15 what is a proportion? If a, b, c, d are in Proportion then write invertendo, componendo and dividendo of the proportion.
Answers
Answer:
invertendo property, alternendo property, componendo Property, dividendo property, convertendo property, componendo-dividendo property, addendo property and equivalent ratio property. These properties are explained below with examples.
I. Invertendo Property: For four numbers a, b, c, d if a : b = c : d, then b : a = d : c; that is, if two ratios are equal, then their inverse ratios are also equal.
If a : b :: c : d then b : a :: d : c.
Proof:
a : b :: c : d
⟹ abab = cdcd
⟹ baba = dcdc
⟹ b : a :: d : c
Example: 6 : 10 = 9 : 15
Therefore, 10 : 6 = 5 : 3 = 15 : 9
II. Alternendo Property: For four numbers a, b, c, d if a : b = c : d, then a : c = b : d; that is, if the second and third term interchange their places, then also the four terms are in proportion.
If a : b :: c : d then a : c :: b : d.
Proof:
a : b :: c : d
⟹ abab = cdcd
⟹ abab ∙ bcbc = cdcd ∙ bcbc
⟹ acac = bdbd
⟹ a : c :: b : d
Example: If 3 : 5 = 6 : 10 then 3 : 6 = 1 : 2 = 5 : 10
III. Componendo Property: For four numbers a, b, c, d if a : b = c : d then (a + b) : b :: (c + d) : d.
Proof:
a : b :: c : d
⟹ abab = cdcd
Adding 1 to both sides of abab = cdcd, we get
⟹ abab + 1 = cdcd + 1
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