरक्षा / Cyber Safety and Security, कंप्यूटर सॉफ़्टवेयर / Computer Software, ऑपरेटिंग सिस्टम / Operating System, कंप्यूटर नेटवर्क / Computer Networks, वर्ड प्रोसेसिंग (MS Word) / Word Processing (MS Word), Terms & Abbreviations, प्रेज़ेंटेशन (MS PowerPoint) / Presentation Software (MS PowerPoint)
  15. Section 3 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Maintenance and Configuration Management, Software Testing
  16. Section 3 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  17. Section 3 | OOPS, C++, JAVA, HTML: OOPS with C++, Java, HTML Basics, CSS
  18. Section 3 | Java: Java Fundamentals, Array, String, IO & References, OOPS with Java
  19. Section 3 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  20. Section 3 | Digital Electronics: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  21. Section 3 | Multimedia: Multimedia
  22. BPSC Previous Year Papers:
  23. BPSC Full Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lesson on Ratio and Proportion, presented by Yash Jain, focuses on two key algebraic properties: the Addendo Property and the Equivalent Ratio Property. The video begins by defining a ratio as a comparison of two quantities before introducing the Addendo Property, which states that if multiple ratios are equal (a/b = c/d = e/f), their common value is preserved when the sums of their antecedents and consequents are taken. This is demonstrated numerically using 2/3 = 6/9 = 8/12, where (2+6+8)/(3+9+12) simplifies to 16/24, which reduces back to 2/3. The instructor then provides a formal algebraic proof by setting the equal ratios to a constant k, expressing numerators as multiples of denominators (a=bk, c=dk), and showing that the combined fraction simplifies to k. A generalized version using variables p, q, r, s, t, u is also presented to show that multiplying terms by constants maintains equivalence. The lesson applies the Addendo Property to solve a problem where p:q = r:s = t:u = 2:3, finding (p+r+t)/(q+s+u) to be 2/3. The second half introduces the Equivalent Ratio Property, which states that if a:b :: c:d, then (a±c):(b±d) :: a:b. This is proven by setting a/b = c/d = k and showing that (a+c)/(b+d) simplifies to k. The property is extended to forms like (pa+qc)/(pb+qd). A final problem asks for the value of (ab+cd)/(b²+d²) given a:b = c:d = 2:3, which is solved by recognizing the expression as an application of the Equivalent Ratio Property with p=b and q=d, yielding 2/3. The lesson concludes with a thank-you slide.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The lesson opens with a title slide for Ratio and Proportion, defining ratio as the comparison of two quantities. The instructor introduces the Addendo Property with the rule: if a:b = c:d = e:f, then each ratio equals (a+c+e):(b+d+f). A numerical example using 2/3 = 6/9 = 8/12 is shown, where (2+6+8)/(3+9+12) = 16/24 simplifies to 2/3. Red annotations highlight corresponding terms and circle the final combined fraction, reinforcing the visual connection between individual ratios and their sum.

  2. 2:00 – 5:00 02:00-05:00

    The instructor transitions from the numerical example to a formal algebraic proof of the Addendo Property. By setting a/b = c/d = e/f = k, the numerators are expressed as multiples of denominators: a=bk, c=dk, e=fk. Substituting these into (a+c+e)/(b+d+f) yields (bk+dk+fk)/(b+d+f), which factors to k(b+d+f)/(b+d+f) and cancels to k. This proves that the combined fraction equals the original ratio, establishing the extended proportion a/b = c/d = e/f = (a+c+e)/(b+d+f).

  3. 5:00 – 10:00 05:00-10:00

    The proof is generalized using new variables p, q, r, s, t, u to represent terms in equivalent fractions ma/mb, nc/nd, oe/of. The combined fraction (p+r+t)/(q+s+u) is shown to equal (ma+nc+oe)/(mb+nd+of), confirming that multiplying terms by constants preserves the ratio. The Addendo Property is then applied to a problem: if p:q = r:s = t:u = 2:3, then (p+r+t)/(q+s+u) = 2/3. The lesson transitions to the Equivalent Ratio Property, stating that if a:b :: c:d, then (a±c):(b±d) :: a:b.

  4. 10:00 – 15:00 10:00-15:00

    The Equivalent Ratio Property is proven by setting a/b = c/d = k (k ≠ 0) and showing that (a+c)/(b+d) simplifies to k(b+d)/(b+d) = k. The property is extended algebraically to forms like (pa+qc)/(pb+qd). A problem on an orange slide asks: if a:b = c:d = 2:3, what is (ab+cd)/(b²+d²)? The instructor re-expresses the target fraction as (ba+dc)/(bb+dd), identifying it with p=b and q=d, so the value is 2/3 by the Equivalent Ratio Property.

  5. 15:00 – 15:45 15:00-15:45

    The lesson concludes with a white slide headed 'Equivalent Ratio Property' that recaps two examples, boxing (ab+cd)/(b²+d²) and ending with a generalized form (4a-3c+9e)/(4b-3d+9f). A final 'THANKS FOR WATCHING' slide closes the video, signaling the end of the instructional content on ratio properties.

The lesson systematically builds from concrete examples to abstract proofs, a pedagogical approach that helps students internalize ratio properties. The Addendo Property is first demonstrated numerically, then proven algebraically using a constant k, and finally generalized with additional variables. This three-step progression (example → proof → generalization) is a key teaching pattern worth noting for exam preparation. The Equivalent Ratio Property follows a similar structure, with the proof emphasizing that (a±c)/(b±d) preserves the common ratio k. The final problem on (ab+cd)/(b²+d²) is particularly instructive because it requires recognizing that the expression fits the (pa+qc)/(pb+qd) form with p=b and q=d, a non-obvious substitution that tests conceptual understanding rather than rote application. Students should focus on identifying when a given fraction can be rewritten to match the property's standard form, as this is the core skill being assessed.

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