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Properties of Proportions

  1. If a:b :: c:d, then b:a :: d:c by invertendo.
    So, if Description: 4847.png then by invertendo Description: 4841.png

    Description: 4835.png

  2. If a:b :: c:d, then a:c :: b:d by alternendo.
    If Description: 4829.png then by alternendo Description: 4823.png

    Description: 4817.png 

  3. If a:b :: c:d, then (a + b):b :: (c + d):d by componendo.
    If Description: 4810.png, then by componendo Description: 4804.png.



    Description: 4798.png.

  4. If a:b :: c:d, then (a – b) : b :: (c – d) : d by dividendo.
    If Description: 4792.png, then by dividendo Description: 4785.png


    Description: 4779.png

  5. If a:b::c:d, then (a + b):(a – b)::(c + d):(c –d) by componendo and dividend.
    If Description: 4773.png, then by componendo and dividendo, Description: 4767.png.


    Description: 4761.png

  6. If a:b::c:d::e:f, then a:b::c:d::e:f::(a + c + e):(b + d + f) by addendo.
    If Description: 4755.png, then by addendo Description: 4749.png


    Description: 4743.png


  7. If a:b::c:d::e:f, then a:b::c:d::e:f::(a – c – e):(b – d – f) by subtrahendo.
    If Description: 4737.png, then by subtrahendo Description: 4730.png


    Description: 4724.png


    What must be added to each of 6, 17, 27 and 59, so that the sums are in proportion?
    Let x be the number to be added:
    (6 + x ) : (17 + x ) :: (27 + x ) : (59 + x ) Description: 4717.png
    Using dividendo, we get
    Description: 4710.png 
    If we had not used dividendo, we would have had to cross-multiply and solve as shown below: (6 + x ) (59 + x ) = (27 + x ) (17 + x ) ⇒ x = 5

Note: In the above example, the second method looks easier. But we might end up with a quadratic/cubic equation after cross-multiplication. Hence, use of componendo and dividendo usually saves a lot of time by eliminating the need to solve the equation in a quadratic form.

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