Invertendo, Alternendo, Componendo & Dividendo Property

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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.